Research Seminar on Analysis, Dynamical Systems and Mathematical Physics

11. April 2024 (April 11th)
Peter Stollmann (Chemnitz)
Regularity results for locally integrable subharmonic functions on local Dirichlet spaces
^{Contact: Daniel Lenz}

18. April 2024 (April 18th)
Tobias Weich
SRB Measures and Ruelle-Taylor resonances for higher
rank Anosov actions (abstract)
^{Contact: Tobias Jäger}

25. April 2024 (April 25th)
Mathematisches Kolloquium:
David Kerr (Münster)
Suspending the pigeonhole principle: amenability and the Rokhlin lemma (abstract)
^{Contact: Tobias Jäger}

3. Mai 2024 (May 3rd)
Workshop on Mathematical Physics:
Mathematical Physics in the Heart of Germany IIIExternal link:

• Marcel Schmidt (Jena): Criticality Theory
• Jens Hoppe (Braunschweig): Relativistic membranes and the fast non-commutative sharp drop
• Marcel Griesemer (Stuttgart): Infraparticle scattering in the confined massless Nelson mode
• Luca Fresta (Bonn): Dynamics of Extended Fermi Gases at High Density
• Jobst Ziebell (Jena): Wetterich's Equation and Renormalisation

^{Contact: David Hasler}

16. Mai 2024 (May 16th)
Mathematisches Kolloquium: 15:00Uhr (3pm) SR 3517, Ernst-Abbe-Platz 2
Anna Wienhard
Beyond hyperbolic geometry (abstract)
^{Contact: Tobias Jäger}

23. Mai.2024 (May 23rd)
Zeyu Kang
Rational approximations and singular continuous spectrum of dynamical systems
^{Contact: Tobias Jäger}

30. Mai 2024 (May 30th)
Christoph Richard
Spectrum of weak model sets with Borel windows
^{Contact: Daniel Lenz}

6. bis 8. Juni.2024 (June 6th until 8th)
Workshop "Operator Theoretic Aspects of Ergodic Theory" in Leipzig
for a program see: https://www.ergodic.de/External link
^{Contact: Daniel Lenz}

20.Juni 2024 (June 20th)
Jena-Leipzig Seminar "Dynamics and Geometry" 
for a program see: https://www.math.uni-leipzig.de/~eisner/Jena-Leipzig-Seminar.htmlExternal link

12:15-13:15: Elzbieta KrawczykExternal link (Krakau): Amorphic complexity and tameness of constant length substitution shifts (abstract)
13:15-14:30: Mittagspause
14:45-15:45: Héctor Jardón-SánchezExternal link (Leipzig): "The cost of a group action" (abstractExternal link)

anschließend:
15:45--16:30 Kaffepause
Mathematisches Kolloquium: 16:30Uhr (3pm) SR 3517, Ernst-Abbe-Platz 2
Andreas Knauf: Phase space compactification for n bodies

^{Contact: Tobias Jäger}

08. August 2024 (August 8th)
12pm c.t. in SR 3517, EAP2
Till Hauser
About mean equicontinuous factor maps
^{Contact: Tobias Jäger}

22. August 2024 (August 22nd)
Nicolae Strungaru
Diffraction of weak model sets
^{Contact: Daniel Lenz}

26.10.2023
Florentin Münch (Leipzig)
Log Sobolev inequality and discrete Ricci curvature Abstract
^{Contact: Daniel Lenz}

10.11.2023
Ivan Veselić (Dortmund) 
Uncertainty relations and parabolic observability for Schroedinger operators with unboundedly growing potentials
^{Contact: Daniel Lenz}

16.11.2023
Tobias Jäger 
Some remarks on Thue Morse Substitutions
^{Contact: Tobias Jäger}

30.11.2023
Gerhard Keller (Erlangen)
B-free numbers – a view through the window
^{Contact: Tobias Jäger}

11.01.2024
Jonas Breitenbücher (Jena)
Topomorphic extensions of equicontinuous dynamical systems
^{Contact: Tobias Jäger}

12.01.2024
Miguel Ballesteros (Mexico-City)
Stationary Scattering Theory on the Discrete Real Line and Levinson's Theorem
^{Contact: David Hasler}

19.01.2024
Ralph Chill (Dresden)
Domination of semigroups generated by forms
^{Contact: Daniel Lenz}

25.01.2024
Jaime Gomez Ortiz
Topo-Isomorphisms of Irregular Toeplitz Subshifts for Residually Finite Groups
^{Contact: Tobias Jäger}

26.01.2024
Christoph Richard (Erlangen) (entfällt wegen Bahnstreik)
Spectrum of weak model sets with Borel windows
^{Contact: Tobias Jäger}

01.02.2024
Marta Imke (Jena)
Diskrete harmonische Räume (held in english) abstract
^{Contact: Daniel Lenz}

02.02.2024
Jena-Leipzig Seminar
Speakers: Philipp Gohlke (Lund) and Hans Crauel (Frankfurt)
^{Contact: Daniel Lenz}

09.06.2023
Michael Hinz (Bielefeld)
A tensor product approach to non-local differential products
^{Contact: Martina Zähle/Tobias Jäger}

15.06.2023
Aihua Fan (Amiens, Wuhan)
A topological version of Furstenberg-Kesten theorem
^{Contact: Tobias Jäger}

22.06.2023
Tobias Jäger (Jena)
KAM Theory and the Anosov-Katok method
^{Contact: Tobias Jäger}

29.06.2023
Lino Haupt (Jena) 
Isomorphic extensions with full fibres over irrational rotations via the Anosov-Katok method
^{Contact: Tobias Jäger}

04.07.2023, Paderborn University, Lecture Hall H4
Workshop 'Mathematical Physics in the Heart of Germany'
Organizers: Volker Bach, David Hasler, Benjamin Hinrichs
For more information please go to the workshop homepageExternal link.

06.07.2023
Maik Gröger (Krakau/Bremen)
TBA
^{Contact: Tobias Jäger}

Thu., 13.04.2023, 12:00, SR384, Carl-Zeiss-Str. 3
Daniel Sell (Toruń)
Finitely many asymptotic classes in Toeplitz subshifts Abstract
^{Contact: Daniel Lenz}

Thu., 20.04.2023, 12:00, SR384, Carl-Zeiss-Str. 3
Daniel Lenz (Jena)
Diffraction and dynamics on translation bounded measures I

Thu., 27.04.2023, 12:00, SR384, Carl-Zeiss-Str. 3
Daniel Lenz (Jena)
Diffraction and dynamics on translation bounded measures II

Thu., 04.05.2023, 12:00, SR384, Carl-Zeiss-Str. 3
Daniel Lenz (Jena)
Diffraction and dynamics on translation bounded measures III

11.05.2023
Henrik Kreidler (Wuppertal)
The Halmos-von Neuman Theorem I
^{Contact: Tobias Jäger}

12.05.2023
Patrick Hermle (Wuppertal)
The Halmos-von Neumann Theorem II
^{Contact: Tobias Jäger}

Thu., 03.11.2022, 12:15, HS4 Abbeanum
Lior Tenenbaum (Potsdam)
Periodic approximation of substitution subshifts Abstract
^{Contact: Daniel Lenz}

Thu., 10.11.2022, 12:15, HS4 Abbeanum
Ian Zimmermann (Jena)
Nonlinear characterizations of stochastic completeness Abstract
^{Contact: Daniel Lenz}

Thu., 01.12.2022, 12:00, HS4 Abbeanum
Melchior Wirth (IST Austria)
Generators of GNS-symmetric Quantum Markov Semigroups Abstract
^{Contact: Marcel Schmidt}

Fri., 02.12.2022, 11:15, R3517, Ernst-Abbe-Platz 2
Steffen Polzer (Geneva)
Renewal approach for the energy-momentum relation of the Fröhlich polaron Abstract
^{Contact: Benjamin Hinrichs}

Fri., 16.12.2022, 11:15, R3517, Ernst-Abbe-Platz 2
Oliver Siebert (Tübingen)
Thermal Area Law for Lattice Bosons Abstract
^{Contact: Benjamin Hinrichs}

Thu., 12.01.2023, 12:00, HS4 Abbeanum
Lino Haupt (Jena)
Introduction to the Hierarchy of Topological Dynamical Systems with Discrete Spectrum Abstract
^{Contact: Tobias Jäger}

Thu., 19.01.2023, 12:00, HS4 Abbeanum
Christian Rose (Potsdam)
Characterizing Gaussian upper heat kernel bounds on graphs with unbounded geometry Abstract
^{Contact: Marcel Schmidt}

Fri., 20.01.2023, 11:15, R3517, Ernst-Abbe-Platz 2
Wolfgang Spitzer (Hagen)
Entanglement entropy of the free Fermi gas Abstract
^{Contact: David Hasler}

Thu., 26.01.2023, 12:00, HS4 Abbeanum
Philipp Kunde (Kraków)
Non-classifiability of ergodic flows up to time change Abstract
^{Contact: Tobias Jäger}

Fri., 03.02.2023, 11:15, R3517, Ernst-Abbe-Platz 2
Siegfried Beckus (Potsdam)
Dry Ten Martini Problem for Sturmian Dynamical Systems Abstract
^{Contact: Daniel Lenz}

Thu., 09.02.2023, 12:00, HS4 Abbeanum
Tobias Jäger (Jena)
Tame minimal systems are regular extensions of group rotations Abstract

June 9, 2022

14:00 -16:00 (DSMP),

Dr. Timo Spindeler (Bielefeld)

Eigenmeasures with respect to the Fourer transform

Abstract: Several classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on R^d. In particular, we classify all periodic eigenmeasures on R, which gives an interesting connection with the discrete Fourier transform, as well as all eigenmeasures on R with uniformly discrete support.

(Contact: D. Lenz)

June 23, 2022

14:00 -16:00 (DSMP),

Dr. Markus Lange (Trieste, Italien)

Quantum Systems at The Brink: Existence of Bound States, Critical Potentials and Dimensionality

Abstract: The existence of bound states plays a crucial role for the properties of quantum systems. I will present a necessary and sufficient condition for Schrödinger operators to have a zero energy bound state. In particular I will show that the asymptotic behavior of the potential is the crucial ingredient.
The existence and non-existence result complement each other and exhibit a strong dependence on the dimension.
This is based on joint work with Dirk Hundertmark and Michal Jex.

(Contact: D. Hasler)

June 30, 2022

14:00 -16:00 (A&G),

Prof. Dr. Daniel Hug (KIT, Karlsruhe)

Curvature measures and soap bubbles beyond convexity

Abstract: A fundamental result in differential geometry states that if a smooth hypersurface in a Euclidean space encloses a bounded domain and one of its mean curvature functions is constant, then it is a Euclidean sphere. This statement has been referred to as the soap bubble theorem. Major contributions are due to Alexandrov (1958) and Korevaar--Ros (1988).

While the smoothness assumption is seemingly natural at first thought, based on the notion of curvatures measures of convex bodies Schneider (1979) established a characterization of Euclidean spheres among general convex bodies by requiring that one of the curvature measures is proportional to the boundary measure. We describe extensions in two directions: (1) The role of the Euclidean ball is taken by a nice gauge body (Wulff shape) and (2) the problem is studied in a non-convex and non-smooth setting. Thus we obtain characterization results for finite unions of Wulff shapes (bubbling) within the class of mean-convex sets or even for general sets with positive reach. Several related results are established. They include the extension of the classical Steiner--Weyl tube formula to arbitrary closed sets in a uniformly convex normed vector space, formulas for the derivative of the localized volume function of a compact set and general versions of the Heintze--Karcher inequality.

