Because of the existence of approximate p-power roots, a perfectoid algebra over ℚp admits no continuous derivations, and thus the natural Kahler tangent space of a perfectoid space over ℚp is identically zero. However, it turns out that many perfectoid spaces (or more general diamonds) arising from constructions involving rigid analytic varieties and their cohomology can be equipped with the extra structure of a Banach-Colmez Tangent Bundle, and using these Tangent Bundles natural period maps can be differentiated. As a first example, Tangent Bundles for infinite level basic EL local Shimura varieties have been computed by Ivanov and Weinstein using a heuristic construction due to Fargues and Scholze. However, in this talk, we are interested primarily in examples that go beyond the purview of the Fargues-Scholze heuristic. We will focus on the first non-trivial case of the infinite level modular curve, where we can describe the main points without assuming too much background with perfectoid rings or p-adic geometry. In this case, the existence of the Tangent Bundle is a consequence of an explicit quotient presentation of a closely related space derived from (a simple case of) the p-adic Simpson correspondence. Using this presentation, we will explain how to differentiate certain functions along Vector Fields in order to recover the annihilation property of Pan’s geometric Sen morphism, how to differentiate the Hodge-Tate period map, and how to relate this Tangent Bundle to a local construction via the Fargues-Scholze heuristic after restriction to good reduction residue disks. Time permitting, we will finish with a brief description of how the theory can be extended beyond infinite level modular curves.
I will discuss a recursive formula for the homotopy type of the space of Legendrian embeddings of sufficiently positive cables with the maximal Thurston-Bennequin invariant. Via this formula, we identify infinitely many new components within the space of Legendrian embeddings in the standard contact 3-sphere that satisfy an injective h-principle. These components include those containing positive Legendrian torus knots with the maximal Thurston-Bennequin invariant.
Given a Lagrangian link with k components it is possible to define an associated Hofer norm on the braid group with k strands. In this talk we are going to detail this definition, and explain how it is possible to prove non-degeneracy if k = 2 and certain area conditions on the Lagrangian link are met. The proof is based on the construction, using Quantitative Heegaard-Floer Homology, of a family of quasimorphisms which detect linking numbers of braids on the disc.
In the talk, I will introduce a distance-like function on the zero section of the cotangent bundle using symplectic embeddings of standard balls inside an open neighbourhood of the zero section. I will provide some examples which illustrate the properties of such a function. The main result that I will present is a relationship between the length structure associated to the introduced distance and the usual Riemannian length. Time permitting, I will explain a connection with the strong Viterbo conjecture for certain domains.
Lagrangian cobordisms induce exact triangles in the Fukaya category. But how many exact triangles can be recovered by Lagrangian cobordisms? One way to measure this is by comparing the Lagrangian cobordism group to the Grothendieck group of the Fukaya category. In this talk, we discuss the setting of exact conical Lagrangian submanifolds in Liouville manifolds and compute Lagrangian cobordism groups of Weinstein manifolds. We conclude that in this setting not always all exact triangles come from Lagrangian cobordisms.
I will discuss how to build small symplectic caps for contact manifolds as a step in building small closed symplectic 4-manifolds. As an application of the construction, I will give explicit handlebody descriptions of symplectic embeddings of rational homology balls into ℂP2.
My plan is to explain how complex projective spaces can be identified with components of totally elliptic representations of the fundamental group of a punctured sphere into PSL2(ℝ). I will explain how this identification realizes the pure mapping class group of the punctured sphere as a subgroup of the group of Hamiltonian diffeomorphisms of the complex projective space.
Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with ℂP1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson's 'four-six' question and the related Stabilizing Conjecture by Ruan.
Étale cohomology of 𝔽p-local systems does not behave nicely on general smooth p-adic rigid-analytic spaces; e.g., the 𝔽p-cohomology of the 1-dimensional closed unit ball is infinite. However, it turns out that the situation is much better if one considers only proper rigid-analytic spaces. These spaces have finite 𝔽p-cohomology groups and these groups satisfy Poincaré Duality if X is smooth and proper. I will explain how one can prove such results using the concept of almost coherent sheaves that allows to 'localize' such questions in an appropriate sense and reduce to some local computations.
The dynamics associated with mechanical Hamiltonian flows with smooth potentials that include sharp fronts may be modelled, at the singular limit, by Hamiltonian impact systems: a class of generalized billiards by which the dynamics in the domain’s interior are governed by smooth potentials and at the domain’s boundaries by elastic reflections. Results on persisting vs non-persisting dynamics of such systems will be discussed. In some cases, called quasi-integrable, the limit systems have fascinating behaviour: their energy surfaces are foliated by 2-dimensional level sets. The motion on each of these level sets is conjugated to a directed motion on a translation surface. The genus of the iso-energy level sets varies - it is only piecewise constant along the foliation. The metric data of the corresponding translation surfaces and the direction of motion along them changes smoothly within each of the constant-genus families. Ergodic properties and quantum properties of classes of such systems are established.
