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Lenhard Ng: New Algebraic Invariants of Legendrian Links

25th March, 2024

For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. 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.

Rudolf Zeidler: Metric inequalities under lower scalar curvature bounds

25th March, 2024

We will explain geometric situations where a lower bound on the scalar curvature of a Riemannian manifold leads to quantitative distance estimates and rigidity results. The study of these has been prompted by several conjectures of Gromov from the recent years. Intuitively, these results can be seen as analogues for scalar curvature of comparison geometry statements such as the Bonnet-Myers theorem for Ricci curvature. However, unlike classical comparison geometry involving stronger curvature conditions, such results for scalar curvature typically rely on an additional topological assumption such as the non-existence of positive scalar curvature metrics on certain submanifolds. Along the way we will thus also provide a brief introduction to obstructions to the existence of positive scalar curvature metrics on closed manifolds.

Padmavathi Srinivasan: Towards a Unified Theory of Canonical Heights on Abelian Varieties

18th March, 2024

p-adic heights have been a rich source of explicit functions vanishing on rational points on a curve. In this talk, we will outline a new construction of canonical p-adic heights on abelian varieties from p-adic adelic metrics, using p-adic Arakelov theory developed by Besser. This construction closely mirrors Zhang's construction of canonical real valued heights from real-valued adelic metrics. We will use this new construction to give direct explanations (avoiding p-adic Hodge theory) of the key properties of p-adic height pairings needed for the quadratic Chabauty method for rational points.

Pravesh Kothari: Sum-of-Squares Proofs, Efficient Algorithms, and Applications

18th March, 2024

Any non-negative univariate polynomial over the reals can be written as a sum of squares. This gives a simple-to-verify certificate of non-negativity of the polynomial. Rooted in Hilbert's 17th problem, there's now more than a century's work that finds the right multivariate generalizations called Positivstellensatz theorems (due to Krivine, Stengle, and Putinar). Beginning in the late 1980s, researchers (initially independent) in optimization, quantum information, and proof complexity theory found an algorithmic counterpart, the sum-of-squares algorithm, of these results. Over the past decade, this algorithmic theory has matured into a powerful tool for designing efficient algorithms for basic problems in algorithm design.

In this talk, I will outline a couple of highlights from these recent developments:

1) Algorithmic Robust Statistics: In the 1960s, Tukey and Huber observed that most statistical estimators are brittle -- they break down with almost no guarantee if the model postulated for data has minor misspecification (say because of 1% outliers). In response, they initiated the field of robust statistics. Over the past five years, a new blueprint, based on the sum-of-squares algorithm, has emerged for efficient robust statistics in high dimensions with new connections to finding efficiently verifiable certificates of concentration and anti-concentration properties of high dimensional probability distributions.

2) The Kikuchi Matrix Method: Finding (or proving that there is none) a solution that satisfies 99% of a given system of k-sparse linear equations (i.e., k non-zero coefficients in each equation) over finite fields is a basic NP-hard problem and thus, unlikely to admit efficient algorithms. In the early 2000s, motivated by whether the hard instances are "pathological", researchers explored whether "semirandom" equations - arbitrary systems with right-hand sides generated uniformly and independently at random - could admit efficient algorithms that output efficiently verifiable certificates of unsatisfiability.

Recently, a restricted class of sum-of-squares proofs was at the heart of efficient algorithms for such semirandom sparse linear equations. Surprisingly, these algorithms have led to new progress in extremal combinatorics and coding theory.

Richard Hind: The Shape Invariant for Toric Domains

18th March, 2024

We discuss the shape invariant, a sort of set valued symplectic capacity defined by the Lagrangian tori inside a domain of ℝ4. Partial computations for convex toric domains are sometimes enough to give sharp obstructions to symplectic embeddings, but in general the shape is far from a complete invariant. We then consider continuous families of Lagrangian embeddings, and describe a seemingly close relation to stabilized symplectic embeddings.