(Based on joint work with Mario Santilli)

(Contact: J. Hörrmann)

July 14, 2022

14:00 -16:00 (A&G),

Anne Marie Svane (Assistenzprofessorin aus Aalborg in Dänemark)

Analyzing point processes using Ripley’s K-function and persistence homology

Abstract: The first half of this talk will give an introduction to two tools for analyzing point patterns, namely Ripley’s K-function and persistence homology. A brief introduction to some of the most common stochastic models for point patterns will also be given. When analyzing the goodness of fit of a given point pattern to a hypothesized stochastic point process model, the K-function and persistence Betti numbers can be used as summary statistics. In the second half of the talk, I will present functional central limit theorems for both the K-function and persistence Betti numbers and show how these results can be used for doing statistics of point processes.

This is joint work with Christophe Biscio, Nicholas Chenavier, and Christian Hirsch.

(Contact: J. Hörrmann)

July 21, 2022

14:00 -16:00 (A&G),

Chiara Meroni (Max-Planck Institut Leipzig)

Intersection bodies using algebra

Abstract: I will discuss the notion of intersection bodies, important constructions in convex geometry. The idea is to approach them using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is a semialgebraic set and provide an algorithm for its computation. This is a joint work with Katalin Berlow, Marie-Charlotte Brandenburg and Isabelle Shankar.

(Contact: J. Hörrmann)

August 1, 2022

14:00 -16:00 (A&G),

Manuel Quaschner (Friedrich-Alexander-Universität Erlangen-Nürnberg)

Non-collision singularities in n-body problems

Abstract: The existence of non-collision singularities in the $n$-body problem was already conjectured by Painlevé in 1895. Even before the existence was proven in the 1990s, the question came up, whether the set of all initial conditions leading to non-collision singularities is a set of measure 0. A first result of this kind was proven for $n=4$ particles in $d\geq 2$ dimensions by Saari (1977). Using the so called Poincaré surface method, Fleischer (2018) could improve this for $n=4$ particles in $d \geq 2$ dimensions by extending the result to a wider class of potentials. But the problem is still open for more than four particles. \\

After an overview of these works, we consider trajectories that are close to the case of four particles but with some small perturbing forces given by additional particles. In order to apply the Poincaré surface method, we need to prove that these forces cannot break the bounds derived for the four-particle case. If time allows, we will sketch in the end how one could use these results to prove the improbability of larger systems that can be suitably decomposed into diverging systems of up to four particles each.

February 10, 2022

13:15 and 14:30 (DS&MP),  Online via ZoomExternal link

https://uni-jena-de.zoom.us/j/62215856917External link

Meeting ID: 622 1585 6917
Passcode: 632172

 

13:15 Henrik Kreidler (Bergische Universität Wuppertal)

The Furstenberg-Zimmer structure theorem revisited

Abstract:
The Furstenberg-Zimmer theorem is a key structural result in ergodic the- ory. It allows, e.g., to show multiple recurrence for measure-preserving systems which, by Furstenberg's correspondence principle, is equivalent to the theorem of Szemerédi on arithmetic progressions. In this talk we propose a new operator theoretic approach to this classical result. We show that, in essence, the Furstenberg- Zimmer theorem is a consequence of operator theory on so-called Kaplansky- Hilbert modules which are natural relative versions of classical Hilbert spaces.

This functional analytic perspective provides a systematic approach to extensions of measure-preserving systems. In addition, it allows to drop any countability assumptions yielding the structure theorem for actions of arbitrary groups on arbitrary probability spaces. It therefore contributes to a recent endeavour by Asgar Jamneshan and Terence Tao to remove such assumptions from classical results of ergodic theory. This is a joint work with Nikolai Edeko (Zurich) and Markus Haase (Kiel).

 

14:30 Yonatan Gutman (Institute of Mathematics of the Polish Academy of Sciences, Warsaw)

Maximal pronilfactors and a topological Wiener-Wintner theorem

Abstract:
For strictly ergodic systems, we introduce the class of CF-Nil(k) systems: systems for which the maximal measurable and maximal topological k-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such systems are abundant in a precise sense.

We show that the CF-Nil(k) systems are precisely the class of minimal systems for which the k-step nilsequence version of the Wiener-Wintner average converges everywhere. As part of the proof, we establish that pronilsystems are coalescent both in the measurable and topological categories. In addition, we characterize a CF-Nil(k) system in terms of its (k+1)-th dynamical cubespace. In particular, for k=1, this provides for strictly ergodic systems a new condition equivalent to the property that every measurable eigenfunction has a continuous version.

Joint work with Zhengxing Lian.

(Contact: T. Jäger / M. Dafinger)

December 16, 2021

14:30 -16:00 (A&G), online via Zoom

Prof. Dr. Lorenz Schwachhöfer (TU Dortmund)

What Lie algebras can tell us about Jordan algebras

(Contact: M. Dafinger / V. Matveev)

December 9, 2021

14:30 -16:00 (A&G), online via Zoom

Dr. Georgios Moschidis (Princeton University, New Jersey)

Weak turbulence and formation of black holes in general relativity

Abstract:
The AdS instability conjecture provides an example of weak turbulence appearing in the dynamics of the Einstein equations in the presence of a negative cosmological constant. The conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will first introduce the setup of the initial-boundary value problem for the Einstein equations and discuss how non-trivial geometric features such as black holes can appear dynamically in the evolution of those equations. I will then present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. If time permits, I will also discuss possible paths for extending these ideas to the vacuum case.

(Contact: M. Dafinger / V. Matveev)

December 2, 2021

14:30 -16:00 (A&G), online via Zoom

Prof. Thomas Barthelmé (Queen's University, Kingston, Canada)

Geometry and flexibility in a fixed conformal class of a surface

Abstract:
(joint work with Alena Erchenko) Classical works from the 80s and onwards gave famous inequalities between different geometrical or dynamical invariants for negatively curved metrics on surfaces, as well as rigidity results.
While the proofs often used conformal classes, the setting for the results was the space of negatively (or non-positively) curved metrics on a fixed surface. In particular, the geometry of the set of non-positively curved Riemannian metric in a fixed conformal class on a surface was not studied as such, and a number of fairly basic questions were left open.
In this talk, I will explain how one can get a coarse understanding of the geometry inside that space, giving some new bounds on certain geometrical and dynamical invariants, as well as finding the flexibility of others, in the sense of Katok’s flexibility program.

(Contact: M. Dafinger / V. Matveev)

November 25, 2021

14:30 -16:00 (A&G), online via Zoom

Prof. Colin Guillarmou (Universite Paris-Saclay)

Geodesic stretch and marked length spectrum rigidity problem

Abstract:
I will explain recent results obtained with Lefeuvre and with Lefeuvre-Knieper on the question of determination of a Riemannian metric with Anosov flow from its marked length spectrum.

(Contact: M. Dafinger / V. Matveev)

November 18, 2021

14:30 -16:00 (A&G), online via Zoom

Prof. Dr. Katharina Neusser (Masaryk University, Brno)

Cone structures and parabolic geometries (based on joint work with J.-M. Hwang)

Abstract:
A cone structure on a complex manifold M is a closed submanifold C of the projectivized tangent bundle of M which is submersive over M. Such structures arise naturally in differential and algebraic geometry, and when they do, they are typically equipped with a conic connection that specifies a distinguished family of curves on M in direction of C. In differential geometry, as we will see, cone structures with conic connections arise from so-called holomorphic parabolic geometries, a classical example of which is the null cone bundle of a holomorphic conformal structure with the conic connection given by the null-geodesics. In algebraic geometry, we have the cone structures consisting of varieties of minimal rational tangents (VMRT) given by minimal rational curves on uniruled projective manifolds. In this talk we will discuss various examples of cone structures and will introduce two important invariants for conic connections. As an application of the study of these invariants, we obtain a local-differential-geometric version of the global algebraic-geometric recognition theorem of Mok and Hong–Hwang, which recognizes certain generalized flag varieties from their VMRT-structures.

(Contact: M. Dafinger / V. Matveev)

November 4, 2021

14:30 -16:00 (A&G), online via Zoom

Frederic Weber (Universität Ulm)

Non-local Bakry-Émery theory and related functional inequalities

Abstract:
The curvature-dimension condition of Bakry and Émery constitutes a powerfull tool to show various functional inequalities for Markov semigroups and their generators. However, a key assumption in the classical Bakry-Émery theory is the validity of certain chain rules, which do not hold in genreal for non-local operators. In this talk we identify a curvature-dimension condition for a large class of non-local operators that implies modified logarithmic Sobolev inequalities under positive curvature bounds, logarithmic entropy-information inequalities under positive curvature and finite dimension bounds and Li-Yau inequalities under non-negative curvature and finite dimension bounds. While we mainly focus on generators of continuous-time Markov chains, we also discuss the fractional Laplacian as an important example.

(Contact: D. Lenz)

October 28, 2021

14:30 -16:00 (A&G), online via Zoom

Prof. Christos Saroglou (University of Ioannina, Greece)

A non-existence result for the L_p-Minkowski problem

(Contact: Th. Wannerer)

October 21, 2021

14:15-15:45 (A&G), Online via Zoom

Dr. Alexey Bolsinov (Loughborough University)

Frobenius pencils and compatible non-homogeneous Poisson structures

Abstract:
This work, joint with A.Konyaev and V.Matveev, is another application of Nijenhuis geometry in the theory of infty-dimensional integrable systems. We study compatible (differential geometric) non-homogeneous Poisson structures of the form B + A, where B and A are homogeneous Darboux-Poisson structures of order 3 and 1 respectively. The problem is reduced to an algebraic problem, namely to classification of pairs of compatible Frobenius algebras. We solve it completely, under some minor genericity conditions, by methods of differential geometry. As an application in mathematical physics we construct new interesting examples of multicomponent integrable (PDE-)dynamical systems

(Contact: M. Dafinger / V. Mateev)

 

16:15-17:45 (DS&MP), SR 3517

Jobst Ziebell (FSU Jena, Theoretisch-Physikalischen Institut)

The Wetterich equation of a real scalar field

Abstract:
In the theoretical physics community the Wetterich equation has become an essential tool in the study of renormalisation group flows of quantum field theories. Its cousin, the Polchinski equation, has seen several applications in mathematical physics but so far the mathematical step to Wetterich's equation had not been taken. I will present a simple regularisation scheme that is designed to "keep everything smooth" and allows a rigorous derivation of the equation.

(Contact: D. Hasler)

October 14, 2021

14:30 -16:00 (A&G), online via Zoom

Prof. Dmitry Faifman (Tel Aviv University)

The Funk metric, between convex and projective geometry.

Abstract:
The Funk metric in the interior of a convex set is a lesser-known cousin of the Hilbert metric. The latter generalizes the Beltrami-Klein model of hyperbolic geometry, and both have straight segments as geodesics, thus constituting solutions of Hilbert's 4th problem alongside normed spaces. Unlike the Hilbert metric, the Funk metric is not projectively invariant. I will explain how, nevertheless, the Funk metric gives rise to many projective invariants, which moreover enjoy a duality extending results of Holmes-Thompson and Alvarez Paiva on spheres of normed spaces and Gutkin-Tabachnikov on Minkowski billiards. Time permitting, I will also discuss how extremizing the volume of metric balls in Funk geometry yields extensions of the Blaschke-Santalo inequality and Mahler conjecture.

