For more information: https://scgp.stonybrook.edu/archives/35509
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Speaker: Annalisa Grossi
Title: Terminalizations of quotients of HK manifolds via symplectic actions.
Abstract: Terminalizations of symplectic quotients of HK manifolds are sources of new deformation classes of irreducible holomorphic symplectic varieties. In this talk I will describe all terminalizations of quotients by groups of automorphisms on Hilbert scheme of K3 surfaces and generalized Kummer varieties induced by symplectic automorphisms of the underlying K3 surface or abelian surface. I will sketch how to determine the second Betti number of these varieties and the fundamental group of the regular locus of these varieties. This is a joint work in progress with Bertini, Capasso, Debarre, Mauri and Mazzon.
Title: Uniformization of metric surfaces by solving Plateau’s problem
Speaker: Damaris Meier [University of Fribourg]
Abstract: The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the unit disk, complex plane or Riemann sphere. The non-smooth uniformization problem now asks for the strongest possible extension after replacing smooth surfaces by metric spaces.
On the way to answering this question in the setting of metric surfaces of locally finite Hausdorff 2-measure, we encounter the outstanding uniformization results of Bonk-Kleiner and Rajala. Moreover, we will outline how the solution of Plateau‘s problem can be used to provide such a uniformization theorem. Recall that Plateau‘s problem consists in finding a surface of minimal area spanning a given closed curve.
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Title: On KAM-rigidity of affine $Z^k$-actions on the torus.
Speaker: Bassam Fayad [University of Maryland]
Abstract: In which ways can one perturb the action on the torus of a commuting pair of affine automorphisms?
It is a general knowledge today that higher rank actions tend to be locally rigid while $Z$ actions are usually not locally rigid. The aim of the talk is to address the general classification of $Z^k$-actions on the torus by affine automorphisms in terms of local rigidity.
The talk will be based on a joint work with D. Damjanovic and M. Saprykina.
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Speaker: Amihay Hanani
Title: TBA
Title: TBA
Speaker: Simon Brendle [Columbia University]
Abstract:
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Title: Hodge-to-singular correspondence
Speaker: Mirko Mauri [IST Vienna]
Abstract: We show that the cohomology of moduli spaces of Higgs bundles decomposes in elementary summands depending on the topology of the symplectic singularities on a (fixed!) master object and/or the combinatorics of certain posets and lattice polytopes. This is based on a joint work with Luca Migliorini and Roberto Pagaria.
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Title: Reversing the direction of application, from physics to mathematics
Abstract: Traditionally mathematics is applied to physics. Yet, humans tend to be better at physics than at mathematics. When an apple falls from a tree, there are more people who can catch it—they know physically how the apple moves—than those who can compute its trajectory from a differential equation. Applying physical ideas to discover and make sense of mathematical results is therefore natural, even if it has seldom been tried in the history of science. (The exceptions include Archimedes, some old Russian sources, a recent book of Levi’s, as well as my articles and lectures.). A variety of surprising examples will be presented.
*Speaker will present via zoom. A viewing will be held in the Della Pietra Family Auditorium, room 103 of the Simons Center. Please visit our website for more information
Speaker: Amihay Hanani
Title: TBA
Title: Quantum entropy thermalization
Abstract: In an isolated quantum many-body system undergoing unitary evolution, the entropy of a subsystem (smaller than half the system size) thermalizes if at long times, it is to leading order equal to the thermodynamic entropy of the subsystem at the same energy. We prove entropy thermalization for a nearly integrable Sachdev-Ye-Kitaev model initialized in a pure product state. The model is obtained by adding random all-to-all $4$-body interactions as a perturbation to a random free-fermion model. Joint work with Aram W. Harrow
Speaker: Chris Brav
Title: From Calabi-Yau categories to the symplectic geometry of moduli spaces
Abstract: First, we survey some ideas from the non-commutative geometry of differential graded categories, in particular so-called Calabi-Yau structures on categories, and how they can be used to construct symplectic structures on various moduli spaces appearing in algebraic geometry, representation theory, and symplectic topology. Second, we use deformation theory of Calabi-Yau structures to show how cyclic homology classes on 2d Calabi-Yau categories give rise to Hamiltonians on such moduli spaces and a calculation of Poisson brackets between them. Examples include Goldman’s Hamiltonians on character varieties, Hitchin’s Hamiltonians on the moduli space of Higgs bundles, and Hamiltonians on Nakajima quiver stacks generalising the Calogero-Moser Hamiltonians from integrable systems. These results give chain-level refinements of more classical results on the Goldman Lie bracket of loops and the necklace Lie algebra of a quiver, put into a general framework and admitting generalizations to higher dimensions, where one obtains shifted symplectic structures and chain-level string brackets generalising those of Chas-Sullivan. This first part is joint work with Tobias Dyckerhoff and the second part with Nick Rozenblyum.
Title: A Proof that CAT(k) Surfaces Have Bounded Integral Curvature
Speaker: Saajid Chowdhury, Hechen Hu, Adam Tsou [Stony Brook University]
Abstract: The field of Alexandrov Geometry studies the geometric properties of metric spaces where properties such as angle are well-defined. In this talk, we consider CAT(κ) surfaces. These are defined as surfaces in which small geodesic triangles are thinner than triangles drawn with matching edge lengths in a surface of constant curvature. This condition gives an upper curvature bound on the surface. A separate notion of curvature bound on a surface is to have a uniform upper bound on the angle excess of any finite collection of non-overlapping triangles contained in a given compact neighborhood. This condition is known as having bounded integral curvature. We give a proof of a folklore theorem that a CAT(κ) surface is also a surface of bounded integral curvature. Our proof uses a new tool called vertex-edge triangulations, which are triangulations obtained by repeatedly subdividing a base triangulation. Their usefulness comes from the observation that angle excess and model area of a triangulation behave monotonically under subdivision. By a Gauss–Bonnet type argument, we derive a uniform upper bound on angle excess of a vertex-edge triangulation refining an arbitrary finite collection of triangles. This research was carried out as part of the 2022 Stony Brook REU program.
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Title: TBA
Speaker: Stefano Marmi [Scuola Normale Superiore di Pisa]
Abstract:
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