Artem Chernikov: Recognizing groups in Erdős geometry and model theory

15th March, 2024

Erdős-style geometry is concerned with difficult questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be viewed as asking for the possible number of intersections of a given algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups. Techniques from model theory - variants of Hrushovski’s group configuration and of Zilber’s trichotomy principle - are very useful in recognizing these groups, and led to far reaching generalizations of Elekes-Szabó in the last decade. I will overview some of the recent developments in this area, in particular explaining how all of this is not just about polynomials and works for definable sets in o-minimal structures.

Joshua Sabloff: Relative Calabi-Yau Structures for Legendrian Contact Homology

11th March, 2024

Legendrian Contact Homology (LCH) was among the first, and is still among the most important, non-classical invariants of Legendrian knots. In this talk, I will tell a story that builds up ever more sophisticated analogues of Poincare Duality in LCH. Despite the algebraic nature of the talk, I promise pictures and examples.

Rohil Prasad: Invariant Sets in Three-Dimensional Energy Surfaces

8th March, 2024

Let H be any smooth function on ℝ4 and let Y be any compact and regular level set. I'll explain a proof that Y admits an infinite family of proper compact subsets that are invariant under the Hamiltonian flow, which moreover have dense union in Y. This improves on a recent result by Fish-Hofer.

Thomas Mark: Constraints on Contact Type Hypersurfaces in Symplectic 4-Manifolds

4th March, 2024

In joint work with Bulent Tosun, it was shown that Heegaard Floer theory provides an obstruction for a contact 3-manifold to embed as a contact type hypersurface in standard symplectic 4-space. As one consequence, no Brieskorn homology sphere admits such an embedding (regardless of the contact structure). I will review the ideas that lead to these results, and discuss recent extensions that can obstruct suitably convex embeddings in closed symplectic 4-manifolds, particularly rational complex surfaces.

Daniel Cristofaro-Gardiner: Hofer-Wysocki-Zehnder’s Conjecture on Two or Infinitely Many Orbits

26th February, 2024

In their 2001 paper, Hofer, Wysocki and Zehnder conjectured that every autonomous Hamiltonian flow has either two or infinitely many simple periodic orbits on any compact star-shaped energy level; in the same paper, the authors prove this assuming in addition that the flow is non-degenerate and the stable and unstable manifolds of all hyperbolic orbits intersect transversally, a condition which holds generically. I will explain recent joint work resolving this conjecture. Our results also apply to show that every Finsler metric on the two-sphere has either two or infinitely many prime closed geodesics, answering a question attributed to Alvarez Paiva, Bangert and Long.

Tess Bouis: Motivic Cohomology of Mixed Characteristic Schemes

14th February, 2024

I will present a new theory of motivic cohomology for general (qcqs) schemes. It is related to non-connective algebraic K-theory via an Atiyah-Hirzebruch spectral sequence. In particular, it is non-A1-invariant in general, but it recovers classical motivic cohomology on smooth schemes over a Dedekind domain after A1-localisation. The construction relies on the syntomic cohomology of Bhatt-Morrow-Scholze and the cdh-local motivic cohomology of Bachmann-Elmanto-Morrow, and generalises the construction of Elmanto-Morrow in the case of schemes over a field.

Yasha Savelyev: Gromov-WItten Invariants of Riemann-Finsler Manifolds

12th February, 2024

I will give a construction of certain ℚ-valued deformation invariants of (in particular) complete non-positively curved Riemannian manifolds. These are obtained as certain elliptic Gromov-Witten curve counts. As one immediate application we give the (possibly) first generalization to non-compact fibrations, of Preissman's now classical theorem on non-existence of negative sectional curvature metrics on compact products. One additional goal of the talk is to use the above theory to motivate a very elementary but deep open problem in Riemannian geometry/dynamics concerning existence of Reebable and geodesible sky catastrophes. I will give a partial answer to this problem for surfaces.