(Contact: Th. Wannerer)

July 15, 2021

16:15 -17:45 (A&G)

Prof. Raffaele Vitolo (Univesitá del Salento, Italy)

Hamiltonian operators, quasilinear PDEs and projective geometry

Abstract: The Hamiltonian formulation of quasilinear first-order partial differential equations, or hydrodynamic-type systems, has well-known invariance properties with respect to differential-geometric transformations of coordinates. In this talk we will discuss recent results that show the presence of projective-geometric invariance in both Hamiltonian operators and quasilinear first-order PDEs.

(Contact: V. Matveev / M. Dafinger)

July 1, 2021

16:15 -17:45 (DS&MP)

Prof. Mostafa Sabri (Cairo University)

Spectral analysis of large quantum graphs

Abstract: Quantum graphs are continuum analogs of discrete (combinatorial) graphs in which the adjacency matrix is replaced by a differential operator on the edges. They arise naturally in chemistry, physics and engineering when one considers propagation of waves through a quasi-one-dimensional system, and they are also interesting for purely mathematical reasons.

This talk is intended for a wide audience; I will first introduce quantum graphs then discuss four problems : how to define an appropriate notion of convergence for sequences of quantum graphs ? Does this notion guarantee the convergence of empirical spectral measures ? How about the nature of the spectrum at the limit infinite quantum graph ? Can we show that in some cases it exhibits absolutely continuous spectrum ? And if yes, what if we consider a sequence of finite quantum graphs converging to such model, are the eigenvectors delocalized in the sense that they are essentially uniformly distributed over the graph ?

This is based on several joint works with Nalini Anantharaman, Maxime Ingremeau and Brian Winn.

(Contact: D. Lenz)

June 24, 2021

16:15 -17:45 (DS&MP)

Dr. Reza Mohammadpour Bejargafsheh (University of Bordeaux)

Ergodic optimization and multifractal formalism of Lyapunov exponents

In this talk, we discuss ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles over mixing subshifts of finite type, and that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show both the continuity of the entropy at the boundary of the Lyapunov spectrum for such cocycles and the continuity of the minimal Lyapunov exponent under the assumption that linear cocycles satisfy a cone condition.

We consider a subadditive potential $\Phi$. We obtain that for $t\rightarrow \infty$ any accumulation point of a family of equilibrium states of $t\Phi$ is a maximizing measure, and that the Lyapunov exponent and entropy of equilibrium states for $t\Phi$ converge in the limit $t\rightarrow \infty$ to the maximum Lyapunov exponent and entropy of maximizing measures.

(Contact: D. Lenz)

June 17, 2021

16:15 -17:45 (A&G)

Prof. Gabriele Mondello (Sapienza – Università di Roma)

On spherical surfaces of genus 1 with 1 conical point

Abstract: A spherical surface with conical points is real 2-dimensional manifold that can be obtained from a disjoint union of convex spherical triangles by isometric identification of pairs of edges. Thus, to every spherical surface we can associate an underlying Riemann surface with marked points. Once the angles are fixed (and the Gauss-Bonnet constraint permits), the analogous procedure with hyperbolic or flat surfaces produces a bijection between constant curvature metric structures with conical singularities and conformal structures with marked points, which generalizes the uniformization theorem. In the spherical case the situation looks quite different.

In this talk I will first review some known results about the topology of the moduli space of spherical surfaces of genus g with n conical points. Then I will describe a synthetic approach to the case of genus 1 with 1 conical point in more detail, which is joint work with Eremenko-Panov.

(Contact: V. Matveev / M. Dafinger)

June 10, 2021

16:15 -17:45 (A&G)

Prof. Ian Anderson (Utah State University)

Spacetimes with Symmetry

Abstract: The mathematical physicist A. Z. Petrov, who is best known for his classification of spacetimes according to the algebraic character of the Weyl tensor, also gave a remarkable classification of local spacetimes which admit a non-trivial Lie algebra of Killing vector fields. In this talk I will focus on two open problems arising from Petrov’s classification of spacetimes with symmetry.

The first open problem deals with spacetimes whose isometry group is simply transitive. Using the Newman-Penrose formalism, I will describe a complete algebraic enumeration of all such spacetimes and a detailed solution to the local equivalence problem. The effectiveness of this classification is demonstrated by solving some long-standing problems in the field of exact solutions of the Einstein equation.

The second problem focuses on the classification of spacetimes with intransitive isometry groups for which there is no slice. I’ll describe a new method for the local classification of these spacetimes.

(Contact: V. Matveev / M. Dafinger)

June 3, 2021

16:15 -17:45 (A&G)

Prof. Vsevolod Shevchishin (University of Warmia and Mazury in Olsztyn)

Polynomially superintegrable metrics on surfaces admitting a linear integral.

Abstract:We give a complete local classification of superintegrable metrics on surfaces admitting two polynomial integrals one of which is linear.
We also describe the Poisson algebra of polynomial invariants of such metrics presenting a natural set of generators and polynomial relations between them and giving expressions of Poisson brackets of those generators.
We also give explicit formulas in some interesting cases.

(Contact: V. Matveev / M. Dafinger)

May 27, 2021

16:15 -17:45  (DS&MP)

Max Kämper (TU Dortmund)

Approximating the integrated density of states of random Schrödinger operators - results from empirical process theory

Abstract: Random Schrödinger operators are a model for metals with random impurities and this talk will present them for the special case of the Anderson operator on a lattice. We will introduce the integrated density of states and its uniform approximation by eigenvalue counting functions and show how results from empirical process theory can be used to improve quantitative results for this approximation.
This talk is based on joint work with Christoph Schumacher, Fabian Schwarzenberger and Ivan Veselic.

(Contact: D. Lenz)

May 20, 2021

16:15 -17:45 (A&G)

Prof. Gérard Besson (Institut Fourier, Universität Grenoble Alpes)

Finiteness Theorems for Gromov-Hyperbolic Groups

This is a joint work with G. Courtois, S. Gallot and A. Sambusetti.
We shall prove that, given two positive numbers $\delta$ and $H$, there are finitely non cyclic torsion-free $\delta$-hyperbolic marked group $(\Gamma , \Sigma)$ satisfying ${\rm Ent} (\Gamma , \Sigma) \le H$, up to isometry (of marked groups). Here a marked group is a group $\Gamma$ together with a symmetric generating set $\Sigma$ and ${\rm Ent}$ is the entropy of the marked group. These notions will be defined precisely.

(Contact: V. Matveev / M. Dafinger)

May 12, 2021

12:15-13:45 (A&G)

Nicolas Boumal (Ecole Polytechnique Fédérale de Lausanne (EPFL))

Riemannian geometry for numerical optimization: the tools we use and some geometry questions that arise

(Contact: V. Matveev / M. Dafinger)

May 6, 2021

16:15 -17:45 (A&G)

Davide Parise (University of Cambridge)

Convergence of the self-dual U(1)-Yang-Mills-Higgs energies to the (n - 2)-area functional

Abstract: We overview the recently developed level set approach to the existence theory of minimal submanifolds and present some joint work with A. Pigati and D. Stern.

The underlying idea is to construct minimal hypersurfaces as limits of nodal sets of critical points of functionals. In the first part of the talk we will give a general overview of the codimension one theory. We will then move to the higher codimension setting, and introduce the self-dual Yang-Mills-Higgs functionals. These are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points have long been studied as a basic model problem in gauge theory. We will explain to what extend the variational theory of these energies is related to the one of the (n - 2)-area functional and how one can interprete the former as a relaxation/regularization of the latter. Time permitting we will mention some elements of the proof, with special emphasis on the gradient flow of the Yang-Mill-Higgs energies.

(Contact: V. Matveev / M. Dafinger)

April 29, 2021

16:15 -17:45 (A&G)

Dr. Dong Cheng & Prof. Dimitri Burago (Pennsylvania State University)

Open problems in geometry

(Contact: V. Matveev / M. Dafinger)

April 22, 2021

16:15 -17:45 (A&G)

Prof. Alexey Glutsyuk (CNRS, ENS de Lyon; HSE University (Moscow))

On polynomially integrable billiards on surfaces of constant curvature

(Contact: V. Matveev / M. Dafinger)

April 15, 2021

16:15 -17:45 (A&G)

Dr. Louis Merlin  (RWTH Aachen University)

On the relations between the universal Teichmuller space and Anti de Sitter geometry

Abstract: Anti de Sitter (AdS) space is the Lorentzian cousin of the hyperbolic 3-space: it is a symmetric space with constant curvature -1. In this talk, we will consider surface group representations in the isometry group of AdS space, called quasi-Fuchsian representations. There is 2 classical objects associated to those representations and one of the goal is to understand their interplay: the limit set which is a quasi-circle in the boundary at infinity of AdS space and a convex set inside AdS which is preserved by the group action and bounded by two pleated surfaces. I will conclude the talk by a report on a work in common with Jean-marc Schlenker where we extend the "Teichmüller" situation to the "universal Teichmüller".

(Contact: V. Matveev / M. Dafinger)

April 8, 2021

16:15 -17:45 (A&G)

Marcos Cossarini  (Ecole Polytechnique Fédérale de Lausanne (EPFL))

Discrete surfaces with length and area and minimal fillings of the circle

Abstract: We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of crossings with the walls, and the area of the surface is the number of crossings between the walls themselves. We show how to approximate any Riemannian (or self-reverse Finsler) metric by a wallsystem.

This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere has minimum area among orientable Riemannian surfaces that fill isometrically a closed curve of given length. (A surface fills its boundary curve isometrically if the distance between each pair of boundary points measured along the surface is not less than the distance measured along the curve.) We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n(n-1)/2 squares. This conjecture is equivalent to the continuous FAC for surfaces with self-reverse Finsler metric.

(Contact: V. Matveev / M. Dafinger)

April 1, 2021

16:15 -17:45 (A&G)

Prof.  Andrey Mironov (Sobolev Institute of Mathematics, Novosibirsk, Russia)

On integrable magnetic geodesic flows on 2-torus

Abstract: We will discuss the problem of integrability of geodesic and magnetic geodesic flows on 2-torus. The talk is based on joint results with Misha Bialy (Tel-Aviv) and Sergey Agapov (Novosibirsk).

