Events

It is known that Fukaya categories are not filtered $A_\infty$ categories for arbitrary choices of the parameters needed for their construction, but only weakly-filtered. In this talk we will present a trick to construct classes of such parameters so that the associated Fukaya categories are filtered. Then, we will discuss how different choices of parameters will affect the persistence structure of Fukaya categories at the derived level. If time permits we will show some applications of the filtered structures on Fukaya categories.

I will discuss ongoing work with Davesh Maulik in which we formulate a generalization of the GW/DT conjectures to the setting of simple normal crossings pairs. When the divisor in the pair is smooth, the formulation of the conjecture necessitates the study of the cohomology of the Hilbert scheme of points on a surface. The formulation of the logarithmic GW/DT conjecture requires new geometry coming from a logarithmic Hilbert scheme of points. We prove a strengthened logarithmic degeneration formula on both sides of the correspondence and prove that the new conjectures are compatible with the old ones via degeneration. I’ll discuss this circle of ideas, and explain which parts of the conjectures are within reach.

A short abstract of your talk: Ever wondered what category theory is and what we use it for? In this talk, we will introduce affine group schemes which an excellent example for representable functors, and discuss a special type of categories (Tannakian categories) which generalize the idea of tensor products for vector spaces.

Abstract: In this talk, I will present a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric d by d matrices A_1,...,A_n each with operator norm at most 1 and rank at most n/\log^3 n, one can efficiently find \pm 1 signs x_1,... ,x_n such that their signed sum has spectral norm \|\sum_{i=1}^n x_i A_i\|_op= O(\sqrt{n}). This result also implies a (\log n - 3 \log \log n) qubit lower bound for quantum random access codes encoding n classical bits with advantage >> 1/\sqrt{n}. Our proof uses the recent refinement of the non-commutative Khintchine inequality in [Bandeira, Boedihardjo, van Handel, 2022] for random matrices with correlated Gaussian entries. This talk is based on a joint work with Nikhil Bansal and Raghu Meka, available at https://arxiv.org/abs/2208.11286.

Hyperbolic structures on a surface are equivalent to injective homomorphisms with discrete image from its fundamental group to PSL(2,R). These are the objects of study of classical Teichmüller theory. Replacing PSL(2,R) by a higher rank Lie group, such as PSL(n,R) for n at least 3, leads to the notion of a higher rank Teichmüller space. In this talk, we will define higher rank Teichmüller spaces, discuss their existence and properties, and see an example of how they arise as geometric structures on the surface. No prior knowledge on Lie groups or hyperbolic geometry is required.

In this series of talks, I will summarize some recent progress on degenerations and mirror symmetry. The first talk is to relate relative Gromov--Witten theory with absolute orbifold Gromov--Witten theory. The second talk is about structures in relative Gromov--Witten theory. The third talk is about relative mirror symmetry and applications.

Gaussian random fields (Gaussian processes) are beautiful mathematical objects with many applications. In machine learning, they are widely accepted as models of choice in scenarios where decision making under uncertainty is required e.g. in black-box optimization and reinforcement learning. In this talk I will briefly overview Gaussian processes in this setting and my own work on Gaussian processes for modeling functions on non-Euclidean domains, including Riemannian manifolds and graphs.

Gaussian processes play an important role in statistics for making inference about spatial or spatiotemporal data. Traditionally, the dependence structures of these random processes (in space or space-time) are defined via their covariance kernels. Since the computational costs of these kernel-based approaches for applications such as predictions are, in general, cubic in the number of data points, a vibrant research area has evolved, where various methods for “big data” are proposed. In the last decade, the Stochastic Partial Differential Equation (SPDE) approach has proven to be very efficient for tackling the conflict between limited computing power and desired modeling capabilities. Motivated by a well-known relation between the Gaussian Matérn class and fractional-order SPDEs, the key idea of this approach is to define Gaussian processes as solutions to appropriate SPDEs and to use efficient numerical methods, such as the Finite Element Method (FEM) or wavelets, for approximating them. In this talk I will give an introduction to the (spatial) SPDE approach and discuss several recent developments, in particular with regard to the quality and computational costs of FEM approximations in statistical applications. Finally, I will give an outlook on spatiotemporal models which are based on SPDEs involving fractional powers of parabolic space-time differential operators. This talk is based on joint works with David Bolin, Sonja Cox, Lukas Herrmann, Mihály Kovács, Christoph Schwab and Joshua Willems.

Consider a rooted tree whose vertices will be interpreted as free parking spots, each spot accommodating at most one car. On top of that tree, we consider a non-negative integer labeling representing the number of cars arriving on each vertex. Each car tries to park on its arrival vertex, and if the spot is occupied, it travels downwards in direction of the root of the tree until it finds an empty vertex to park. If there is no such vertex on the path towards the root, the car exits the tree, contributing to the flux of cars at the root. This models undergoes an interesting phase transition which we will analyze in detail. After an overview of the case where the underlying tree is a critical Bienaymé—Galton—Watson tree, we will concentrate on the case where the underlying tree is the infinite binary tree, where the phase transition which turns out to be “discontinuous”. The talk is based on a joint work with David Aldous, Nicolas Curien and Olivier Hénard.

We construct a machine which takes as input a locally small symmetric closed complete multicategory V. And its output is again a locally small symmetric closed complete multicategory V-Cat, the multicategory of small V-categories and multi-entry V-functors. An example of such V is provided by short spaces (vector spaces with a system of seminorms) and short maps. When the ground multicategory V is set we obtain strict 2-categories by iterating the construction of categories.