The Breuil-Mezard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group G. Most of the progress thus far has been concentrated on the case G = GL2, which has several special features. I will talk about joint work with Bao Le Hung on a new approach to the Breuil-Mezard Conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). It is based on the intuition that the Breuil-Mezard conjecture is analogous to homological mirror symmetry.
Translational tiling is a covering of a space (such as Euclidean space) using translated copies of one building block, called a "translational tile'', without any positive measure overlaps. Can we determine whether a given set is a translational tile? Does any translational tile admit a periodic tiling? A well-known argument shows that these two questions are closely related. In the talk, we will discuss this relation and present some new developments, joint with Terence Tao, establishing answers to both questions.
In the recent years there have been some spectacular applications of the theory of o-minimality (a branch of Model Theory) to some problems in Diophantine Geometry. It culminated in the unconditional proof of the Andre-Oort conjecture on the Zariski closure of sets of special points on Shimura varieties. We will present ideas and methods surrounding this proof.
Cohomology of classifying space/stack of a group G is the home which resides all characteristic classes of G-bundles/torsors. In this talk, we will try to explain some results on Hodge/de Rham cohomology of BG where G is a p-power order commutative group scheme over a perfect field of characteristic p, in terms of its Dieudonné module.
In the study of Hamiltonian systems, integrable dynamics play a crucial role. Integrability, however, appears to be a delicate property that is not expected to persist under generic small perturbations. Understanding the essence of this fragility presents a compelling task, which turns out to be relevant across different contexts. In this talk, I shall present some results aimed at shedding more light on this issue, within the framework of symplectic twist maps of the 2-dimensional annulus. Specifically, I shall investigate the persistence and the properties of invariant Lagrangian tori that are foliated by periodic points and discuss how their fragility underpins the rigidity of completely integrable twist maps.
The C0 distance on the space of contact forms on a contact manifold has been studied recently by different authors. It can be thought of as an analogue for Reeb flows of the Hofer metric on the space of Hamiltonian diffeomorphisms. In this talk, I will explain some recent progress on the stability properties of the topological entropy with respect to this distance obtained in collaboration with M. Alves, L. Dahinden, and A. Pirnapasov. Our main result states that the topological entropy for closed contact 3-manifolds is lower semi-continuous in the C0 distance for C∞-generic contact froms. Applying our methods to geodesic flows of surfaces, we obtain that the points of lower-semicontinuity of the topological entropy include non-degenerate metrics. In particular, given a geodesic flow of such a metric with positive topological entropy, the topological entropy does not vanish for sufficiently C0-small perturbations of the metric.
Polarised abelian surfaces vary in 3-dimensional families. In contrast, the derived category of an abelian surface A has a 6-dimensional space of deformations; moreover, based on general principles, one should expect to get 'algebraic families' of their categories over 4-dimensional bases. Generalized Kummer varieties (GKV) are hyperkähler varieties arising from moduli spaces of stable sheaves on abelian surfaces. Polarised GKVs have 4-dimensional moduli spaces, yet arise from moduli spaces of stable sheaves on abelian surfaces only over 3-dimensional subvarieties.
I present a construction that addresses both issues. We construct 4-dimensional families of categories that are deformations of Db(A) over an algebraic space. Moreover, each category admits a Bridgeland stability condition, and from the associated moduli spaces of stable objects one can obtain every general polarised GKV, for every possible polarisation type of GKVs. Our categories are obtained from ℤ/2-actions on derived categories of K3 surfaces.
Recent interactions between condensed mathematics and K-theory have led us to revisit the topic of (nonconnective) algebraic K-theory of topological algebras. In this talk, among recent developments, I will focus on the ring of continuous functions on a compact Hausdorff space valued in a local field (or a local division ring). This work resolves a previously unconfirmed claim about negative K-theory made by Rosenberg in 1990. The method employed is inspired by the resolution of Weibel's conjecture. The main result provides new counterexamples in K-theory by importing pathology from general topology.
Results concerning rigidity of Lagrangian submanifolds lie at the heart of symplectic topology, and have been intensively studied since the 1990s. An example for this phenomenon is the concept of Lagrangian Barriers, a form of symplectic rigidity introduced by Biran in 2001, which involves obligatory intersections of symplectic embeddings with Lagrangian submanifolds not derived from mere topology. In this joint work with Richard Hind and Yaron Ostrover, we present what appears to be the first illustration of Symplectic Barriers (and in particular not Lagrangian). The key point being that Lagrangian submanifolds are not the sole barriers, and there exist situations where a symplectic submanifold does not exhibit flexibility. In our work, we also tackle a question by Sackel–Song–Varolgunes–Zhu and provide bounds on the capacity of the ball after removing a codimension 2 hyperplane with a prescribed Kähler angle.