(Contact: V. Matveev / M. Dafinger)

25. März 2021

16:15 -17:45 (A&G)

Prof. Richard Montgomery (University of California, Santa Cruz)

Four Open questions in the N-body problem

Abstract: The classical N-body problem is alive and well. I begin with a pictorial survey of some of its solution curves. I then describe four open questions within the problem and recent progress on these questions.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

18. März 2021

14:30-16:00 (A&G)

Prof. Emilio Musso (Politecnico di Torino)

The Cr-strain functional for Legendrian curves in the 3-dimensional Sphere

Abstract: Let S³ be the unit 3-sphere with its standard Cauchy–Riemann (CR) structure. We consider the CR geometry of Legendrian curves in S3, thought of as homogeneous space of its CR-transformations group. More specifically, the focus is on the simplest cr-invariant variational problem for Legendrian curves and on its closed critical curves. The Liouville integrability of such a variational problem is considered. We discuss the admissible contact isotopy classes of closed critical curves with constant bending. Subsequently, we label closed critical curves with non-constant bending with three numerical invariants (quantum numbers). We exhibit that each critical curve can be explicitly reconstructed once that one knows these numerical invariants. We analyze the geometrical meaning of the quantum numbers.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

11. März 2021

14:30-16:00  (A&G)

Prof. Boris Kruglikov (UiT the Arctic University of Norway)

Dispersionless integrability: different approaches and examples

Abstract: I will discuss what is the integrability of dispersionless systems via hydrodynamic reductions, Lax pairs, Zakharov pairs, and background geometry. Several examples of equations of mathematical physics will be considered. No special preliminary knowledge will be assumed.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

4. März 2021

(nicht stattgefunden)

16:15 -17:45 (A&G)

Prof. Richard Montgomery (University of California, Santa Cruz)

Four Open questions in the N-body problem

Abstract: The classical N-body problem is alive and well. I begin with a pictorial survey of some of its solution curves. I then describe four open questions within the problem and recent progress on these questions.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

18. Februar 2021

14:30-16:00 (A&G)

Marc Mars, Miguel Manzano (Fundamental Physics Department, Universidad de Salamanca)

Null shells: general matching across null boundaries and connection with cut-and-paste formalism

Abstract: Null shells are a useful geometric construction to study the propagation of infinitesimally thin concentrations of massless particles or impulsive waves. In this talk, I will present the necessary and sufficient conditions that allow for the matching of two spacetimes with respective null embedded hypersurfaces as boundaries. Whenever the matching is possible, it is shown to depend on a diffeomorphism between the set of null generators in each boundary and a scalar function, called step function, that determines a shift of points along the null generators. Generically there exists at most one possible matching but in some circumstances this is not so. When the null boundaries are totally geodesic, the point-to-point identification between them introduces a freedom whose nature and consequences are detailed. The expression for the energy-momentum tensor of a general null shell is also shown.
Finally, the most general shell (with non-zero energy, energy flux and pressure) that can be generated by matching two Minkowski regions across a null hyperplane is presented. This connects the original Penrose’s cut-and-paste construction with the standard matching formalism.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

11. Februar 2021

14:30-16:00 (A&G)

Prof. Dr. Daniel Grieser (Oldenburg)

The geodesic flow on singular spaces

Abstract: From a dynamical systems point of view the geodesic flow on a complete Riemannian manifold is rather boring when considered for short times only. The situation changes if we allow the underlying space to have singularities. Then the short time behavior of geodesics near the singularities can be quite interesting, even in the case of pretty simple singularities, like cones and (incomplete) cusps. I will explain results obtained in collaboration with Vincent Grandjean and how the study of these questions naturally involves blow-ups, Hamiltonian systems with degenerate symplectic form and normally hyperbolic dynamical systems.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

4. Februar  2021

16:15 -17:45 (DS&MP)

Markus Lange (University of British Columbia)

Exactness of Linear Response in the Quantum Hall Effect

Abstract: In general, linear response theory expresses the relation between a driving and a physical system’s response only to first order in perturbation theory. In the context of charge transport, this is the linear relation between current and electromotive force expressed in Ohm’s law. In this talk I will present, that in the case of the quantum Hall effect, the linear responds is the full responds of the system and all higher order corrections vanish.

(Contact: D. Hasler/ M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

21. Januar 2021

14:30-16:00 (A&G)

Dr. Omid Makhmali (Institute of Mathematics of the Polish Academy of Sciences, Poland)

Causal structures and their space of null geodesics

Abstract: We define generalized causal structures as a field of projective hypersurfaces over a manifold which can be considered as a Finslerian extension of conformal pseudo-Riemannian geometry. We solve the local equivalence problem for such structures using Cartan's method of equivalence. We investigate geometric structures induced on the space of "null geodesics" of certain classes of causal structures which will involve CR structures of hypersurface type and Finsler metrics of constant flag curvature.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

17. Dezember  2020

14:30-16:00 (A&G)

Prof. Paolo Piccione  (São Paulo, Brasilien)

Minimal spheres in ellipsoids

Abstract: In 1987, Yau posed the question of whether all minimal 2-spheres in a 3-dimensional ellipsoid inside R⁴ are planar, i.e., determined by the intersection with a hyperplane. While this is the case if the ellipsoid is nearly round, Haslhofer and Ketover have recently shown the existence of an embedded non-planar minimal 2-sphere in sufficiently elongated ellipsoids, with min-max methods. Using bifurcation theory and the symmetries that arise in the case where at least two semi-axes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal 2-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint work with R. G. Bettiol.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

26. November 2020

14:30-16:00 (A&G)

Prof. Karin Hanley Melnick (University of Maryland)

A D'Ambra Theorem in conformal Lorentzian geometry

Abstract: D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

19. November 2020

14:30-16:00 (A&G)

Dr. Shaosai Huang (University of Wisconsin)

Topological rigidity of the first Betti number and Ricci flow smoothing

Abstract: The infranil fiber bundle is a typical structure appeared in the collapsing geometry with bounded sectional curvature. In this talk, I will discuss a topological condition on the first Betti numbers that guarantees a torus fiber bundle structure (a special type of infranil fiber bundle) for collapsing manifolds with only Ricci curvature bounded below. The main technique applied here is smoothing by Ricci flows. This covers my joint with Bing Wang.

(Contact: V. Matveev / M. Dafinger)
Everyone who wants to participate and does not have a password yet, please contact Markus Dafinger by e-mail.

12. November 2020

14:30-16:00 (A&G)

Silvan Bernklau (Universität Jena)

The spectral mapping theorem for C₀-semigroups

Abstract: Two known proofs of the spectral mapping theorem for eventually norm continuous semigroups are presented, one exploiting individual properties of subcomponents of the spectrum and one relying on a representation of the spectrum in abelian Banach algebras.

(Contact: V. Matveev)

11. April 2019

16:00 (DS&MP), seminar room 3517

Francisco Lopez Hernandez (Universidad Autonoma de San Luis Potosi)

Some aspects of dynamics on solenoids

We will define solenoids from different points of view in order to describe some dynamical properties of solenoidal homeomorphisms.

(Contact: T. Jäger)

 

25. April 2019

14:30 (DS&MP), seminar room 3517

Tobias Weihrauch (Universität Leipzig)

Resistance Forms of Graphs

We review basic properties of Resistance Forms as introduced by Kigami and investigate the special case of energy forms of infinite graphs. In particular, we discuss the connection of their associated resistance metric to the graphs random walk.

(Contact: M. Schmidt)

 

09. May 2019

14:30 (DS&MP), seminar room 3517

Anna Muranova (Universität Bielefeld)

Two approaches to the notion of effective impedance

Abstract: It is known that electrical networks with resistors are related to the Laplace operator and Dirichlet problem on weighted graphs. In this talk we consider more general electrical networks with coils, capacitors, and resistors. Such electrical networks give rise to complex-weighted graphs. The corresponding Dirichlet problem with complex-valued coefficients does not necessary have a solution, and if it has, it may be not unique. This creates difficulties in definition of the effective impedance. We present two approaches to overcoming this difficulty. In the first approach we show that, in the case of multiple solutions, all they have the same energy and, therefore, the effective impedance is well-defined. In the second approach we consider weights of the edges as rational functions of λ=iω, here ω is a frequency of the current and use the fact that rational functions form an ordered field. Then we develop a theory of weighted graphs with weights from an ordered field and prove that the effective impedance is always well defined in this case.

(Contact: D. Lenz)

 

23. May 2019

14:30 (DS&MP), seminar room 3517

Martin Schneider (TU Dresden)

Concentration of measure and its applications in topological dynamics

Abstract:  The phenomenon of measure concentration, in its modern formulation, was isolated in the late 1960s and early 1970s by Vitali Milman, extending an idea going back to Paul Levy's work on the geometry of (high-dimensional) Euclidean spheres, and has since led to numerous interesting applications in geometry and combinatorics. In their groundbreaking 1983 joint work, Gromov and Milman linked this phenomenon with topological dynamics: they proved that any Levy group, i.e., any topological group G admitting a dense exhaustion by an increasing sequence of compact subgroups with Levy-Milman concentration of the corresponding normalized Haar measures, is extremely amenable, which means that every continuous action of G on a non-void compact Hausdorff space must have a fixed point. Examples of such Levy (hence extremely amenable) groups include the unitary group of the infinite-dimensional separable Hilbert space equipped with the strong operator topology, the isometry group of the Urysohn space with the topology of point-wise convergence, and the automorphism group of the non-atomic standard Lebesgue space with the weak topology.
In my talk, I will survey some very recent developments concerning applications of measure concentration in topological dynamics. This will include a generalization of the above-mentioned result by Gromov and Milman, a correspondence principle connecting Gromov's metric measure geometry with dynamics and ergodic theory of large topological groups, as well as some new manifestations of concentration of measure.

(Contact: T. Hauser, T. Jäger)

 

16:30 Mathematical Colloquium Jena, Carl-Zeiß-Straße 3, SR 307

Prof. Dr. Edriss S. Titi (University of Texas, Weizmann Intitute of Science, University of Cambridge, zurzeit Einstein-Visiting-Fellow FU Berlin)

Is dispersion a stabilizing or destabilizing mechanism? Landau-damping induced by fast background  flows

Abstract: In this talk I will present a unifed approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some new results concerning two- and three-dimensional turbulent  flows with high Reynolds numbers in periodic domains, which exhibit "Landua-damping" mechanism due to large spatial average in the initial data.

 

6. June 2019

14:30 (DS&MP), seminar room 3517

Nikolai Edeko (Universität Tübingen)

Equicontinuous factors of flows on locally path-connected compact spaces

Given a quotient map f of topological spaces X and Y, it is generally difficult to relate geometric properties of X to those of Y without assumptions on the quotient map f. A useful assumption is monotonicity of f, i.e., connectedness of its fibers, since it ensures that a quotient map between appropriate spaces induces a surjective homomorphism on the level of fundamental groups. Motivated by a result of T. Hauser and T. Jäger proving the monotonicity of the maximal equicontinuous factor of a flow on a locally connected compact space, we provide a general functional analytic criterion for monotonicity of factors of topological dynamical systems which allows to understand the monotonicity of the maximal equicontinuous factor under a different perspective. We then show that this monotonicity result and the above-mentioned geometric property of monotone quotient maps allow, in the absence of non-trivial fixed functions, to represent equicontinuous factors of (abelian) flows on compact manifolds as rotations on compact abelian Lie groups. We discuss how this can be used to link the spectral theory of such flows with the first Betti number of the underlying manifold.

(Contact: T. Hauser, T. Jäger)

 

16:30 Mathematical Colloquium Jena, Carl-Zeiß-Straße 3, SR 307

Peter Bürgisser (TU Berlin)

On the number of real zeros of structured random polynomials

We plan to report on two recent results stating that structured (systems of) random polynomials typically only have few real zeros.

The first result is on random fewnomials: it says that a system of polynomials in $n$ variables with a prescribed set of $t$ terms and independent centered Gaussian coefficients has an expected number of positive real zeros bounded by $2 {t \choose n}$.

The second result is on Koiran's Real Tau Conjecture, which claims that the number of real zeros of a sum of $m$ products of $k$ real sparse univariate polynomials, each with a fixed set of at most $t$ terms, is bounded by a polynomial in $m,k,t$. The Real Tau Conjecture implies Valiant's Conjecture $VP \ne VNP$. We have confirmed the conjecture on average: if the coefficients in these structured polynomials are independent standard Gaussians, then theexpected number of real zeros is bounded by O(mkt).

