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Alberto Abbondandolo: Symplectic Capacities of Domains Close to a Ball and Geodesics in the Space of Contact Forms

13th November, 2023

An old open question in symplectic geometry asks whether all normalized symplectic capacities coincide for convex domains in the standard symplectic vector space. I will show that this question has a positive answer for smooth convex domains which are C2-close to a Euclidean ball. This is related to the question of existence of minimizing geodesics in the space of contact forms on a closed contact manifold equipped with a Banach-Mazur-like metric. On the other hand, there are smooth domains which are arbitrarily C1-close to the ball for which the ball capacity is strictly smaller than the cylindrical capacity.

Dalimil Mazac: Sphere Packings, Spectral Gaps and the Conformal Bootstrap

8th November, 2023

I will discuss infinite-dimensional linear programs producing bounds on the spectral gap in various settings. This includes new bounds on the spectral gap of hyperbolic manifolds as well as the Cohn-Elkies bound on the density of sphere packings. The bounds allow us to essentially determine the complete set of spectral gaps achieved by hyperbolic 2-orbifolds. The linear programs involved have been the subject of intense study by mathematical physicists in the context of the conformal bootstrap.

I will review the method of analytic extremal functionals, introduced by the speaker to prove sharp bounds in the conformal bootstrap. When used within the Cohn-Elkies linear program, this method reproduces the groundbreaking solution of Viazovska et al of the sphere packing problem in dimensions 8 and 24, as well as the interpolation basis used in the proof of universal optimality of the E8 and Leech lattice. The connections covered in this talk offer a broader framework for studying optimality in infinite-dimensional linear programs.

Juan Esteban Rodriguez Camargo: The Analytic de Rham Stack

8th November, 2023

In this talk, we introduce the analytic de Rham stack for rigid varieties over ℚp (and more general analytic stacks). This object is an analytic incarnation of the (algebraic) de Rham stack of Simpson, and encodes a theory of analytic D-modules extending the theory of D̂-modules of Ardakov and Wadsley. We mention how a very general six functor formalism can be construct in this set up, as well as other features such as Kashiwara equivalence and Poincaré duality for smooth maps.

Annette Werner: On the Birational Geometry of Matroids

6th November, 2023

The theory of matroids provides a unified abstract treatment of the concept of dependence in linear algebra and graph theory. In this talk we explain Bergman fans of matroids, and we investigate isomorphisms of Bergman fans for different fan structures. In particular, we introduce and study Cremona automorphisms.

Marco Mazzucchelli: Locally Maximal Closed Orbits of Reeb Flows

6th November, 2023

A compact invariant set of a flow is called locally maximal when it is the largest invariant set in some neighborhood. In this talk, based on joint work with Erman Cineli, Viktor Ginzburg, and Basak Gurel, I will present a 'forced existence' result for the closed orbits of certain Reeb flows on spheres of arbitrary odd dimension:

- If the contact form is non-degenerate and dynamically convex, the presence of a locally maximal closed orbit implies the existence of infinitely many closed orbits. 

- If the locally maximal closed orbit is hyperbolic, the assertion of the previous point also holds without the non-degeneracy and with a milder dynamically convexity assumption.

These statements extend to the Reeb setting earlier results of Le Calvez-Yoccoz for surface diffeomorphisms, and of Ginzburg-Gurel for Hamiltonian diffeomorphisms of certain closed symplectic manifolds.

Oliver Edtmair: The Subleading Asymptotics of Symplectic Weyl Laws

3rd November, 2023

Spectral invariants defined via Embedded Contact Homology (ECH) or the closely related Periodic Floer Homology (PFH) satisfy a Weyl law: Asymptotically, they recover symplectic volume. This Weyl law has led to striking applications in dynamics (smooth closing lemma) and symplectic geometry (simplicity conjecture). In this talk, I will report on work in progress concerning the subleading asymptotics of symplectic Weyl laws. I will explain the connection to symplectic packing problems and the algebraic structure of groups of Hamiltonian diffeomorphisms and homeomorphisms.

