Weekly Bulletin

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.

In Neolithic archaeological sites, there are often large quality of pottery fragments. A foundational work is to establish pottery chronology which provides a relative time-line for different sites in a region. Combining with Carbon-14 dating, one can obtain an absolute time-line. The above traditional method is extremely time consuming and uses less than 10% of the materials. One can easily imagine that the remaining 90% material may contain important information which even changes the conclusion of archaeological study. In the talk, we will present an AI-based technique to date individual pieces of pottery fragments, developed by a team of archaeologist in Sichuan, China in collaborations with mathematician/statistician. The highlight of the project is to build a large dataset consisting of over 200,000 images from more than 20 sites.

More information: https://eth-its.ethz.ch/activities/its-fellows--seminar/yongbin-ruan.htmlcall_made

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.

The sample complexity of learning in a convex class and with respect to the squared loss is arguably the most important question in Statistical Learning Theory. The state-of-the-art estimates in this setting rely on Rademacher complexities, and those are generally difficult to control. I will explain why (contrary to prevailing belief) and under minimal assumptions, the Rademacher complexities are not really needed: the sample complexity is actually governed by the behaviour of the limiting gaussian process. In particular, all such learning problems that have the same L_2 structure - even those with heavy-tailed distributions - share the same sample complexity as if the problem were light-tailed. At the heart of the proof is the construction of uniform mean estimation procedures for some natural function classes. I will show how such uniform mean estimation procedures can be derived by combining optimal mean estimation techniques for real-valued random variables with Talagrand's generic chaining method.

This is joint work with Jon DeWitt. We study skew products over area-preserving Anosov diffeomorphisms on T^2×G, where G is a compact Lie group, given by (x,g)?(f(x),h(x)·g). We establish smooth rigidity; that is, if two such skew products are C^0 conjugate, then they are smoothly conjugate, unless h:T^2->G is cohomologous to a constant and the skew product is, in fact, a product with a translation on G. Interestingly, on twisted principal G-bundles, our approach gives exception-free rigidity.

How does counting curves in a complete intersection relate to the geometry of the ambient space? In this talk, I will introduce a tropical decomposition formula as an explicit answer to this Quantum Lefschetz type problem in the Gromov-Witten setting. More precisely, the tropical decomposition formula shows that Gromov-Witten of complete intersections can be approximated using ambient data, with correction terms given by virtual counts of (generalized) special linear series. The tropical decomposition formula is derived from studying the boundary structures of logarithmic Gauged Linear Sigma Models. This is a joint work in progress with Felix Janda and Yongbin Ruan. 

Quasipositive knots occur in complex geometry as transverse intersections of smooth algebraic curves in the complex plane ℂ² with the 3-sphere. A complex cobordism is a surface that arises as a transverse intersection of a smooth algebraic curve with the region bounded between two 4-balls of different radius with common center in ℂ². The two knots bounded by a complex cobordism are necessarily quasipositive, and such a cobordism is necessarily optimal (defined in the talk). Feller asked whether these two necessary conditions for the existence of a complex cobordism between two knots are sufficient. In a joint work with Maciej Borodzik we answer this in the negative for cobordisms of any genus <i>g</i> ≥ 0. In the case of genus <i>g</i> = 0, we improve our result to strongly quasipositive knots. In the talk, we will define the relevant terms and provide some context for our results.

Black hole stability is a central topic in mathematical relativity that has seen numerous advancements in recent years. Both the Kerr-de Sitter and the Kerr black hole spacetimes have been proven to be stable in the slowly-rotating regime. However, the methods used have been markedly different, as well as the decay rates proven. Perturbations of Kerr-de Sitter converge exponentially back to a nearby Kerr-de Sitter black hole, while perturbations of Kerr only converge polynomially back to the family. In this talk, I will speak about wave behavior that is uniform in the cosmological constant by considering solutions to the Teukolsky equations in Kerr(-de Sitter). The main point is a careful handling of the relevant estimates on the region of the spacetime far from the black hole. This provides a first step into understanding the uniform (in the cosmological constant) stability of black hole spacetimes. This is joint work with Jeremie Szeftel and Arthur Touati.

It is well known that finite element approximations of the Helmholtz equation suffer from the pollution effect for large wavenumbers k>0. This degeneracy can be avoided by the application of high order FEMs, with polynomial degree p chosen proportional to log k. The key ingredient of the respective analysis [1] is a so-called regularity splitting, which decomposes the solution of the Helmholtz equation with a L^2 right hand-side into an analytical part and k-well behaved H^2 part. The generalization of this technique for nonconstant coefficients and other boundary conditions is technical and nontrivial, but has received much attention lately. In this talk I show how the classical Schatz technique can be adapted to circumvent the necessity of any regularity splitting, which significantly simplifies the analysis. In the second part of the talk I discuss the application of this approach to heterogeneous media and Maxwell-impedance problems. [1] M. Melenk and S. Sauter, Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79(272):1871–1914, 2010.

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to interacting particle systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. In this talk, I will provide a self-contained introduction to this fascinating question, and then describe a recent partial answer based on entropy and curvature. Joint work with Francesco Pedrotti.

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 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.