Vlasov-Poisson type systems are well known as kinetic models for plasma. The precise structure of the model differs according to which species of particle it describes, with the `classical' version of the system describing the electrons in a plasma. The model for ions, however, includes an additional exponential nonlinearity in the equation for the electrostatic potential, which creates several mathematical difficulties. For this reason, the theory of the ionic system has so far not been as fully explored as the theory for the electron equation. A plasma has a characteristic scale, the Debye length, which describes the scale of electrostatic interaction within the plasma. In real plasmas this length is typically very small, and in physics applications frequently assumed to be very close to zero. This motivates the study of the limiting behaviour of Vlasov-Poisson type systems as the Debye length tends to zero relative to the observation scale--known as the 'quasi-neutral' limit. In the case of the ionic model, the formal limit is the kinetic isothermal Euler system; however, this limit is highly non-trivial to justify rigorously and known to be false in general without very strong regularity conditions and/or structural conditions. I will present a recent work, carried out in collaboration with Mikaela Iacobelli, in which we prove the quasi-neutral limit for the ionic Vlasov-Poisson system for a class of rough (L^\infty) data: that is, data that may be expressed as perturbations of an analytic function, small in the sense of Monge-Kantorovich distances. The smallness of the perturbation that we require is much less restrictive than in the previously known results.

The Ruelle zeta function is a natural function associated with the periods of closed orbits of an Anosov flow, and it is known to have a meromorphic extension to the whole complex plane. The order of vanishing of the Ruelle zeta function at zero is expected to carry interesting topological and dynamical information and can be computed in terms of certain resonant spaces of differential forms for the action of the Lie derivative on suitable spaces with anisotropic regularity. In this talk I will explain how to compute these resonant spaces for any transitive Anosov flow in 3D, with particular emphasis in the dissipative case, that is, when the flow does not preserve any absolutely continuous measure. A prototype example is given by the geodesic flow of an affine connection with torsion and we shall see that for such a flow the order of vanishing drops by 1 in relation to the usual geodesic flow due to the Sinai-Ruelle-Bowen measure having non-zero winding cycle. This is joint work with Mihajlo Cekic.

Coxeter groups are finitely generated reflection groups, classified roughly by the type of space they act on. This talk illustrates the geometric nature of Coxeter groups by means of the reflection length. Besides introductory definitions and results, we look at some hyperbolic pictures.

In bandit with distribution shifts, one aims to automatically adapt to unknown changes in reward distribution, and restart exploration when necessary. While this problem has received attention for many years, no adaptive procedure was known till a recent breakthrough of Auer et al (2018, 2019) which guarantees an optimal regret (LT)^{1/2}, for T rounds and L stationary phases. However, while this rate is tight in the worst case, we show that significantly faster rates are possible, adaptively, if few changes in distribution are actually severe, e.g., involve no change in best arm. This is arrived at via a new notion of 'significant change', which recovers previous notions of change, and applies in both stochastic and adversarial settings (generally studied separately). If time permits, I’ll discuss the more general case of contextual bandits, i.e., where rewards depend on contexts, and highlight key challenges that arise. This is based on ongoing work with Joe Suk.

Der Vortrag startet mit einem klassischen Modell zur Beschreibung der Ausbreitung von Infektionskrankheiten, dem sogenannten SIR-Modell (Susceptible, Infected, Recovered). Ausgehend vom Basis-Modell werden einige Erweiterungen des Modells am Beispiel "HIV in der Schweiz" beschrieben, wie auch der Miteinbezug von genetischen Informationen des Virus. Der Vortrag soll einen Einblick geben, welche Fragestellungen mit relativ einfachen mathematischen Mitteln untersucht werden können, aber auch verdeutlichen, wo die Schwierigkeiten und Tücken bezüglich der Interpretation liegen.

We learn from data that volatility is mostly path-dependent: at least 85-90% of the variance of the implied volatility of equity indexes is explained endogenously by past index returns, and around 60% for (noisy estimates of) future daily realized volatility. The path-dependency that we uncover is remarkably simple: a linear combination of a weighted sum of past daily returns and the square root of a weighted sum of past daily squared returns with different time-shifted power-law weights capturing both short and long memory. This simple model, which is homogeneous in volatility, is shown to consistently outperform existing models across equity indexes for both implied and realized volatility. It suggests a simple continuous-time path-dependent volatility (PDV) model that may be fed historical or risk-neutral parameters. The weights can be approximated by superpositions of exponential kernels to produce Markovian models. In particular, we propose a 4-factor Markovian PDV model which captures all the important stylized facts of volatility, produces very realistic price and volatility paths, and jointly fits SPX and VIX smiles remarkably well. We thus show, for the first time, that a continuous-time Markovian parametric stochastic volatility (actually, PDV) model can practically solve the joint SPX/VIX smile calibration problem.

The theory of quantum chaos predicts strong bounds for the sup-norm of eigenfunctions of the Laplacian on certain negatively curved spaces, for example quotients of the upper half plane. On arithmetic quotients, the original breakthrough of Iwaniec and Sarnak (1995), and the work of many since then, has lead to much progress in producing bounds for Hecke-Maass forms, uniform in the eigenvalue, the volume of the space, or both (hybrid bounds). In higher rank, despite many improvements in the eigenvalue aspect, there is a notable scarcity of results in the volume or level aspect. In this talk, I aim to describe a soft technique for tackling the counting problem at the heart of the Iwaniec-Sarnak method. It relies on algebraic rigidity arguments and produces the first hybrid bounds on a family of compact arithmetic spaces of unbounded rank associated to division algebras.

An important problem in algebraic geometry is the study of rationality of algebraic varieties. I will describe progress on the understanding of rationality in families of smooth projective varieties. This includes the specialization of rationality in families, the existence of families of smooth projective varieties with varying rationality, and associated invariants. Extensions will also be described, e.g., to the orbifold setting.