The proofs are based on the Rice formula and methods fromintegral geometry. 

This is joint work with Alperen Erguer, Josue Tonelli-Cueto and Irenee Briquel.

 

(Contact: T. Wannerer)

 

13. June 2019

16:30 Mathematical Colloquium Jena, Carl-Zeiß-Straße 3, SR 307

Aleksandr Koldobskiy (University of Missouri-Columbia / Max Planck Institute for Mathematics Bonn)

Slicing and distance inequalities for convex bodiesSlicing inequalities provide estimates for the volume of a solid in terms of areas of its plane sections.
One of the problems is the Busemann-Petty problem asking whether convex bodies with uniformly
smaller areas of their central hyperplane sections necessarily have smaller volume. Another is the
problem of Bourgain asking whether every symmetric convex body of volume one has a hyperplane
section whose area is greater than an absolute constant. We show new estimates of this kind depending
on the (outer volume ratio) distance from the body to the class of intersection bodies or to the class of
unit balls of subspaces of L_p. Many of the results hold for arbitrary measures in place of volume. (Contact: T. Wannerer)

 

27. June 2019

14:30 (DS&MP), seminar room 3517

Nicolae Strungaru (MacEwan University, Edmonton)

On the diffraction of Meyer sets

In this talk we will review some recent progress in the diffraction of measures with Meyer set support. We will prove the existence of a generalized Eberlein decomposition, and the norm almost periodicity of the diffraction, and of each of the three spectral components.

(Contact: D. Lenz)

 

29. August 2019

14:30 (DS&MP), seminar room 3517

Welington da Silva Cordeiro (Polish Academy of Sciences, Warsaw)

The L-shadowing property

We will explore the L-shadowing property. First of all, we will discuss basic properties of this notion and compare it with topological hyperbolicity. We will show that systems with expansivity and the shadowing property satisfy the L-shadowing property, and that here are examples with the L-shadowing property which are not exhaustive. Then we will prove a spectrum decomposition theorem for systems with the L-shadowing property.

(Contact: B. Carvalho, T. Jäger)

22. November 2018

16:00 (DS&MP), seminar room 3517

Alejandro Kocsard (Universidade Federale Fluminense, Niteroi/Rio de Janeiro)

Rotational deviations for periodic point free homeomorphisms

(Contact: T. Jäger)

 

 

13. December 2018

14:30 (DS&MP), seminar room 3517

Arne Mosbach (Universität Bremen)

Approaching rigid rotations with beta-transformations

(Contact: D. Lenz)

 

17. January 2019

16:00 (DS&MP), seminar room 3517

Rudolf Hilfer (Universität Stuttgart, Institute for Computational Physics)
Ergodicity breaking, stationarity and local equilibrium

Abstract: The presentation reveals a fundamental dichotomy in ergodic theory between subsets of vanishing and
non-vanishing measure with respect to their induced automorphisms. The observation seems to be related
to fundamental open problems of statistical physics such as local equilibrium in time and emerges from
a scaling limit.

(Contact: M. Zähle)

 

24. January 2019

14:30 (DS&MP), seminar room 3517

Bernardo Melo de Carvalho (Universidade de Minas Gerais/FSU Jena)

Beyond Topological Hyperbolicity

Abstract:  In this talk we will discuss the dynamics of systems admitting some sort of hyperbolicity on non-trivial continua. They are called Continuum-wise Hyperbolic. We plan to introduce interesting examples of these systems and to characterize the possible dynamic phenomena which can occur.

(Contact: T. Jäger)

 

16:00 (A&G), seminar room 3517

Martin Henk (Technische Universität Berlin)

The dual Minkowski problem

Abstract: The (classical) Minkowski problem asked for sufficient and necessary conditions such that a finite Borel measure on the sphere is the surface area measure of a convex body.  Its solution, based on works by  Minkowski, Aleksandrov and Fenchel&Jessen, is one of the centerpieces of the classical Brunn-Minkowski theory.

There are two far-reaching extensions of the classical Brunn-Minkowski theory, the L_p-Brunn-Minkowski theory and  the dual Brunn-Minkowski
theory. In the talk we will discuss the analog of the (classical) Minkowski problem within the dual Brunn-Minkowski theory, i.e., the characterization problem of the dual curvature measures. These measures were recently introduced by Huang, Lutwak, Yang and Zhang and are  the counterparts to the surface area measures within the dual theory.

(Contact: T. Wannerer)

 

7. February 2019

14:30 (A&G), seminar room 3517

Christoph Thäle (Ruhr-Universität Bochum)

Monotonicity for random polytopes

Abstract: Random polytopes are classical objects studied at the crossroad of convex geometry and probability. In this talk we discuss several monotonicity questions for random polytopes. As a special case we consider the expected f-vector of random projections of regular polytopes.

(Contact: T. Wannerer)

 

28. March 2019

14:30 (A&G), seminar room 3517
Antonio Lerario (SISSA, Trieste)
Probabilistic Enumerative Geometry

Abstract: Enumerative geometry deals with the problem of counting ("enumerating") geometric objects satisfying some constraint on their arrangement. For example: "how many lines in three-space intersect at the same time four given lines?" The answer is two if we are allowed to look for complex lines, but it depends on the four given lines if we search for real lines. In the complex framework this question (and similar) can be answered using a beautiful, sophisticated technique called Schubert calculus: it is the study of the way cycles intersect in complex Grassmannians. Unfortunately, over the reals this technique loses its power: this is the old problem of finding real solutions to real equations, for which the number of complex solutions only gives upper bounds. In this talk I will present a probabilistic approach to this problem, trying to address questions like: "how many lines in three-space intersect four given random lines?"
The answer to this question comes through the study of integral geometry in real Grassmannians and has surprising connections to convex geometry and representation theory...

(Contact: T. Wannerer)

 

17. May 2018

14:30 (A&G), seminar room 3517
Gil Solanes (Universitat Autònoma de Barcelona)

Integral geometry of isotropic spaces

Abstract: The kinematic formula of Blaschke and Santaló measures the set of euclidean motions that bring a convex body to intersect another one. Similar formulas exist for other kinds of objects (e.g. smooth submanifolds), and more general ambients, including rank one symmetric spaces. The explicit determination of the kinematic formulas in these spaces is an ongoing program, which has seen important progress based on fundamental results by S. Alesker in the theory of valuations. After reviewing the classical case of real space forms, we will see how the kinematic formulas have been obtained in complex space forms, and other isotropic spaces.

(Contact person: T. Wannerer)

24. May 2018

14:30 (A&G), seminar room 3517
Nguyen-Bac Dang (École Polytechnique)

A positive cone in the space of continuous translation invariant valuations

Abstract: I will discuss a joint work with Jian Xiao. In this talk, I will exploit some ideas coming from complex geometry to define a cone in the space of continuous translation invariant valuations. This cone allows us to define a subspace of valuations V' and a norm on it. I will then explain how the convolution of valuations on smooth valuations in V' extends continuously with respect to the topology induced by this norm so that V' has a structure of graded Banach algebra. Finally, I will give two applications of our construction.

(Contact person: T. Wannerer)

31. May 2018

16:00 (DS&MP), seminar room 3517
Felipe Ramos-Garcia (Universidad Autónoma de San Luis Potosí)

Title: Weak forms of equicontinuity

Abstract: We will show how to use different forms of equicontinuity to classify dynamical systems with discrete spectrum (or Kronecker).

(Contact person: T. Jäger)

7. June 2018

16:00 (DS&MP), seminar room 3517
Michael Hartl (Imperial College London)

Asymptotically autonomous random dynamical systems

Abstract: Random dynamical systems (RDS), i.e. dynamical systems on metric spaces driven by some noise, have been broadly studied in the past. They emerge naturally in many situations, such as solutions of SDE's or a model iterated function systems. Due to the influence of the noise, those systems are per se non-autonomous, but using a dynamical model for the noise, one can write them as autonomous skew product flows defined on the product of a probability space and the metric phase space.

We study random dynamical systems, where even the deterministic part is non-autonomous already. By this we mean noise driven systems whose skew product flow has an explicit dependence on time. Especially we are interested in systems that are asymptotically autonomous, which means that the skew product flow converges as time tends to infinity.

In this talk I will present a general framework for autonomous and non-autonomous RDS. Then I will focus on  stochastic approximations, a particular class of asymptotically autonomous RDS, including examples and a bifuraction result. At the end I will address some open questions concerning generalizations of results for autonomous RDS to the non-autonomous setting.

(Contact person: T. Jäger)

14. June 2018

16:30 Mathematical Colloquium Jena, CZ3, SR 308
Michael Dellnitz (Universität Paderborn)

Glimpse of the Infinite – on the Approximation of the Dynamical Behavior for Delay and Partial Differential Equations

Abstract: Over the last years so-called set oriented numerical methods have been developed for the analysis of the long-term behavior of finite-dimensional dynamical systems. The underlying idea is to approximate the corresponding objects of interest – for instance global attractors or related invariant measures – by box coverings which are created via multilevel subdivision techniques. That is, these techniques rely on partitions of the (finite-dimensional) state space, and it is not obvious how to extend them to the situation where the underlying dynamical system is infinite-dimensional.

In this talk we will present a novel numerical framework for the computation of finitedimensional dynamical objects for infinite-dimensional dynamical systems. Within this framework we will extend the classical set oriented numerical schemes mentioned above to the infinite-dimensional context. The underlying idea is to utilize appropriate embedding techniques for the reconstruction of global attractors in a certain finitedimensional space. We will also illustrate our approach by the computation of global attractors both for delay and for partial differential equations; e. g. the Mackey-Glass equation or the Kuramoto-Sivashinsky equation.

(Contact person: T. Jäger)

21. June 2018

16:00 (A&G), seminar room 3517
Florian Besau (Goethe-Universität Frankfurt)

Weighted Floating Bodies

Abstract: Imagine for a moment a solid body floating in water. From a physics point of view, Archimedes’ principle states that the buoyant force acting on the body in an upward direction is equal to the weight of the water displaced by it. In other words, the body is floating if the weight of the body is equal to the weight of the water displaced.

If the body rolls around, then there is a part inside of it that will always be below the water surface. This kernel is the floating body. To be more precise, for a convex body, i.e., a compact convex subset of R^d with non-empty interior, we may define the floating body as the subset that is obtained by cutting away all caps of volume equal to a given positive constant δ. This classic construction can be traced back to C. Dupin in the 19th century.

Remarkably, the floating body behaves covariant with respect to affine transformations. Of particular interest has therefore been the volume difference between the body and its floating body as δ goes to zero. This gives rise to an (equi-)affine invariant, namely Blaschke’s affine surface area. This affine surface area was introduced by Blaschke in the 1920s for smooth convex bodies in dimension two and three and an extension to all convex bodies in all dimensions was established by C. Schütt and E. M. Werner in 1990 using the floating body.  Independently E. Lutwak and also K. Leichtweiß gave extensions of the affine surface area to all convex bodies around the same time and all three extensions later turn out to be equivalent. C. Schütt also established that for a convex polytope the derivative of the volume difference between the polytope and its floating body gives rise to the total number of full flags of the polytope—an important combinatoric invariant.

