## Seminars and Colloquia by Series

### Algebraic/Arithmetic properties of curves and Galois cohomology

Series
Job Candidate Talk
Time
Wednesday, February 2, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Wanlin LiCRM Montreal

A lot of the algebraic and arithmetic information of a curve is contained in its interaction with the Galois group. This draws inspiration from topology, where given a family of curves over a base B, the fundamental group of B acts on the cohomology of the fiber. As an arithmetic analogue, given an algebraic curve C defined over a non-algebraically closed field K, the absolute Galois group of K acts on the etale cohomology of the geometric fiber and this action gives rise to various Galois cohomology classes. In this talk, we discuss how to use these classes to detect algebraic/arithmetic properties of the curve, such as the rational points (following Grothendieck's section conjecture), whether the curve is hyperelliptic, and the set of supersingular'' primes.

https://bluejeans.com/270212862/6963

### On Gapped Ground State Phases of Quantum Lattice Models

Series
Job Candidate Talk
Time
Monday, January 31, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Amanda YoungTechnical University Munich

Quantum spin systems are many-body physical models where particles are bound to the sites of a lattice. These are widely used throughout condensed matter physics and quantum information theory, and are of particular interest in the classification of quantum phases of matter. By pinning down the properties of new exotic phases of matter, researchers have opened the door to developing new quantum technologies. One of the fundamental quantitites for this classification is whether or not the Hamiltonian has a spectral gap above its ground state energy in the thermodynamic limit. Mathematically, the Hamiltonian is a self-adjoint operator and the set of possible energies is given by its spectrum, which is bounded from below. While the importance of the spectral gap is well known, very few methods exist for establishing if a model is gapped, and the majority of known results are for one-dimensional systems. Moreover, the existence of a non-vanishing gap is generically undecidable which makes it necessary to develop new techniques for estimating spectral gaps. In this talk, I will discuss my work proving non-vanishing spectral gaps for key quantum spin models, and developing new techniques for producing lower bound estimates on the gap. Two important models with longstanding spectral gap questions that I recently contributed progress to are the AKLT model on the hexagonal lattice, and Haldane's pseudo-potentials for the fractional quantum Hall effect. Once a gap has been proved, a natural next question is whether it is typical of a gapped phase. This can be positively answered by showing that the gap is robust in the presence of perturbations. Ensuring the gap remains open in the presence of perturbations is also of interest, e.g., for the development of robust quantum memory. A second topic I will discuss is my research studying spectral gap stability.

URL for the talk: https://bluejeans.com/602513114/7767

### Is there a smallest algebraic integer?

Series
Job Candidate Talk
Time
Thursday, January 27, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Vesselin DimitrovUniversity of Toronto

The Schinzel-Zassenhaus conjecture describes the narrowest collar width around the unit circle that contains a full set of conjugate algebraic integers of a given degree, at least one of which lies off the unit circle. I will explain what this conjecture precisely says and how it is proved. The method involved in this solution turns out to yield some other new results whose ideas I will describe, including to the closest interlacing of Frobenius eigenvalues for abelian varieties over finite fields, the closest separation of Salem numbers in a fixed interval, and the distribution of the short Kobayashi geodesics in the Siegel modular variety.

https://bluejeans.com/476147254/8544

### Dimension-free analysis of k-means clustering, stochastic convex optimization and sample covariance matrices in log-concave ensembles

Series
Job Candidate Talk
Time
Tuesday, January 25, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/958288541/0675
Speaker
Nikita ZhivotovskiyETH Zurich

The first part of the talk is devoted to robust algorithms for the k-means clustering problem where a quantizer is constructed based on N independent observations. I will present recent sharp non-asymptotic performance guarantees for k-means that hold under the two bounded moments assumption in a general Hilbert space. These bounds extend the asymptotic result of D. Pollard (Annals of Stats, 1981) who showed that the existence of two moments is sufficient for strong consistency of an empirically optimal quantizer. In the second part of the talk I discuss a dimension-free version of the result of Adamczak, Litvak, Pajor, Tomczak-Jaegermann (Journal of Amer. Math. Soc, 2010) for the sample-covariance matrix in log-concave ensembles. The proof of the dimension-free result is based on a duality formula between entropy and moment generating functions. Finally, I will briefly discuss a recent bound on an empirical risk minimization strategy in stochastic convex optimization with strongly convex and Lipschitz losses.