Alexander Petrov: On Local Systems of Geometric Origin

1st November, 2023

I will discuss the following conjecture: an irreducible ℚ̅ℓ-local system L on a smooth complex algebraic variety S arises in cohomology of a family of varieties over S if and only if L can be extended to an etale local system over some descent of S to a finitely generated subfield of complex numbers. I will describe the motivation for this conjecture coming from relative p-adic Hodge theory, known partial results, and possible approaches (not very successful so far) to formulating a purely p-adic (and thus hopefully more tractable) version of this conjecture. A large part of the talk will be expository, including material based on the ideas of Hélène Esnault, Raju Krishnamoorthy, and Josh Lam.

Gabriele Benedetti: Rigidity and Flexibility of Periodic Hamiltonian Flows

30th October, 2023

An old problem in classical mechanics is the existence of periodic flows within specific classes of Hamiltonian systems such as geodesic and magnetic flows, and central forces. In the last years, interest in this problem has been revitalized since recent research has unveiled a deep relationship between periodic Hamiltonian flows and systolic questions in symplectic and contact geometry. While only trivial examples of periodic flows among magnetic and central systems exist, Zoll and, later, Guillemin have shown that there are many exotic examples among geodesic flows on the two-sphere. Following Guillemin's approach, the goal of this talk is to show how the Nash-Moser implicit function theorem can be used to construct magnetic flows on the two-torus which are periodic for a single value of the energy.

Lukas Nakamura: A Metric on the Contactomorphism Group of an Orderable Contact Manifold

27th October, 2023

We discuss some properties of a pseudo-metric on the contactomorphism group of a strict contact manifold M induced by the maximum/minimum of Hamiltonians. We show that it is non-degenerate if and only if M is orderable and that its metric topology agrees with the interval topology introduced by Chernov and Nemirovski. We also discuss analogous results on isotopy classes of Legendrian submanifolds and on universal covers.

Han Lou: On the Hofer Zehnder Conjecture for Semipositive Symplectic Manifolds

27th October, 2023

Arnold's conjecture says that the number of 1-periodic orbits of a Hamiltonian diffeomorphism is greater than or equal to the dimension of the Hamiltonian Floer homology. In 1994, Hofer and Zehnder conjectured that there are infinitely many periodic orbits if the equality doesn't hold. In this talk, I will show that the Hofer-Zehnder conjecture is true for semipositive symplectic manifolds with semisimple quantum homology.

Zeev Dvir: High-Dimensional Variants of the Finite Field Kakeya Problem

23rd October, 2023

The finite field Kakeya problem asks about the size of the smallest set in (𝔽q)n containing a line in every direction. Raised by Wolff in 1999 as a 'toy' version of the Euclidean Kakeya conjecture, this problem is now completely resolved using the polynomial method. In this talk I will describe recent progress on its higher-dimensional variant in which lines are replaced with k-dimensional flats. It turns out that, unlike in the 1-dimensional case, when k ≥ 2, one can prove that there are no 'interesting' constructions (with size smaller than trivial) even if one asks for sets that only have large intersection with a flat in every direction. This theorem turns out to have surprising applications in questions involving lattice coverings and linear hash functions.

Russell Avdek: CH, GW and the geography of tight convex hypersurfaces

20th October, 2023

While convex hypersurfaces are well understood in 3d contact topology, we are just starting to explore their basic properties in high dimensions. I will describe how to compute contact homologies (CH) of their neighborhoods, which can be used to infer tightness in any dimension. Then I’ll give a general construction of high-dimensional convex hypersurfaces in the style of Gompf’s fiber sum. For these convex hypersurfaces, relative Gromov-Witten can often compute CH in the style of Diogo-Lisi. We’ll work through some interesting examples.

Linquan Ma: Test Ideals in Mixed Characteristic via the p-adic Riemann-Hilbert Correspondence

18th October, 2023

Multiplier ideals in characteristic zero and test ideals in positive characteristic are fundamental objects in the study of commutative algebra and birational geometry in equal characteristic. We introduced a mixed characteristic version of the multiplier/test ideal using the p-adic Riemann-Hilbert correspondence of Bhatt-Lurie. Under mild finiteness assumptions, we show that this version of test ideal commutes with localization and can be computed by a single alteration up to small perturbation.

