Seminars in Geometry

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Emanuele Macrì: Deformations of t-structures

14th November, 2024

Bridgeland stability conditions were introduced about 20 years ago, with motivations from algebraic geometry, representation theory, and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk, I will present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari, and Zhao. As an application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.

Alberto San Miguel Malaney: Partial Resolutions of Affine Symplectic Singularities II

9th September, 2024

We will continue to discuss partial resolutions of conical affine symplectic singularities, particularly their deformation theory and Springer theory. First we will explain the construction of the universal deformations of symplectic singularities and their partial resolutions, generalizing the Grothendieck-Springer resolution. Then we will use these universal deformations to study the Springer theory of symplectic singularities and their partial resolutions, using recent work of McGerty and Nevins. In particular, we will compute the cohomology of the fibres of the partial resolutions under suitable conditions, generalizing a result of Borho and MacPherson for the nilpotent cone. Finally, we will use partial resolutions to construct and study symplectic resolutions of symplectic leaf closures, generalizing the Springer maps from cotangent bundles of partial flag varieties to nilpotent orbit closures.

Alberto San Miguel Malaney: Partial Resolutions of Affine Symplectic Singularities I

26th August, 2024

Symplectic singularities are a generalization of symplectic manifolds that have a symplectic form on the smooth locus but allow for certain well-behaved singularities. They have a strong relationship to representation theory and include nilpotent cones of semisimple Lie algebras, quiver varieties, affine Grassmannian slices, and Kleinian singularities. There is a combinatorial description for partial resolutions of conical affine symplectic singularities, stemming from Namikawa's 2013 result that a symplectic resolution is also a relative Mori Dream Space. In this talk we will explore these partial resolutions in more detail, exploring their birational geometry, deformation theory, and Springer theory. In particular, we will review the definition of the Namikawa Weyl group for conical affine symplectic singularities and use birational geometry to define a generalization for their partial resolutions. We will also use this Namikawa Weyl group to classify the Poisson deformations of the partial resolutions. We will then describe how these partial resolutions fit into the framework of Springer Theory for symplectic singularities, following Kevin McGerty and Tom Nevins' recent paper, Springer Theory for Symplectic Galois Groups. Finally, we will discuss some ongoing research that stems from these ideas, inspired by parabolic induction and restriction.

Bertrand Toën: Geometric quantization for shifted symplectic structures

4th July, 2024

The purpose of this talk is to present an ongoing work on geometric quantization in the setting of shifted symplectic structures. I will start by recalling the various notions involved as well as the results previously obtained by James Wallbridge, who constructed the prequantized (higher) categories of a given integral shifted symplectic structure. I will then explain our main result so far: the construction of the shifted analogues of the Kostant–Souriau prequantum operators, which will be realized as a "Poisson module over a Poisson category" (a categorification of the notion of a Poisson module over a Poisson algebra). This will be obtained by means of deformation theory arguments for categories of sheaves in the setting of (derived) differential geometry. If time permits, I will discuss further aspects associated to the notion of polarizations of shifted symplectic structures.

Rita Fioresi: Quantum principal bundles on quantum projective varieties

24th June, 2024

In non-commutative geometry, a quantum principal bundle over an affine base is recovered through a deformation of the algebra of its global sections: the property of being a principal bundle is encoded by the notion of Hopf-Galois extension, while the local triviality is expressed by the cleft property. We examine the case of a projective base X in the special case X = G/P, where G is a complex semisimple group and P is a parabolic subgroup. The quantization of G will then be interpreted as the quantum principal bundle on the quantum base space X, obtained via a quantum section.

Lenhard Ng: New Algebraic Invariants of Legendrian Links

24th May, 2024

For the past 25 years, Legendrian contact homology has played a key role in contact topology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids.

Viktor Ginzburg: Invariant Sets and Hyperbolic Periodic Orbits

10th May, 2024

The presence of hyperbolic periodic orbits or invariant sets often has an affect on the global behaviour of a dynamical system. In this talk we discuss two theorems along the lines of this phenomenon, extending some properties of Hamiltonian diffeomorphisms to dynamically convex Reeb flows on the sphere in all dimensions. The first one, complementing other multiplicity results for Reeb flows, is that the existence of a hyperbolic periodic orbit forces the flow to have infinitely many periodic orbits. This result can be thought of as a step towards Franks’s theorem for Reeb flows. The second result is a contact analogue of the higher-dimensional Le Calvez-Yoccoz theorem proved by the speaker and Gurel and asserting that no periodic orbit of a Hamiltonian pseudo-rotation is locally maximal.

Sam Raskin: The Geometric Langlands Conjecture

6th May, 2024

I will describe the main ideas that go into the proof of the (unramified, global) geometric Langlands conjecture. All of this work is joint with Gaitsgory and some parts are joint with Arinkin, Beraldo, Chen, Faergeman, Lin, and Rozenblyum. I will also describe recent work on understanding the structure of Hecke eigensheaves (where the attributions are varied and too complicated for an abstract).

Egor Shelukhin: Symplectic Aspects of the Hilbert-Smith Conjecture and p-adic Actions

6th May, 2024

I will discuss a recent proof of new cases of the Hilbert-Smith conjecture for actions by homeomorphisms of symplectic nature. In particular, it rules out faithful actions of the additive p-adic group in this setting and provides further obstructions to group actions in symplectic topology. The proof relies on a new approach to this circle of questions combined with power operations in Floer cohomology and quantitative symplectic topology.