Link to the online talk: https://bluejeans.com/958288541/0675

### Coloring hypergraphs of small codegree, and a proof of the Erdős–Faber–Lovász conjecture

Series
Job Candidate Talk
Time
Thursday, January 20, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Thomas KellyUniversity of Birmingham

A long-standing problem in the field of graph coloring is the Erdős–Faber–Lovász conjecture (posed in 1972), which states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$, or equivalently, that a nearly disjoint union of $n$ complete graphs on at most $n$ vertices has chromatic number at most $n$.  In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large $n$.  Recently, we also solved a related problem of Erdős from 1977 on the chromatic index of hypergraphs of small codegree.  In this talk, I will survey the history behind these results and discuss some aspects of the proofs.

### Long-time dynamics of dispersive equations

Series
Job Candidate Talk
Time
Tuesday, January 18, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Gong ChenUniversity of Toronto

Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”.  Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations.  After an informal introduction to dispersive equations, I will survey some of my recent results towards understanding the long-time behavior of dispersive waves and the soliton resolution using techniques from both partial differential equations and inverse scattering transforms.

### Turbulent Weak Solutions of the 3D Euler Equations

Series
Job Candidate Talk
Time
Thursday, January 13, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Matthew NovackIAS

The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory of turbulence.

### Thresholds

Series
Job Candidate Talk
Time
Wednesday, December 15, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/487699041/8823
Speaker
Jinyoung ParkStanford University

Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will first introduce the Kahn-Kalai Conjecture with some motivating examples and then talk about the recent resolution of a fractional version of the Kahn-Kalai Conjecture due to Frankston, Kahn, Narayanan, and myself. Some follow-up work, along with open questions, will also be discussed.

### Regularity lemma: discrete and continuous perspectives

Series
Job Candidate Talk
Time
Monday, December 13, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/774516207/3993
Speaker
Fan WeiPrinceton University

Szemerédi's regularity lemma is a game-changer in extremal combinatorics and provides a global perspective to study large combinatorial objects. It has connections to number theory, discrete geometry, and theoretical computer science. One of its classical applications, the removal lemma, is the essence for many property testing problems, an active field in theoretical computer science. Unfortunately, the bound on the sample size from the regularity method typically is either not explicit or enormous. For testing natural permutation properties, we show one can avoid the regularity proof and yield a tester with polynomial sample size. For graphs, we prove a stronger, "L_\infty'' version of the graph removal lemma, where we conjecture that the essence of this new removal lemma for cliques is indeed the regularity-type proof. The analytic interpretation of the regularity lemma also plays an important role in graph limits, a recently developed powerful theory in studying graphs from a continuous perspective. Based on graph limits, we developed a method combining with both analytic and spectral methods, to answer and make advances towards some famous conjectures on a common theme in extremal combinatorics: when does randomness give nearly optimal bounds?

These works are based on joint works with Jacob Fox, Dan Kral',  Jonathan Noel, Sergey Norin, and Jan Volec.

### Canonical measures and equidistribution in the arithmetic of forward orbits

Series
Job Candidate Talk
Time
Thursday, December 9, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
online
Speaker
Nicole LooperBrown University

This talk is about the arithmetic of points of small canonical height relative to dynamical systems over number fields, particularly those aspects amenable to the use of equidistribution techniques. Past milestones in the subject include the proof of the Bogomolov Conjecture given by Ullmo and Zhang, and Baker-DeMarco's work on the finiteness of common preperiodic points of unicritical maps. Recently, quantitative equidistribution techniques have emerged both as a way of improving upon some of these old results, and as an avenue to studying previously inaccessible problems, such as the Uniform Boundedness Conjecture of Morton and Silverman. I will describe the key ideas behind these developments, and raise related questions for future research.

https://bluejeans.com/788895268/8348