Vadim Vologodsky: Prismatic F-gauges and Fontaine-Laffaille Modules

11th October, 2023

With every bounded prism Bhatt and Scholze associated a cohomology theory of formal p-adic schemes. The prismatic cohomology comes equipped with the Nygaard filtration and the Frobenius endomorphism. The Bhatt-Scholze construction has been advanced further by Drinfeld and Bhatt-Lurie who constructed a cohomology theory with values in a stable ∞-category of prismatic F-gauges. The new cohomology theory is universal, meaning that, for every bounded prism, the associated prismatic cohomology theory factors through the category of prismatic F-gauges.

In this talk, I will explain how a full subcategory of the category of prismatic F-gauges formed by objects whose Hodge-Tate weights lie in the interval [0,p-2] is equivalent to the derived category of Fontaine-Laffaille modules with a similar weight constraint. In the geometric context, this means that the prismatic F-gauge associated with a formally smooth scheme over p-adic integers of dimension less than p-1 can be recovered from its Hodge filtered de Rham cohomology equipped with the Nygaard refined crystalline Frobenius endomorphism.

If time permits, I will explain a generalization of the above statement to the case of prismatic F-gauges over a smooth p-adic formal scheme.

Bulent Tosun: Contact Surgeries and Symplectic Fillability

9th October, 2023

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, extending an earlier work of the speaker with Conway and Etnyre, we will discuss some new results about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (n) surgery along a Legendrian knot to yield a weakly fillable contact manifold, for some integer n > 0. When specialized to knots in the three sphere with its standard tight structure, this result can be effectively used to find many examples of fillable surgeries along with various obstructions and surprising topological applications. For example, we prove that a knot admitting lens space surgery must have slice genus equal to its 4-dimensional clasp number.

Kalyani Kansal: Irregular Loci in the Emerton-Gee Stack for GL2

9th October, 2023

Let K be a finite extension of ℚp. The Emerton-Gee stack for GL2 is a stack of etale (φ, Γ)-modules of rank two. Its reduced part, X, is an algebraic stack of finite type over a finite field, and can be viewed as a moduli stack of two-dimensional mod p representations of the absolute Galois group of K. By the work of Caraiani, Emerton, Gee and Savitt, it is known that in most cases, the locus of mod p representations admitting crystalline lifts with specified regular Hodge-Tate weights is an irreducible component of X. Their work relied on a detailed study of a closely related stack of etale phi-modules which admits a map from a stack of Breuil-Kisin modules with descent data. In our work, we assume K is unramfied and further study this map with a view to studying the loci of mod p representations admitting crystalline lifts with small, irregular Hodge-Tate weights. We identify these loci as images of certain irreducible components of the stack of Breuil-Kisin modules and obtain several inclusions of the non-regular loci into the irreducible components of X.

Jacob Lurie: What is… p-adic geometry?

9th October, 2023

Let p be a prime number. Roughly speaking, rigid analytic geometry is a counterpart of complex analysis where one replaces the field ℂ of complex numbers by the field ℚp of p-adic rational numbers (or some extension thereof).

In this talk, I'll try to explain some of the motivations which led to the development of this theory, and to give some flavour of the recent progress which make it a timely subject for our Special Year. No prior knowledge of p-adic geometry will be assumed.

Mohammed Abouzaid: Bordism and Floer theory

5th October, 2023

In the late 1980s Andreas Floer revolutionized low-dimensional and symplectic topology by discovering the existence an extension of Morse theory to an infinite-dimensional setting where the standard methods of variational calculus fail. While he foresaw that his theory should be able to encompass generalized homology theory (bordism, K-theory, ...), severe foundational difficulties prevented any significant progress on this question until two years ago. I will explain the advances that have been made on two fronts: (I) defining concrete models, in terms of equivariant vector bundles, for the moduli spaces that appear in Floer theory, and (II) understanding the geometric consequences of lifting Floer homology to generalized homology theories. I will end by formulating how the notion of derived orbifold bordism provides a universal receptacle for Floer's invariants, and its descendants.