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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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