Sahana Vasudevan: Triangulated Surfaces in Moduli Space

29th April, 2024

Triangulated surfaces are Riemann surfaces formed by gluing together equilateral triangles. They are also the Riemann surfaces defined over the algebraic numbers. Brooks, Makover, Mirzakhani and many others proved results about the geometric properties of random large genus triangulated surfaces, and similar results about the geometric properties of random large genus hyperbolic surfaces. These results motivated the question: how are triangulated surfaces distributed in the moduli space of Riemann surfaces, quantitatively? I will talk about results related to this question.

Matthew Morrow: Algebraic K-theory and p-adic arithmetic geometry

22nd April, 2024

To any unital, associative ring R one may associate a family of invariants known as its algebraic K-groups. Although they are essentially constructed out of simple linear algebra data over the ring, they see an extraordinary range of information: depending on the ring, its K-groups can be related to zeta functions, corbordisms, algebraic cycles and the Hodge conjecture, elliptic operators, Grothendieck's theory of motives, and so on.

Our understanding of algebraic K-groups, at least as far as they appear in algebraic and arithmetic geometry, has rapidly improved in the past few years. This talk will present some of the fundamentals of the subject and explain why K-groups are related to the ongoing special year in p-adic arithmetic geometry. The intended audience is non-specialists.

Leonid Polterovich: Big Fibre Theorems, Ideal-Valued Measures, and Symplectic Topology

22nd April, 2024

I will discuss an adaptation of Gromov's ideal-valued measures to symplectic topology. It leads to a unified viewpoint at three 'big fibre theorems': the Centerpoint Theorem in combinatorial geometry, the Maximal Fibre Inequality in topology, and the Non-displaceable Fibre Theorem in symplectic topology, and yields applications to symplectic rigidity.

Shira Tanny: From Gromov-Witten Theory to the Closing Lemma

19th April, 2024

An old question of Poincaré concerns creating periodic orbits via perturbations of a flow/diffeomorphism. While pseudoholomorphic methods have successfully addressed this question in dimensions 2-3, the higher-dimensional case remains less understood. I will describe a connection between this question and Gromov-Witten invariants, which goes through a new class of invariants of symplectic cobordisms.

Yan-Lung Leon Li: Equivariant Lagrangian Correspondence and a Conjecture of Teleman

12th April, 2024

It has been a continuing interest, often with profound importance, in understanding the geometric and topological relationship between a Hamiltonian G-manifold Y and a symplectic quotient X. In this talk, we shall provide precise relations between their (equivariant) Lagrangian Floer theory. In particular, we will address a conjecture of Teleman, motivated by 3d mirror symmetry, on the 2d mirror construction of X from that of Y, which generalises Givental-Hori-Vafa mirror construction for toric varieties. The key technical ingredient is the Kim-Lau-Zheng’s equivariant extension of Fukaya’s Lagrangian correspondence tri-modules over equivariant Floer complexes.

Austin Christian: Persistent Legendrian Contact Homology

12th April, 2024

This talk will report on an REU whose goal was to introduce the notion of persistence into Legendrian contact homology. The LCH of a Legendrian knot is computed as the homology of the knot's Chekanov-Eliashberg DGA and is a well-studied invariant of Legendrian isotopy types. For a given Legendrian embedding, the Chekanov-Eliashberg DGA admits a natural filtration, allowing for the computation of persistent homology. The purpose of this REU was to initiate the study of the resulting filtered homology.

Jacob Lurie: Rationalized Syntomic Cohomology

10th April, 2024

A few years ago, Bhatt-Morrow-Scholze introduced an invariant of p-adic formal schemes called syntomic cohomology, which has a close relationship to (étale-localized) algebraic K-theory. In a recent paper, Antieau-Mathew-Morrow-Nikolaus showed that, after inverting p, syntomic cohomology admits a concrete description in terms of more familiar invariants, such as de Rham and crystalline cohomology. In this talk, I'll explain an alternative perspective on their result, which avoids the use of K-theoretic methods.

Shaoyun Bai: Gromov-Witten Invariants and Complex Cobordism

8th April, 2024

Since the beginning of the subject, it has been speculated that Gromov-Witten invariants should admit refinements in complex cobordism. I will propose a resolution of this question based on joint work-in-progress with Abouzaid, building on recent advances in Symplectic Topology (FOP perturbations developed jointly with Xu) and functorial resolution of singularities from algebraic geometry.

Paul Minter: An Overview of Geometric Measure Theory, Area Minimising Currents, and Recent Progress

8th April, 2024

Structures which minimise area appear in numerous geometric contexts often related to degeneration phenomena. In turn, in many situations these structures also reflect the ambient geometry in some way (they are ‘calibrated’) and so they may provide a way to study the interplay between geometry and topology, as has historically been the case for variational methods in geometry.

Almgren developed a theory which established that these area minimising structures are manifolds away from a codimension 2 ‘singular set’. The singular set itself, however, remained rather mysterious, including whether it necessarily has locally finite measure, unique tangent cones, or geometric structure (rectifiability).

In this talk I will attempt to give an overview of these ideas, as well as of recent work (joint with Camillo De Lellis and Anna Skorobogatova) answering some of the questions above related to singularities of area minimizers.

Sally Gilles: The v-Picard Group of Stein Spaces

3rd April, 2024

In this talk, I will present a computation of the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties. When the variety is the affine space, Heuer has proved that this image is equal to the group of closed differential forms. In general, we will see that the image always contains such forms but the quotient can be non-trivial: it contains a ℤp-module that maps, via the Bloch-Kato exponential map, to integral classes in the proétale cohomology.