In the construction of the floating body one measures the volume of caps, or in other words, one assumes a uniform density. A natural generalization is to introduce a positive continuous weight function which leads to the notion of weighted floating bodies. We started investigating the limit behavior of weighted floating bodies of convex bodies and polytopes. These weighted floating bodies can also be related to intrinsic floating body constructions in spaces of constant curvature, e.g. the Euclidean unit sphere or hyperbolic space.

In this talk I will present a short overview of our results for weighted floating bodies and our applications in spherical and hyperbolic spaces. (Based on joint works in part with Monika Ludwig, Carsten Schütt and Elisabeth M. Werner.)

(Contact person: T. Wannerer)

28. June 2018

16:30 (MP&DS), seminar room 3517
Daniel Karrasch (TU München)

Lagrangian Coherent Structues

Abstract: In this talk, I give an introduction to Lagrangian Coherent Structures (LCSs). LCSs have been introduced as distinguished surfaces of trajectories in a dynamical system that exert a major influence on nearby trajectories over a finite time interval. Of particular interest are Lagrangian coherent vortices in turbulent fluid flows, which can be viewed as finite-time analogues to regular islands in a surrounding chaotic sea.

Usually, LCSs are considered in a purely advective (i.e., deterministic dynamics) framework. I will indicate issues with this approach and instead provide an advection–diffusion-based framework. This opens new avenues to analytical, computational and visualization approaches to study, for instance, turbulent fluid flows.

19. October 2017

16:00 (DS&MP), seminar room 3517
Alexander Adam (Institut Mathématique de Jussieu, UMPC, Paris)

Expansion of horocycle ergodic average

(Contact person: A. Pohl)

 

26. October 2017

Dies Academicus

 

14:30 (A&G), seminar room 3517
Jun Jason Luo (Chongqing University/Friedrich Schiller University Jena)

Topological properties of self-similar fractal squares

I will present some basic topological properties of a class of self-similar sets arising from a unit square. First we give a complete classification of their topology.  Secondly we discuss their Lipschitz equivalence. The main approach used is to construct a Gromov hyperbolic graph on the symbolic space of the self-similar set and study its hyperbolic boundary properties. Further works on more general self-similar sets and even self-affine sets will be discussed as well.

(Contact person: M. Zähle)

 

2. November 2017

16:00 (DS&MP), seminar room 3517

Christian Weiß (Hochschule Ruhr West)

Ergodic Quasi-Monte Carlo Methods

(Contact person: A. Pohl)

 

9. November 2017

14:30 (A&G), seminar room 3517
Louis Soares (Jena)

Resonances for Large Covers of Hyperbolic Surfaces

(Contact person: L. Soares, A. Pohl)

 

16:30 Mathematical Colloquium Jena, CZ3, SR 114
Vlada Limic (Université Paris Sud 11)

Near-critical random graphs and multiplicative coalescents

(Contact person: T. Jäger)

 

16. November 2017

16:00 (DS&MP), seminar room 3517
Maxim Kirsebom (Aarhus University)

The statistics of shrinking target problems in homogeneous dynamics

Abstract: It is a common question in dynamics to ask how orbits of the system hit a target, i.e. a certain subset of the system. In the last couple of decades a new variation of this question emerged known as shrinking target problems. In this case we ask how orbits hit a sequence of sets of decreasing measure. Typical results in this direction are known as quantitative Poincaré recurrence, logarithm laws, dynamical Borel-Cantelli lemmas, hitting/return time statistics and extreme value distributions.

In this talk I will give an overview of these type of results, what information each result provides and how they are connected. I will also survey recent results in this direction. Finally, I will present own results on extreme value distributions for one-parameter subgroups acting on homogeneous spaces. For the purpose of the talk I will consider the concrete case of extremes for shortest vectors in lattices in SL(d,R)/SL(d,Z).

(Contact person: A. Pohl)

 

23. November 2017

16:30 Mathematical Colloquium Jena, CZ3, SR 308
Andres Koropecki (Universidade Federal FluminenseExternal link/Friedrich Schiller Universität Jena)

Boundary dynamics for surface homeomorphisms

Abstract: I will discuss recent results about the dynamics on the boundary of invariant domains for a surface homeomorphism, and its relationship with the induced dynamics in the prime ends compactifications.

(Contact person: T. Jäger)

 

30. November 2017

14:30 (A&G), seminar room 3517
Eugenia Saorín Gómez (FSU Jena)

The difference body is a prototype of a valuation under volume constraints

(Contact person: V. Matveev)

 

7. Dezember 2017

14:30 (A&G), seminar room 3517
Claudio Gorodski (Universidade Sao Paulo)

Geometry of orbit spaces

(Contact person: V. Matveev)

 

14. Dezember 2017

14:30 (A&G), seminar room 3517
Christian Lange (Universität Köln)

Orbifolds all of whose geodesics are closed

Abstract:  In his thesis Zoll, a student of Hilbert, constructed surfaces all of whose geodesics are closed. We give more elementary proofs for the known facts that on such a Zoll surface all closed geodesics have the same length, and that the metric has to be round in case of the projective plane. For 2-orbifolds all of whose geodesics are closed the length spectrum may be more complicated. We explain why it is still rigid under changes of the metric. Depending on the time we discuss generalizations to periodic Hamiltonian flows and recent results on higher dimensional orbifolds all of whose geodesics are closed.

(Contact person: V. Matveev)

 

21. Dezember 2017

14:30 (A&G), seminar room 3517
Fabio Tal (Universidade Sao Paulo/FSU Jena)

Rotation sets for a standard family of hamiltonian torus diffeomorphisms

Abstract:  We study the rotation sets for  a natural parameter family of torus diffeomorphisms lifted that can be viewed as an analouge to the standard family of twist diffeomorphisms of the annulus. Our main goal is to identify the region O on the parameter space where the rotation set has nonempty interior,  a property that is know to have several important dynamical consequences like the existence of periodic points of arbitrarily large prime period and positive topological entropy.  Numerical evidence shows that the region O has very interesting geometry, and suggests several interesting question. We will present rigorous proofs that confirm some of the more  identifiable aspects of O, as well as asymptotic bounds on its boundary both for large and small parameters. We also show a general result of continuity of the rotation set for families of hamiltonian homeomorphisms. This is joint work w. T. Jäger and A. Koropecki.

(Contact person: T. Jäger)

 

16:30 (Mathematical Colloquium), Lecture Hall 3, CZ3
Harald Helfgott (Universität Göttingen)

Voronoi and Eratosthenes: sieves and the divisor problems

Abstract:  We show how to carry out a sieve of Erastosthenes up to N in space O(N³)$ and essentially linear time. This improves over the usual versions, which take space about O(N) and essentially linear time.  The algorithm -- which, like the one in (Galway, 2000), is ultimately related to diophantine approximation -- can also be used to factorize integers n, and thus to give the values of arithmetical functions such as the Möbius function and the Liouville function for all integers up to N.

(Contact person: A. Pohl)

 

11. January 2018

14:30 (A&G), seminar room 3517
Sabrina Kombrink (Universität zu Lübeck)

Steiner formula for fractal sets

(Contact person: A. Pohl)

 

18. January 2018

14:30 (A&G), seminar room 3517
Christian Seifert (TU Hamburg)

On traces of Dirichlet forms

(Contact person: M. Tautenhahn)

 

16:00 (DS&MP), seminar room 3517
Konstantin Hoffmann (FSU Jena)

Entropy and statistics of DNA sequences

Abstract: The statistical structure of DNA sequences shows long-range dependencies that are reflected in a decrease of the entropy of associated Markov approximations. We describe some statistical tools that can be used to quantify this phenomena.  

(Contact person: T. Jäger)

 

8. Februar 2018

 

16:30 (Mathematical Colloquium), SR 309, CZ3
Bernd Sturmfels (MPI-MIN Leipzig)

Tensors and their Eigenvectors

Abstract: 

Abstract: Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi around 2005. It has numerous applications, and ties in closely with optimization and dynamical systems. We present an introduction that emphasizes algebraic and geometric aspects.

6. April 2017

14:30 (A&G), seminar room 3517
Boris Doubrov (Belarussian State University, Minsk)

Real hypersurfaces in C³ with large symmetry

 

16:00 (DS&MP), seminar room 3517
Tobias Weich (Universität Paderborn)

Ruelle Resonanzen

In a first part of the talk we introduce the notion of Ruelle resonances for dynamical systems and their implications for the decay of correlations. We will in particular focus on the modern spectral theoretical approach and explain how Ruelle resonances are related to poles of meromorphically continued resolvents. In the rest of the talk we will present two recent results on the support of the resonant states and the Ruelle spectrum on locally symmetric spcaes.
 

26. April 2017

15:00 (A&G), seminar room 3319
Alexey Bolsinov (Loughborough University)

Projectively equivalent metrics in small dimensions

 

27. April 2017

16:00 Mathematical Colloquium Jena, CZ3, SR 308 - FÄLLT wegen Krankheit AUS!
Andreas Hamel (Free University of Bozen)

An abstract convexity approach to set relations and set optimization

 

11. May 2017

14:30 (A&G), seminar room 3517
Siegfried Beckus (Israel Institute of Technology)

The space of Delone dynamical systems and related objects

Recent developments show that the space of dynamical systems equipped with the Chabauty-Fell topology plays an important role in the spectral theory of Schrödinger operators as it greatly encodes topological and dynamical properties. Specifically, it is shown that the spectra of these operators behave continuous in the Hausdorff metric if and only if the underlying dynamical systems vary continuously in the Chabauty-Fell topology. This raises further questions whether other related quantities behave well in this topology. During the talk, we focus on the space of Delone dynamical systems in a locally compact, second countable Hausdorff group G acting by translation. Under the additional assumption of unique ergodicity of the limit point, the weak-* convergence of corresponding invariant probability measures on these Delone dynamical systems is proven.
Delone sets model solids in mathematical physics while associated Schrödinger operators describe the long time behavior of an particle in such a solid. Using the previously developed theory, we show that also other spectral quantities of these operators behave well in the Chabauty-Fell topology.

 

16:00 (DS&MP), seminar room 3517
Martin Tautenhahn (TU Chemnitz)

Quantitative unique continuation and application to control theory for the heat equation

 

18. May 2017

14:30 (A&G), seminar room 3517
Zhiyuan Zhang (Université Paris Diderot)

Density of mode-locking for a class of skew-products

16:30 Mathematical Colloquium Jena, CZ3, SR 317
Matthias Kreck (Universität Bonn)

Ein übersehenes Problem: Bettizahlen geschlossener Mannigfaltigkeiten

 

01. June 2017

16:00 (DS&MP), seminar room 3517
Felix Krahmer (Technische Universität München)

On the connection between analog and digital conversion and the roots of Chebyshev polynomials

Sigma-Delta modulation is a popular approach for coarse quantization of audio signals. That is, rather than taking a minimal amount of samples and representing them with high resolution, one considers redundant representations and works with a low resolution. The underlying idea is to employ a feedback loop, incorporating the prior evolution of the sampling error. In this way, the representation of a sample can partially compensate for errors made in previous steps. The design of the filter at the core of the feedback loop is crucial for stability and hence for performance guarantees. Building on work of Güntürk (2003) who proposed to use sparse filters, we optimize the sparsity pattern, showing that a distribution mimicking the roots of Chebyshev polynomials of the second kind is optimal. The focus of this talk will be on the interplay between complex variables, orthogonal polynomials, and signal processing in the proof. This is joint work with Percy Deift and Sinan Güntürk (Courant Institute of Mathematical Sciences, NYU).

