Mon, 16 Jun 2025
14:15
L5

BPS polynomials and Welschinger invariants

Pierrick Bousseau
(University of Georgia)
Abstract
For any smooth projective surface $S$, we introduce BPS polynomials — Laurent polynomials in a formal variable $q$ — derived from the higher genus Gromov–Witten theory of the 3-fold $S \times {\mathbb P}^1$. When $S$ is a toric del Pezzo surface, we prove that these polynomials coincide with the Block–Göttsche polynomials defined in terms of tropical curve counts. Beyond the toric case, we conjecture that for surfaces $S_n$ obtained by blowing up ${\mathbb P}^2$ at $n$ general points, the evaluation of BPS polynomials at $q=-1$ yields Welschinger invariants, given by signed counts of real rational curves. We verify a relative version of this conjecture for all the surfaces $S_n$, and prove the main conjecture for n less than or equal to 6. This establishes a surprising link between real and complex curve enumerations, going via higher genus Gromov-Witten theory. Additionally, we propose a conjectural relationship between BPS polynomials and refined Donaldson–Thomas invariants. This is joint work with Hulya Arguz.



 

Thu, 20 Feb 2025
16:00
L5

E-Gamma Divergence: Its Properties and Applications in Differential Privacy and Mixing Times

Behnoosh Zamanlooy
(McMaster University)
Further Information

Please join us outside the lecture room from 15:30 for refreshments.

Abstract

We investigate the strong data processing inequalities of contractive Markov Kernels under a specific f-divergence, namely the E-gamma-divergence. More specifically, we characterize an upper bound on the E-gamma-divergence between PK and QK, the output distributions of contractive Markov kernel K, in terms of the E-gamma-divergence between the corresponding input distributions P and Q. Interestingly, the tightest such upper bound turns out to have a non-multiplicative form. We apply our results to derive new bounds for the local differential privacy guarantees offered by the sequential application of a privacy mechanism to data and we demonstrate that our framework unifies the analysis of mixing times for contractive Markov kernels.

Thu, 30 Jan 2025
16:00
L5

Market Making with fads, informed and uninformed traders.

Adrien Mathieu
(Mathematical Institute)
Abstract

We characterise the solutions to a continuous-time optimal liquidity provision problem in a market populated by informed and uninformed traders. In our model, the asset price exhibits fads -- these are short-term deviations from the fundamental value of the asset. Conditional on the value of the fad, we model how informed traders and uninformed traders arrive in the market. The market maker knows of the two groups of traders but only observes the anonymous order arrivals. We study both, the complete information and the partial information versions of the control problem faced by the market maker. In such frameworks, we characterise the value of information, and we find the price of liquidity as a function of the proportion of informed traders in the market. Lastly, for the partial information setup, we explore how to go beyond the Kalman-Bucy filter to extract information about the fad from the market arrivals.

Thu, 23 Jan 2025

11:00 - 12:00
L5

A new axiom for Q_p^ab and non-standard methods for perfectoid fields

Leo Gitin
(University of Oxford)
Abstract

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

Mon, 10 Mar 2025
15:30
L5

Uniform spectral gaps above the tempered gap

Vikram Giri
(ETH Zurich)
Abstract
We will explore the possibility of getting uniform spectral gaps for some invariant differential operators on hyperbolic manifolds. We will see a construction of a sequence of hyperbolic 3-manifolds with a uniform spectral gap for the 1-form Laplacian acting on coclosed forms and conclude with an application of having such gaps to torsion homology growth. Based on joint works with A. Abdurrahman, A. Adve, B. Lowe, and J. Zung.
Mon, 03 Mar 2025
15:30
L5

The Gauss-Manin connection in noncommutative geometry

Ezra Getzler
(Northwestern University and Uppsala University)
Abstract

The noncommutative Gauss-Manin connection is a flat connection on the periodic cyclic homology of a family of dg algebras (or more generally, A-infinity categories), introduced by the speaker in 1991.

The problem now arises of lifting this connection to the complex of periodic cyclic chains. Such a lift was provided in 2007 by Tsygan, though without an explicit formula. In this talk, I will explain how this problem is simplified by considering a new A-infinity structure on the de Rham complex of a derived scheme, which we call the Fedosov product; in joint work with Jones in 1990, the speaker showed that this product plays a role in a multiplicative version of the Hochschild-Kostant-Rosenberg theorem, and the point of the present talk is that it seems to be the correct product on the de Rham complex for derived geometry.

Let be an open subset of a derived affine space parametrizing a family of -algebras . We will construct a chain level lift of the Gauss-Manin connection that satisfies a new equation that we call the Fedosov equation: .

Mon, 24 Feb 2025
15:30
L5

Small eigenvalues of hyperbolic surfaces

William Hide
((Oxford University))
Abstract

We study the spectrum of the Laplacian on finite-area hyperbolic surfaces of large volume, focusing on small eigenvalues i.e. those below 1/4. I will discuss some recent results and open problems in this area. Based on joint works with Michael Magee and with Joe Thomas.
 

Mon, 17 Feb 2025
15:30
L5

Koszul duality and Calabi Yau strutures

Julian Holstein
(Universität Hamburg)
Abstract
I will talk about two aspects of Koszul duality. Firstly, Koszul duality for dg categories provides a way of modelling dg categories as certain curved coalgebras. This is a linearization of the correspondence of simplicial categories as simplicial sets (quasi-categories). Secondly, Koszul duality exchanges smooth and proper Calabi-Yau structures for dg categories and curved coalgebras. This is a generalization and conceptual explanation of the following phenomen: For a topological space X with the homotopy type of a finite complex having an oriented Poincaré duality structure (with local coefficients) is equivalent to a smooth Calabi-Yau structure on the dg algebra of chains on the based loop space of X.  This is joint work with Andrey Lazarev and with Manuel Rivera, respectively.
Mon, 10 Feb 2025
15:30
L5

Invariants that are covering spaces and their Hopf algebras

Ehud Meir
(The University of Aberdeen)
Abstract
Different flavours of string diagrams arise naturally in studying algebraic structures (e.g. algebras, Hopf algebras, Frobenius algebras) in monoidal categories. In particular, closed diagrams can be realized as scalar invariants. For a structure of a given type the closed diagrams form a commutative algebra that has a richer structure of a self dual Hopf algebra. This is very similar, but not quite the same, as the positive self adjoint Hopf (or PSH) algebras that were introduced by Zelevinsky in studying families of representations of finite groups. In this talk I will show that the algebras of invariants admit a lattice that is a PSH-algebra. This will be done by considering maps between invariants, and realizing them as covering spaces. I will then show some applications to subgroup growth questions, and a formula that relates the Kronecker coefficients to finite index subgroups of free groups. If time permits, I will also explain some connections with 2 dimensional TQFTs.

 
 
Mon, 03 Feb 2025
15:30
L5

Relative Thom conjectures

Matthew Hedden
(Michigan State University)
Abstract

Gauge theory excels at solving minimal genus problems for 3- and 4-manifolds.  A notable triumph is its resolution of the Thom conjecture, asserting that the genus of a smooth complex curve in the complex projective plane is no larger than any smooth submanifold homologous to it.  Gauge theoretic techniques have also been used to verify analagous conjectures for Kähler surfaces or, more generally, symplectic 4-manifolds.  One can formulate versions of these conjectures for surfaces with boundary lying in a 3-manifold, and I'll discuss work in progress with Katherine Raoux which attempts to extend these "relative" Thom conjectures outside the complex (or even symplectic) realm using tools from Floer homology.

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