Melchior Wirth

Research Interests

Roughly speaking, I am interested in the interplay between analysis, geometry and probability beyond the smooth world. More concretely, this includes the following topics and their interconnections:

• discrete and noncommutative optimal transport
• gradient flows and curvature in non-smooth settings
• analysis and geometry of infinite graphs

Publications and Preprints

Peer-reviewed publications

Domination of quadratic formsExterner Link

(with Daniel Lenz, Marcel Schmidt), Mathematische Zeitschrift, accepted

Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tilesExterner Link

(with Christian Richter), Discrete & Computational Geometry, 63(1), 2019

On the geometry of geodesics in discrete optimal transportExterner Link

(with Matthias Erbar, Jan Maas), Calculus of Variations and Partial Differential Equations, Volume 58 (2018)

Sobolev-Type Inequalities and Eigenvalue Growth on Graphs with Finite MeasureExterner Link

(with Bobo Hua, Matthias Keller, Michael Schwarz), Proceedings of the American Mathematical Society, accepted

Diffusion determines the recurrent graphExterner Link

(with Matthias Keller, Daniel Lenz, Marcel Schmidt), Advances in Mathematics, Volume 269 (2015), 364-398.

Preprints

Self-Adjoint Extensions of Bipartite HamiltoniansExterner Link

(with Daniel Lenz, Timon Weinmann)

A Noncommutative Transport Metric and Quantum Markov Semigroups as Gradient Flows of the EntropyExterner Link

submitted

Geometric properties of Dirichlet forms under order isomorphismsExterner Link

(with Daniel Lenz, Marcel Schmidt), submitted

Stability of Kac regularity under domination of quadratic formsExterner Link

submitted

Uniqueness of form extensions and domination of semigroupsExterner Link

(with Daniel Lenz, Marcel Schmidt), submitted

Master's thesis

Uniqueness of form extensions and domination of semigroupsExterner Link

(FSU Jena 2015, Advisor: Prof. Daniel Lenz)

Bachelor's thesis

Does diffusion determine the graph structure?Externer Link

(FSU Jena 2013, Advisor: Prof. Daniel Lenz)

Talks and Posters

Talks

• June 2019, Conference 'Geometric aspects of harmonic analysis and spectral theoryExterner Link' at Technion, Haifa: 'Sobolev-Type Inequalities and Eigenvalue Growth on Graphs with Finite Measure'

• June 2019, research seminar of the RTG 'Energy, Entropy, and Dissipative DynamicsExterner Link' at RWTH Aachen: 'Entropic gradient flow structure of quantum Markov semigroups'

• July 2018, joint Jena-Leipzig seminarExterner Link: 'The Geometry of Geodesics in Discrete Optimal Transport'

• April 2018, PhD SeminarExterner Link at FSU Jena: 'Covering the plane with triangles'

• March 2018, Closing Workshop RTG Quantum and Gravitational Fields at FSU Jena: 'The Heat Flow as Gradient Flow of the Entropy'

• November 2017, PhD SeminarExterner Link at FSU Jena: 'Gradient flows in Euclidean, Hilbert, and metric spaces'

• August 2017, summer school 'Analysis und Theoretische Numerik' in Siegmundsburg: 'Laplacians and isometries of metric measure spaces'

• March 2017, workshop 'Interface between Commutative and Non-Commutative Stochastic Analysis' at Hokkaido University Sapporo: 'A transport metric for Dirichlet forms and gradient flows of the entropy'

• March 2017, workshop 'Dirichlet Forms and Their GeometryExterner Link' at Tohoku University Sendai: 'Geometric properties of Dirichlet forms under order isomorphisms'

• March 2017, spring school 'From Particle Dynamics to Gradient FlowsExterner Link' at TU Kaiserslautern: 'Dirichlet forms and gradient flows of the entropy'