 

8. June 2017

16:00 Mathematical Colloquium Jena, CZ3, SR 308
Erich Wittmann (TU Dortmund)

Was läuft im Mathmatikunterricht und in der Lehrerbildung falsch? Wie könnte umgesteuert werden?

 

15. June 2017

16:00 (DS&MP), SR 131 CZ
Pablo Ramacher (Marburg)

Equivariant convex and subconvex bounds for eigenfunctions and Hecke-Maas forms

 

22. June 2017

14:30 (A&G), seminar room 3517
Moussa Ndour (TU Dresden)

Qualitative changes in flow patterns

We discuss a global approach for detecting early-warning signals for qualitative changes in flow patterns. As examples, we consider the double gyre and 4-mode quasi-geostrophic circulation models.

 

16:00 (DS&MP), seminar room 3517
Maximilian Engel (Imperial College London)

Quasi-stationary dynamics and bifurcations of random dynamical systems

We look at Markov processes that induce a random dynamical system evolving in a domain with forbidden states constituting a trap. The process is said to be killed when it hits the trap and it is assumed that this happens almost surely. We investigate the behavior of the process before being killed, asking what happens when one conditions the process to survive for a long time.

The topic goes back to the pioneering work by Yaglom in 1947 but in recent years new ideas have been developed. We discuss concepts like quasi-stationary and quasi-ergodic distributions, calling the associated random dynamics quasi-stationary or quasi-ergodic if such distributions exist. Given their existence, we can define average Lyapunov exponents and the Dichotomy spectrum of the random dynamical system with killing and describe the bifurcation behavior of typical examples of stochastic bifurcation theory within this environment. The underlying philosophy is to exhibit the local character of random bifurcations for stochastic differential equations which are usually hidden in the global analysis. We further relate these concepts to dynamical systems with holes.

 

29. June 2017

16:00 Mathematical Colloquium Jena, CZ3, SR 309
Arno Berger (University of Alberta)

Digits and dynamics - A tour of Benford's Law

Benford's Law (BL), a notorious gem of mathematics folklore, asserts that leading digits of numerical data are usually not equidistributed, as might be expected, but rather follow one particular logarithmic distribution. Since first recorded by Newcomb in 1881, this apparently counter-intuitive phenomenon has attracted much interest from scientists and mathematicians. This talk will introduce and discuss some of the intriguing aspects of BL, and relate them to problems in probability and number theory and, above all, dynamics.
In view of their pivotal role as models of many real-world processes, it is natural to ask whether dynamical systems can actually comply with BL in some sense or other and, if so, whether in turn something about dynamics can be learned from this. The talk will answer both questions in the affirmative. Moreover, all real data sets, such as e.g. data recorded from a dynamical system, necessarily are finite, and determining exactly what (and what not) BL means for such data will emerge as a formidable challenge in itself.

 

06. July 2017

14:30 (A&G), seminar room 3517
Pablo Rodriguez-Sanchez (University of Wageningen)

Invasive species: a mathematician among biologistsThis is a story about multidisciplinarity. It starts with a theoretical physicist being hired as a mathematician by a biology department. But, what does exactly a mathematician in such a singular ecosystem? In this talk we'll learn that the relation between biology and mathematics can be traced back to the XIII century. We'll also learn that the survival of plankton communities is strongly related with chaotic attractors, and how differential geometry had an unexpected role in a science communication problem.

 

16:00 (DS&MP), seminar room 3517
Sara Munday (University of Bologna)

A Birkhoff ergodic theorem for infinite measure systems

18. August 2016

14:15, seminar room 3517
Michael Baake (Universität Bielefeld)

Renormalisation theory for primitive substitutions

 

20. October 2016

14:30 (A&G)
Jakub Konieczny (University of Oxford, United Kingdom)

Automatic sequences, generalised polynomials and nilmanifolds

 

16:00 (DS&MP)
Tanja Eisner (Universität Leipzig)

Weighted Ergodic Theorems

 

27. October 2016

14:00 (A&G)
Aapo Kauranen (Jyväskylä/Prague)

Sobolev spaces and Lusin's condition (N) on hyperplanes

 

03. November 2016

14:30 (A&G)
Alexander Teplyaev (University of Connecticut, USA)

Existence, uniqueness and vector analysis on fractals

The talk will describe how the heat kernel estimates, which are mainly due to Grigor'yan and Telcs, and related functional spaces and potential theory, imply the existence and uniqueness of self-similar Dirichlet forms (and hence Laplacians and diffusion processes) on generalized Sierpinski carpets. This is a joint result with Barlow, Bass and Kumagai. The second part of my talk will review recent results on vector analysis on such spaces, such as quasilinear PDEs, Dirac and magnetic Schrödinger operators, spectral triples, Hodge theory, Navier-Stokes equations, and some unusual properties of the classical curl operator. This includes joint results with J.P. Chen, M. Hinz, D. Kelleher, M. Röckner. The motivation for this vector analysis come from physics, such as studying magnetic properties of fractal structures.

 

10. November 2016

14:30 (A&G)
Franz Schuster (Technische Universität Wien, Austria)

Affine vs. Euclidean isoperimetric inequalities

In this talk we explain how every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them - the Petty projection inequality. As an application, a family of sharp Sobolev inequalities for functions of bounded variation is obtained, each of which is stronger than the classical Sobolev inequality. This is joint work with Christoph Haberl.

 

24. November 2016

14:30 (A&G)
Dan Rust (Universität Bielefeld)

Ordered cohomology and co-dimension one cut and project sets

We'll discuss how to study aperiodic tilings in Euclidean space from a topological point of view and how tools from algebraic topology, namely Cech cohomology, can be used to distinguish them. For some classes of tilings, cohomology groups are an extremely strong invariant, but for others, one needs to enrich this invariant in order to extract finer information to distinguish within a class. Tilings coming from cut-and-project methods happen to be such a class where the cohomology on its own isn't very useful. We'll talk about how the top degree Cech cohomology of a tiling comes equipped with a natural order structure. Our main result is that ordered cohomology completely classifies codimension-one cut-and-project tilings up to homeomorphism. Moreover, isomorphism of two ordered cohomology groups is equivalent to a very concrete condition involving the existence of a unimodular integer matrix.

 

01. December 2016

14:30 (A&G)
Matthias Reitzner (Universität Osnabrück)

Random polytopes: limit theorems

Let η be the set of random points of a Poisson point process in R^d, and let K be a convex set of volume 1. Denote by s the mean number of random points in K, and by K_s the convex hull of these points. We are interested in properties of K_s as s tends to infinity: expectations, variances, limit theorems and large deviations for functionals of K_s.

 

16:00 (DS&MP)
Alexey Bolsinov (Loughborough University, United Kingdom)

Stability analysis, singularities and topology of integrable systems

In the theory of integrable systems, there are two popular topics:

1) Topology of integrable systems, which studies stability of equilibria and periodic trajectories, bifurcations of Liouville tori, singularities and their invariants, topological obstructions to the integrability and so on.

2) Theory of compatible Poisson brackets, which studies one of the most interesting mechanisms for integrability based on the existence of a bi-Hamiltonian representation.

The aim of the talk is to construct a bridge between these two areas and to explain how singularities of bi-Hamiltonian systems are related to algebraic properties of compatible Poisson brackets. This bridge provides new stability analysis methods for a wide class of integrable systems.

 

08. December 2016

14:30 (A&G)
Nina Lebedeva (Steklov Institute of Mathematics at St.Petersburg, Russia)

Total curvature of geodesics on convex surfaces

We prove that the total curvature of a minimizing geodesic segment on a convex surface in the 3-dimensional Euclidean space can not be arbitrarily large (joint with Anton Petrunin).

 

16:00 (DS&MP)
Marc Rauch (Universität Jena)

The inverse variational principle

 

15. December 2016

14:30 (A&G)
Jun Luo (Universität Jena/College of Mathematics and Statistics, Chongqing University, China)Self-similar sets, simple augmented trees, and Lipschitz equivalence

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graphs on the IFS has been established recently. In this talk, we introduce a notion of simple augmented tree on the IFS which is a Gromov hyperbolic graph. By using a combinatoric device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this result to consider the Lipschitz equivalence of self-similar sets with or without assuming the open set condition, which is an important topic in fractal geometry and geometric measure theory.

 

05. January 2017

14:30 (A&G)
Felix Dorrek (Technische Universität Wien, Austria)

Minkowski Endomorphisms

A Minkowski endomorphism is a continuous and SO(n)-equivariant map, from the space of convex bodies to itself, such that Φ(K + L) = Φ(K) + Φ(L), for all K, L ∈ K. These endomorphisms were first considered by Schneider in 1974. In this talk a few open questions about Minkowski endomorphisms are going to be discussed. Among other things, it will be shown that Minkowski endomorphisms are uniformly continuous. This, in turn, implies a stronger form of a representation result for Minkowski endomorphisms due to Kiderlen. 

 

19. January 2017

14:30 (A&G)
Andreas Bernig (Goethe-Universität Frankfurt)

Integral geometry of the complex projective space

Two complex submanifolds of the complex projective space of complementary dimension and in general position will intersect in a constant number of points which is given by Bezout's theorem. If we take two real submanifolds of complementary dimension, the number of intersection points will no longer be constant and one may ask about the average number of intersection points. More generally, given two geometric objects A,B in complex projective space (compact submanifolds with boundaries or corners; sets of positive reach or subanalytic sets) and some isometry invariant functional (for instance Euler characteristic or volume), one may ask about the expected value of this functional applied to A intersected with gB, where g is an element of the isometry group. The solution to this old problem was recently obtained, in collaboration with Joseph Fu (Univ. of Georgia) and Gil Solanes (UA Barcelona) in the form of a kinematic formula in complex projective space. 

02. February 2017

14:30 (A&G)
Andreas Knauf (Friedrich-Alexander-Universität Erlangen-Nürnberg)

Symplectic aspects of scattering

 

16:30 (Mathematical Colloquium)
Fabio Tal (Universidade de Sao Paulo/Friedrich-Schiller-Universität Jena)

Entropy zero dynamics in dimension two

 

07. April 2016

14:30 (A&G)
Christoph Richard (Friedrich-Alexander-Universität Erlangen-Nürnberg)

Poisson summation and pure point diffraction

The diffraction formula for regular model sets is equivalent to the Poisson Summation Formula for the underlying lattice. We will explain this result, which is obtained by Fourier analysis of unbounded measures on locally compact second countable abelian groups, as developed by Argabright and de Lamadrid. This is joint work with Nicolae Strungaru (Peterborough).

16:00 (DS&MP)
Dominik Kwietniak (Uniwersytet Jagiellonski, Krakow, Poland & UFRJ, Rio de Janeiro, Brazil)

Dbar metric and invariant measures of hereditary shift spaces

I will describe some properties of the simplex of invariant measures of a hereditary shift space. We say that a shift space over {0,1} is hereditary if it is closed with respect to coordinatewise multiplication by an arbitrary 0-1 sequence. Using dbar pseudometric as a tool we prove that the set of ergodic measures is arcwise connected for every hereditary shift space and it is even entropy-dense for a particular class of hereditary shifts, which include many B-free shifts.