Leonid Positselski: Semi-infinite algebraic geometry of quasi-coherent torsion sheaves

8th February, 2024

This talk is based on the book Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes—quasi-coherent torsion sheaves, the semiderived category, and the semitensor product. I will start with some examples serving as special cases of the general theory, such as the tensor structure on the category of unbounded complexes of injective quasi-coherent sheaves on a Noetherian scheme with a dualizing complex. Then I will proceed to explain the setting of a flat affine morphism of ind-schemes into an ind-Noetherian ind-scheme with a dualizing complex, and the main ingredient concepts of quasi-coherent torsion sheaves, pro-quasi-coherent pro-sheaves, and the semiderived category. In the end, I will spell out the construction of the semi-tensor product operation on the semi-derived category of quasi-coherent torsion sheaves, making it a tensor triangulated category.

Shubhodip Mondal: Dieudonné Theory via Prismatic F-gauges

7th February, 2024

In this talk, I will first describe how classical Dieudonne module of finite flat group schemes and p-divisible groups can be recovered from crystalline cohomology of classifying stacks. Then, I will explain how in mixed characteristics, using classifying stacks, one can define Dieudonné module of a finite locally free group scheme as a prismatic F-gauge (prismatic F-gauges have been recently introduced by Drinfeld and Bhatt-Lurie), which gives a fully faithful functor from finite locally free group schemes over a quasi-syntomic algebra to the category of prismatic F-gauges. This can be seen as a generalization of the work of Anschütz-Le Bras on "prismatic Dieudonne theory" to torsion situations.

Thomas Massoni: Taut Foliations Through a Contact Lens

26th January, 2024

In the late 90s, Eliashberg and Thurston established a remarkable connection between foliations and contact structures in dimension three: any co-oriented, aspherical foliation on a closed, oriented 3-manifold can be approximated by positive and negative contact structures. Additionally, when the foliation is taut, its contact approximations are (universally) tight. In this talk, I will present a converse result concerning the construction of taut foliations from suitable pairs of contact structures. I will also describe a comprehensive dictionary between the languages of foliations and of (pairs of) contact structures. Although taut foliations are usually considered rigid objects, this contact viewpoint reveals some degree of flexibility. As an application, I will show that taut foliations survive after performing large slope surgeries along transverse knots.

Valerio Assenza: On the Geometry of Magnetic Flows

26th January, 2024

A magnetic system is the toy model for the motion of a charged particle moving on a Riemannian manifold endowed with a magnetic force. To a magnetic flow we associate an operator, called the magnetic curvature operator. Such an operator encodes together the geometrical properties of the Riemannanian structure together with terms of perturbation due to magnetic interaction, and it carries crucial informations of the magnetic dynamics. For instance, in this talk, we see how a level of the energy positively curved, in this new magnetic sense, carries a periodic orbit. We also generalize to the magnetic case the classical Hopf's rigity and we introduce the notion of magnetic flatness for closed surfaces.

Soham Chanda: Augmentation Varieties and Disk Potential

26th January, 2024

Dimitroglou-Rizell-Golovko constructs a family of Legendrians in prequantization bundles by taking lifts of monotone Lagrangians. These lifted Legendrians have a Morse-Bott family of Reeb chords. We construct a version of Legendrian Contact Homology (LCH) for Rizell-Golovko's lifted Legendrians by counting treed disks. Our formalism of LCH allows us to obtain augmentations from certain non-exact fillings. We prove a conjecture of Rizell-Golovko relating the augmentation variety assoiciated to the LCH of a lifted Legendrian and the disk potential of the base Lagrangian. As an application, we show that lifts of monotone Lagrangian tori in projective spaces with different disk-potentials, e.g. as constructed by Vianna, produce non-isotopic Legendrian tori in contact spheres.

Johanna Bimmermann: From Magnetically Twisted to Hyperkähler

26th January, 2024

The tangent bundle of a Kähler manifold admits in a neighborhood of the zero section a hyperkähler structure. From a symplectic point of view, this means we have three symplectic structures all compatible with a single metric. Two of the three symplectic structures are easy to describe in terms of the canonic symplectic structure. The third one is harder to describe, but in the case of hermitian symmetric spaces, there is an explicit formula found by Biquard and Gauduchon. In this talk, I will construct a surprising diffeomorphism of the tangent bundle of a hermitian symmetric space that identifies this third symplectic structure with the magnetically twisted symplectic structure, where the twist is given by the Kähler form on the base.