Events

Abstract: Gauged Linear Sigma Models (GLSM) introduced by Witten in the 90's can be viewed as a way of counting curves in critical loci of holomorphic functions called superpotentials.  On the mathematical side, the framework of GLSM has been established  by Fan-Jarvis-Ruan and Chang-Li-Li. Compared to other Gromov-Witten type invariants, non-properness in GLSM is a special feature due to non-constant superpotentials. In this talk, I will report on a joint work with Felix Janda and Yongbin Ruan on logarithmic Gauged Linear Sigma Models (log GLSM), which compactify GLSM using stable logarithmic maps of Abramovich-Chen-Gross-Siebert. A key in our construction is a universal logarithmic modification, called the uniform maximal degeneracy, over which superpotentials become well-behaved along logarithmic boundaries. This leads to a reduced perfect obstruction theory of log GLSM, recovering the Kiem-Li cosection localized virtual cycles of GLSM.

Almost Toric Fibrations (ATFs) are a certain type of integrable system and provide a useful tool to study symplectic four-manifolds, allowing for various explicit constructions of Lagrangian torus knots and interesting symplectomorphisms. The latter gives a counterexample to the Lagrangian Poincaré recurrence conjecture. This talk will give a "users manual" to ATFs, outlining the tools used to manipulate them.

A new type of billiard system, of interest for Celestial Mechanics, is taken into consideration: here, a closed refraction interface separates two regions in which different potentials (harmonic and Keplerian) act. This model, which can be studied both in two and three dimensions, presents strong analogies with the more studied Kepler billiard, where a Keplerian inner potential is associated with a reflecting wall. The seminar aims to provide results on the two models, ranging from the existence of periodic and quasi-periodic orbits to the characterization of integrable and non-integrable boundaries. Joint work with Vivina Barutello and Susanna Terracini.

The main question in quantum chaos is to relate the chaotic properties of a dynamical system (like a billiard in a bounded domain, or the geodesic flow on a closed manifold) to the spectral properties of the corresponding Schrödinger operator in quantum mechanics (the laplacian, in the examples above). This is usually asked for a given dynamical system, but one may try to make the problem more tractable by studying a "random" billiard, or the geodesic flow on a "random manifold". Several years ago, I became interested in the ergodic and spectral properties of random hyperbolic surfaces, in the asymptotic regime where the area of the surface goes to infinity. I will survey the existing techniques and some of the results.

There is a growing body of work on proving hardness results for average-case optimization problems by bounding the low-degree likelihood ratio (LDLR) between a null distribution and a related planted distribution. Such hardness results are now ubiquitous not only for foundational average-case problems but also central questions in statistics and cryptography. This line of work is supported by the low-degree conjecture of Hopkins [Hop18], which in its extended form postulates that a vanishing degree-D LDLR implies the absence of any noise-tolerant distinguishing algorithm with runtime n^{D / polylog(D)} whenever (1) the null distribution is product and (2) the planted distribution is permutation invariant. We disprove this conjecture. Specifically, we show that for all sufficiently small ε > 0, there is a planted distribution on {0, 1}^{n(n-1)/2} that has a vanishing degree-n^{1-O(ε)} LDLR with respect to the uniform distribution on {0, 1}^{n(n-1)/2} even as an n^{O(log n)}-time algorithm solves the corresponding ε-noisy distinguishing problem. Our construction relies on algorithms for list-decoding for noisy polynomial interpolation in the high-error regime. Joint work with Jun-Ting Hsieh, Aayush Jain, and Pravesh Kothari.

Tensors are fundamental objects in mathematics, physics, statistics, and computer science, and they play an important role in a wide range of applied sciences and engineering disciplines. In this talk, we will focus on concentration inequalities for simple random tensors. We establish sharp dimension-free concentration inequalities and expectation bounds for the deviation of the sum of simple random tensors from its expectation. As part of our analysis, we use generic chaining techniques to obtain a sharp high-probability upper bound on the suprema of multi-product empirical processes. In so doing, we generalize classical results for quadratic and product empirical processes to higher-order settings.

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb in 2022. This hyperbolic metric graph is used to study the homeomorphism group of the surface through its action on the graph. Since the introduction of this object, many researchers have worked to establish connections between the dynamics of surface homeomorphisms, and the dynamics of their action by isometries on the fine curve graph. These endeavours have led to the discovery of a connection between this subject and the rich area of rotation theory on surfaces. Rotation theory was first developed in the late 1980s by Misiurewicz and Ziemian, and remains an active field of research to this day. This talk will include an introduction to these theories, as well as statements of interesting results and unanswered questions.

The distribution of future losses related, for example, to climate risk is typically not perfectly known and is subject to ambiguity. In this context, we investigate how to design an optimal sharing scheme among agents (insurers, reinsurers, countries). We first derive the optimal risk-sharing arrangement under mean-variance preferences, considering distributional ambiguity of the aggregate risk to be shared. It is shown that proportional risk-sharing is always optimal and that the presence of ambiguity does not affect the structure of the risk-sharing arrangement, making it robust to such uncertainty. Desirable properties of risk sharing under ambiguity are discussed and several generalizations are also explored. Risk sharing under distortion risk measures will also be discussed. This is joint work with Steven Vanduffel.

The Bergelson conjecture from 1996 asserts that the multilinear polynomial ergodic averages with commuting transformations converge pointwise almost everywhere in any measure-preserving system. This problem was recently solved affirmatively for polynomials with distinct degrees. In this talk, I will review the recent progress on this conjecture, focusing on the multilinear circle method --- a versatile new tool that combines methods from additive combinatorics and Fourier analysis, which are crucial in problems of this kind. This is based on joint work with D. Kosz, S. Peluse and J. Wright.