B-free shifts are defined as follows: Given a set of integers A we identify its characteristic function with an infinite 0-1 sequence x(A). The closure of the orbit of x(A) with respect to the shift generates a symbolic dynamical system X_A. Recall that an integer is B-free if it has no factor in a set B contained in N. For example square-free integers are Bsq-free where Bsq is the set of squares of primes. Abdalauoi, Lemańczyk and De La Rue, extending an idea of Sarnak, studied B-free integers F_B through dynamics of the shift space generated by the characteristic sequence of F_B.

I am going to present how using dbar metric one can obtain results about invariant measures of B-free shifts and their entropy. (This is a joint work with Jakub Konieczny and Michal Kupsa.)

 

14. April 2016

14:30 (A&G)
David Damanik (Rice University, Houston, USA)

Almost periodicity in time of solutions of the KdV equation

We describe joint work with Ilia Binder, Michael Goldstein and Milivoje Lukic, which is motivated by the following conjecture of Percy Deift: Solutions of the KdV equation with almost periodic initial data are almost periodic in time. Our work confirms this conjecture in the so-called Sodin-Yuditskii regime, that is, assuming that the Schrödinger operator whose potential is given by the initial datum has purely absolutely continuous spectrum (along with some assumptions on the topological structure of the spectrum). The talk will explain the overall structure of our approach and some of the key ideas.

 

21. April 2016

Mathematisches Kolloquium

16:30 Carl-Zeiß-Straße 3, SR 309
Israel Michael Sigal (University of Toronto)

Magnetic Vortices, vortex lattices and automorphic functions

 

12. May 2016

14:30 (A&G)
Johannes Kellendonk (Université Claude Bernard Lyon I, France)

Cyclic cohomology for graded Real C*-algebras and with an application to topological insulators

The classification of topological insulators is based on K-theory. In the non-commutative approach (which allows for the inclusion of disorder) one needs to work with Real C*-algebras, their K-theory and an appropriate dual theory. Response coefficients are then obtained as pairings. There are two options for the dual theory and the associated pairings: K-homology with index pairings or cyclic cohomology with Connes pairings. We explore the latter possibility.

 

19. May 2016

Mathematisches Kolloquium

16:30 Carl-Zeiß-Straße 3, SR 309
Eldar Straume (University of Trondheim)

On the collaboration of Felix Klein and Sophus Lie in the early 1870's, seen for the first time in the light of Klein's letter to Lie

 

26. May 2016

16:00 (DS&MP)
Carlos Sierra (MPI Biogeochemie, Jena)

The global carbon cycle as a dynamical system

In this talk I will introduce a research program that aims at conceptualizing models of the global carbon cycle as dynamical systems and use existing mathematical theory to improve our understanding of the interaction between climate and the carbon cycle. This program has four main themes: 1) identification of isomorphic models, 2) characterization of system invariants, 3) assessment of stability, and 4) determination of the statistical behavior of classes of dynamical systems used to model the global carbon cycle. I will provide examples of how existing mathematical results can greatly help in doing synthesis of computer models, and will pose some questions for mathematical results that are needed to solve some outstanding scientific questions.

 

2. June 2016

16:00 (DS&MP)
Katrin Gelfert (Universidade Federal do Rio de Janeiro, Brazil)

Dimensions and critical regularity of hyperbolic graphs

Wir diskutieren die fraktale Struktur invarianter Mengen bestimmter Abbildungen. Im Allgemeinen - insbesondere in Phasenraum mit Dimension > 2 - kann die Struktur nur schwer analysiert werden und ein natürlicher Zugang ist deshalb die Untersuchung schrittweise komplizierterer (z.B. höherdimensionaler) Systeme. Diesem Zugang folgend, betrachten wir hier zunächst Mengen, die als Graphen über bestimmten Mengen in einem 2-dimensionalem Phasenraum aufgefasst werden können; und wir beschränken uns auf Diffeomorphismen mit gleichzeitig partiell-hyperbolischer und hyperbolischer Struktur. Wir beschreiben die (kritische: Lipschitz oder auf allen Skalen Hölder-stetige) Regularität solcher Graphen und ziehen Rückschlüsse über deren Box-counting-Dimension. Die Ergebnisse resultieren teilweise aus gemeinsamer Arbeit mit L. Diaz (PUC-Rio), M. Gröger und T. Oertel-Jäger.

 

9. June 2016

14:30 (A&G)
Gabriel Fuhrmann (Friedrich-Schiller-Universität Jena)

Nonsmooth saddle-node bifurcations in quasiperiodically forced systems

 

Mathematisches Kolloquium

16:30 Carl-Zeiß-Straße 3, SR 309

Manfred Einsiedler (ETH Zürich)

Integer points on spheres and their orthogonal complement

16. June 2016

14:30 (A&G)
Nobuaki Obata (Tohoku University, Sendai, Japan)

Quantum probability and spectral analysis of graphs

Quantum probability is an algebraic extension of classical (Kolmogorovian) probability, tracing back to von Neumann who gave the mathematical formalism of statistical problems in quantum mechanics. Thus, quantum probability is based on the pair $(\mathcal{A},\varphi)$ of a *-algebra and a state on it. The algebraization is also useful for the study of spectra of (growing/random) graphs. I like to overview the method of quantum decomposition, which enables to study symmetric matrix (adjacency matrix) in terms of annihilation and creation operators in Fock space and is closely related to the theory of orthogonal polynomials.If time permits, I like to refer to some concepts of independence and application to graph products.

23. June 2016

14:30 (A&G)
Dmitry Faifman (University of Toronto, Canada)

Some integral-geometric formulas for O(p,q)

A valuation is a finitely additive measure on convex bodies. Valuation theory traces its origins to Hilbert's 3rd problem, and has since become an integral part of convex geometry.
Among the central pieces of the theory are Hadwiger's theorem, classifying all continuous valuations that are invariant under Euclidean motions; and the Poincare-Blaschke-Chern kinematic formulas, which evaluate the averages of such valuations over the different relative positions of two convex bodies. In this talk, we will put those results in the framework of Alesker's theory of valuations. I will then describe what happens when the Euclidean group of motions is replaced with the indefinite orthogonal group, e.g. the Lorentz group.

 

16:00 (DS&MP)
Jing Wang (Friedrich-Schiller-Universität Jena & Nanjing University, China)

Mode-locking in quasiperiodically forced circle maps

30. June 2016

Mathematisches Kolloquium

16:30 Carl-Zeiß-Straße 3, HS 6
Klaus Böhmer (Philipps-Universität Marburg)

Math Goes Public: Interessante Phänome bei Nichtlinearen Problemen, Tau-Tropfen auf Spinnen-Netzen

07. July 2016

16:00 (DS&MP)
Tony Samuel (California Polytechnic State University, San Luis Obispo, USA)

Aperiodic order and Jarnik sets

Sturmian words (subshifts) are combinatorial objects that are quite remarkable just by the fact that they can be formulated in terms of a variety of mathematical framework, for instance:

• Billiards - Movement of a ball on a square billiard table.
• Combinations - Aperiodic words that are balanced.
• Geometry - Digitised straight lines or circle rotations.
• Dynamical Systems - Minimal factors.
• Number Theory - Continued fractions and semi-group morphisms.

Given a θ=[0; a₁, a₂, ...] with unbounded continued fraction entries, we will discuss new characterising relations of Sturmian subshifts with slope θ with respect to the regularity properties of spectral metrics  as introduced by Kellendonk and Savinien, level sets defined in terms of the Diophantine properties of θ and complexity notions which are generalisations and extensions of the combinatorial concepts of linearly repetitive, repulsive and power free.

03. February 2016

Katja Polotzek (Technische Universität Dresden)

Numerical computation of rotations sets of torus homeomorphisms

27. January 2016

Siegfried Beckus (Friedrich-Schiller-Universität Jena)

Spectral Approximation of Schrödinger Operators: Continuous Behavior of the Spectra

20. January 2016

Frederik Witt (Universität Stuttgart)

Holomorphe integrable Systeme, Prymvarietäten und Hyperkählermetriken

Hitchins Higgs-Modulraum ist ein zentrales Beispiel für ein holomorph integrables System (HIS), dem komplexen Gegenstück integrabler Systeme aus der klassischen Mechanik. Insbesondere fasern HIS in komplexe Tori, für den Higgs-Modulraum beispielsweise in sogenannte Prymvarietäten. Darüber hinaus existieren auf HIS Hyperkählermetriken, eine spezielle Klasse Ricci-flacher Riemannscher Metriken. In diesem Vortag möchte ich zunächst Hitchins Konstruktion und Geometrie des Higgs-Modulraum eingehend diskutieren, bevor ich neuere Ergebnisse zur Asymptotik der Geometrie anspreche, die auf gemeinsamer Arbeit mit R. Mazzeo, J, Swoboda und H. Weiß beruhen.

06. January 2016

Mickaël Kourganoff (École normale supérieure de Lyon)

Similarity structures and De Rham decomposition

16. December 2015

Jun Masumune (Friedrich-Schiller-Universität Jena)

On the Liouville property of harmonic functions under certain integrable conditions

09. December 2015

Christian Oertel (Friedrich-Schiller-Universität Jena)

Model sets with positive entropy

25. November 2015

Markus Lange (Friedrich-Schiller-Universität Jena)

On Asymptotic Expansions for Spin Boson Models

18. November 2015

Alexey Glutzuk (CNRS, ENS de Lyon and Higher School of Economics, Moscow)

On periodic orbits in complex planar billiards

A conjecture of Victor Ivrii (1980) says that in every billiard with smooth boundary the set of periodic orbits has measure zero. This conjecture is closely related to  spectral theory. Its particular case for triangular orbits was proved by M. Rychlik (1989, in two dimensions), Ya. Vorobets (1994, in any dimension) and other mathematicians. The case of quadrilateral orbits in dimension two was treated in our joint work with Yu. Kudryashov (2012). We study the complexified version of planar Ivrii's conjecture with reflections from a collection of planar holomorphic curves. We present the classification of complex counterexamples with four reflections and partial positive results. The recent one says that a billiard on one irreducible complex algebraic curve without too complicated singularities cannot have a two-dimensional family of periodic orbits of any period. The above complex results have applications to other problems on real billiards: Tabachnikov's commuting billiard problem and Plakhov's invisibility conjecture.

11. November 2015

Tobias Oertel-Jäger (Friedrich-Schiller-Universität Jena)

Model sets and Toeplitz flows

04. November 2015

Alexander Lyapin (Siberian Federal University)

The Mouivre theorem for multidimensional linear difference equations

28. October 2015

Boris Kruglikov (Universitetet i Tromsø)

On the first gap in symmetry dimensions

I will present an interesting geometric phenomenon: often in finite-type systems there is a gap in dimensions between the most symmetric model and the other (non-flat) ones. I will give some general results and then illustrate this for a variety of geometries. Among the latest advances I will discuss the submaximal symmetric structures in c-projective geometry (joint with Vladimir Matveev and Dennis The) and nearly pseudo-Kähler geometry (joint with Henrik Winther) and CR-geometry.

Antoine Gournay (Technische Universität Dresden)

L^p cohomology, random walks and boundary values

27. October 2015

Henrik Winther (Universitetet i Tromsø)

Strictly nearly pseudo-Kähler manifolds with large symmetry groups

We consider strictly nearly pseudo-Kähler manifolds of dimension 6 and a closely related generalization of these called non-degenerate almost complex structures. We explore the relationship between these and use this to determine the maximal, sub-maximal and sub-submaximal symmetry dimension of such spaces and give a complete list of examples realizing these symmetry dimensions (joint work with Boris Kruglikov).