Date/Time : 17 Apr 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : Shilpak Banerjee, Indian Institute of Technology Tirupati

A fundamental theme in dynamics is the classification of systems up to appropriate equivalence relations. For instance, the equivalence relation of topological conjugacy preserves the qualitative behavior of topological dynamical systems. Smale’s celebrated program proposes to classify topological or smooth dynamical systems up to topological conjugacy. In Ergodic Theory the isomorphism problem dates back to von Neumann’s foundational paper and asks to classify measure-preserving transformations up to measure isomorphism.

These classification problems not only turn out to be hard but sometimes even to be impossible. In this talk, we give an overview of classification as well as anti-classification results and present some related projects.

Add to calendar :

Date/Time : 17 Apr 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Feng Yu, Peking University

Mixing and dissipation enhancement are two closely related concepts in the study of incompressible fluids. They also possess broad applications across various disciplines or even daily lives. In this talk, I will first introduce the idea and recent development of these topics. Then, I will explain how to apply these ideas to studying the advective Cahn-Hillard equation (ACHE). Without the advection, the Cahn-Hillard equation models the phase separation of a binary alloy. Our main result shows that on two and three dimensional tori, if the given flow is sufficiently in mixing (quantified in terms of dissipation time), then phase separation does not occur. Instead, the solution to the ACHE converges to its spatial mean exponentially in the sense of L^2. At the end of the talk, I will outline some open problems in these fields.

Add to calendar :

Date/Time : 17 Apr 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Philipp Kunde, Universität Hamburg

Dating back to the foundational paper by John von Neumann, a fundamental theme in ergodic theory is the isomorphism problem to classify invertible measure-preserving transformations (MPTs) up to isomorphism. In a series of papers, Matthew Foreman, Daniel Rudolph and Benjamin Weiss have shown that such a classification is impossible in terms of computable invariants, even with a very inclusive understanding of “computability”.  Besides isomorphism, Kakutani equivalence is the best known and most natural equivalence relation on ergodic MPTs for which the classification problem can be considered. For ergodic flows {St}t ∈ R and {Tt}t ∈ R Kakutani showed that the two flows have Kakutani equivalent transformations as cross-sections if and only if the flows are isomorphic up to a time change. Here, a time change of a flow is a reparametrization of the orbits of the flow such that each orbit is mapped to itself by an orientation-preserving homeomorphism of the parameter space. 

In joint work with Marlies Gerber we prove that the Kakutani equivalence relation of ergodic MPTs is not a Borel set. This shows in a precise way that the problem of classifying such transformations up to Kakutani equivalence is also intractable. In particular, our results imply the non-classifiability of ergodic flows up to isomorphism after a time change.

Add to calendar :

Date/Time : 17 Apr 2024 10:00
Venue : S17 #04-06 (SR1)
Speaker : Riccardo Brasca, Université Paris Cité

Workshop on Formal Proofs and Lean

Add to calendar :

Date/Time : 15 Apr 2024 10:00
Venue : S17 #04-06 (SR1)
Speaker : Riccardo Brasca, Université Paris Cité

Workshop on Formal Proofs and Lean

Add to calendar :

Date/Time : 12 Apr 2024 16:00
Venue : S17 #04-04 (SR3)
Speaker : Yan Wang, Shanghai Jiao Tong University

A maximum matching in a graph can be found in polynomial time. However, Karp shows that it is NP-hard to find even for 3-uniform hypergraphs. Much effort has been devoted to finding good sufficient conditions for the existence of a large matching in uniform hypergraphs, including Dirac type conditions. In this talk, I will talk about some recent progress on hypergraph matching problems.

Add to calendar :

Date/Time : 11 Apr 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Dougal Davis, University of Melbourne

In a seminal paper from 2011, Kari Vilonen and Wilfried Schmid proposed that the deep theory of mixed Hodge modules can be applied to the old problem of determining the unitary representations of a real reductive Lie group. They conjectured that the unitarity of a representation is completely determined by a canonical filtration, the Hodge filtration, which arises naturally from the geometry of the complex flag variety. In this talk, I will explain a recent paper with Vilonen (arXiv:2309.13215) in which we prove this conjecture. A key role (in both the proof of the conjecture and in applications) is played by a Hodge-theoretic refinement of Beilinson-Bernstein localisation: we show that, for mixed Hodge modules on flag varieties, the Hodge filtration is always globally generated and has vanishing higher cohomology.

Add to calendar :

Date/Time : 10 Apr 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Maciej Dunajski, University of Cambridge

We prove that the intrinsic Riemannian geometry of compact cross-sections of any Einstein extremal horizon must admit a Killing vector field. This establishes the rigidity theorem for the extremal Kerr black hole horizon. This extremal horizon is a special case of a quasi-Einstein structure. We shall discuss other examples of quasi-Einstein structures on compact surfaces, including a global result relevant in projective metrizability.

Add to calendar :

Date/Time : 09 Apr 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Dougal Davis, University of Melbourne

One of the oldest open problems in representation theory is to classify the irreducible unitary representations of a real reductive Lie group G_R. Such representations play a fundamental role in harmonic analysis and the Langlands program and arise in physics as the Hilbert spaces of states of quantum systems with G_R-symmetry. The orbit method, which goes back to the work of Kirillov in the 1960s, takes the physical perspective seriously and proposes to construct all unitary representations as quantisations of orbits of the group on the dual of its Lie algebra. This has proved difficult to implement for reductive groups, however: the best general results use deformation quantisation to construct interesting examples of representations but provide no tools to prove that they are unitary. In this talk, I will explain joint work in preparation with Lucas Mason-Brown overcoming this difficulty: we give a uniform proof that, under suitable hypotheses, the orbit method always produces unitary representations. The main tool is recent joint work of myself and Kari Vilonen, which gives a new characterisation of unitary representations in terms of Hodge theory on complex flag varieties.

Add to calendar :