• January 2017, research seminar 'Discrete spectral geometryExterner Link' at University of Potsdam: 'A transport metric for non-local Dirichlet forms and gradient flows of the entropy'

• December 2016, workshop 'Operator Theory and Indefinite Inner Product SpacesExterner Link' at TU Vienna: 'Uniqueness of form extensions and domination of semigroups'

• November 2016, workshop 'Heat kernels, stochastic processes and functional inequalitiesExterner Link' in Oberwolfach: 'Geometric properties of Dirichlet forms under order isomorphisms'

• September 2016, summer school 'Spectral Theory, Differential Equations and Probability' at University of Mainz: 'Does Diffusion Determine the Geometry?'

• July 2016, summer school 'Analysis und Theoretische Numerik' in Siegmundsburg: 'Uniqueness of form extensions via domination of semigroups'

• July 2016, Studierendenkonferenz der DMV at TU Berlin: 'Uniqueness of form extensions on L²-spacesExterner Link'

• June 2016, PhD SeminarExterner Link at Friedrich Schiller University Jena: 'Groups and C*-Algebras: Amenability, C*-Simplicity and the Unique Trace Property'

• January 2016, workshop 'Spectral GeometryExterner Link' at University of Potsdam: 'Domination of semigroups and uniqueness of form extensions'

• July 2015, PhD SeminarExterner Link at Friedrich Schiller University Jena: 'Uniqueness of extensions of magnetic Schrödinger forms'

• January 2015, one day workshop 'New Directions in Mathematical Physics and beyondExterner Link' at Friedrich Schiller University Jena: 'Does diffusion determine the graph structure?Externer Link'

Posters

• June 2018, graduate summer school 'Mean Field Games and ApplicationsExterner Link' at IPAM, UCLA: 'On the geometry of geodesics in discrete optimal transport'

• November 2017, conference 'Discrete and Continuous Models in the Theory of NetworksExterner Link' at Center for Interdisciplinary Research Bielefeld: 'Does Diffusion Determine the Graph Structure?'

• November 2016, winter school 'Dynamical Methods in Open Quantum SystemsExterner Link' at University of Göttingen: 'Noncommutative Dirichlet forms and gradient flows of the entropyExterner Link'

• September 2015, conference 'Stochastic Analysis and Applications'Externer Link at Tohoku University Sendai: 'Diffusion determines the recurrent graphExterner Link'

Curriculum Vitae

Current position

PhD student, Friedrich Schiller University Jena

Education

2014 - 2015	Master studies in mathematics, FSU Jena
2013 - 2014	Master studies in mathematics, WWU Münster
2011 - 2013	Bachelor studies in mathematics, FSU Jena
2010 - 2011	Bachelor studies in physics, FSU Jena

Prizes

2016 Prize of the President of Jena University for master's thesis

2016 Prize at DMV Students' Conference for master's thesis: participation in workshop at Mathematisches Forschungsinstitut Oberwolfach (MFO)

Funding

since April 2017 PhD studies funded by the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes)

since April 2016 Member of the Research Training group (1523/2) Quantum and Gravitational Fields

2016 Oberwolfach Leibniz Graduate Student (OWLG)

2013 - 2015 Studies funded by the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes)

(Co-)Organized

Scientific Meetings

One Day Workshop 2018Externer Link 'Nonlocal Models in Analysis and Probability', March 22, 2018 (together with Daniel Lenz and Ilya Pavlyukevich)

Young Researchers' SymposiumExterner Link within conference Analysis and Geometry on Graphs and ManifoldsExterner Link at Potsdam University, July 31 to August 3, 2017 (together with Michael Schwarz)

Teaching

Courses

Fall 2019	Analysis III (exercise class)
Fall 2015	Functional Analsis II (exercise class)
Summer 2015	Ordinary Differential Equations (exercise classes)
Summer 2013	Functional Analysis I (exercise class)
Fall 2012	Analysis III (exercise class)
Summer 2012	Analysis II (exercise class)