## Seminars and Colloquia Schedule

### The diffeomorphism group of a 4-manifold

Series
Geometry Topology Seminar
Time
Monday, January 24, 2022 - 14:00 for
Location
Online (Zoom)
Speaker
Danny RubermanBrandeis University

Associated to a smooth n-dimensional manifold are two infinite-dimensional groups: the group of homeomorphisms Homeo(M), and the group of diffeomorphisms, Diff(M). For manifolds of dimension greater than 4, the topology of these groups has been intensively studied since the 1950s. For instance, Milnor’s discovery of exotic 7-spheres immediately shows that there are distinct path components of the diffeomorphism group of the 6-sphere that are connected in its homeomorphism group.  The lowest dimension for such classical phenomena is 5.

I will discuss recent joint work with Dave Auckly about these groups in dimension 4. For each n, we construct a simply connected 4-manifold Z and an infinite subgroup of the nth homotopy group of Diff(Z) that lies in the kernel of the natural map to the corresponding homotopy group of Homeo(Z). These elements are detected by (n+1)—parameter gauge theory. The construction uses a topological technique.  I’ll mention some other applications to embeddings of surfaces and 3-manifolds in 4-manifolds.

### 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

### Inflation of poorly conditioned zeros of systems of analytic functions

Series
Algebra Seminar
Time
Tuesday, January 25, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anton LeykinGeorgia Tech

Given a system of analytic functions and an approximate zero, we introduce inflation to transform this system into one with a regular quadratic zero. This leads to a method for isolating a cluster of zeros of the given system.

(This is joint work with Michael Burr.)

### An adaptation of Kohler-Jobin rearrangement technique with fixed torsional rigidity to the Gaussian space

Series
Analysis Seminar
Time
Wednesday, January 26, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Orli HerscoviciGeorgia Tech

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In this talk, we show an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we present the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of Polya-Szego: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a modified''  torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure.

We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle.

The seminar will be held on Zoom via the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

### 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

### Stability and Instability of the Kelvin-Stuart Cat's Eyes Flow to the 2D Euler's Equation

Series
Dissertation Defense
Time
Friday, January 28, 2022 - 09:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Shasha LiaoGeorgia Tech

The linear stability of a family of Kelvin-Stuart Cat's eyes flows of 2D Euler equation was studied both analytically and numerically. We proved linear stability under co-periodic perturbations and linear instability under multi-periodic perturbations. These results were first obtained numerically using spectral methods and then proved analytically.

### Finite Automata and Transfer Matrices

Series
Algebra Student Seminar
Time
Friday, January 28, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Teams
Speaker
Trevor GunnGeorgia Tech

This talk is a primer on solving certain kinds of counting problems through regular languages, finite automata and transfer matrices. Example problems: count the number of binary strings that contain "0110", count the number of binary strings that contain 0, 1, 2,... copies of "0110," a derivation of the negative binomial distribution function.

The only requirements for this talk is a basic familiarity with directed graphs, matrices and generating functions.

### Realizable Learning is All You Need

Series
ACO Student Seminar
Time
Friday, January 28, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Max HopkinsUCSD

The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. With variants ranging from classical settings like PAC learning and regression to recent trends such as adversarially robust and private learning, it’s surprising we still lack a unifying theory explaining these results.

In this talk, we'll introduce exactly such a framework: a simple, model-independent blackbox reduction between agnostic and realizable learnability that explains their equivalence across a wide host of classical models. We’ll discuss how this reduction extends our understanding to traditionally difficult settings such as learning with arbitrary distributional assumptions and general loss, and look at some applications beyond agnostic learning as well (e.g. to privacy). Finally, we'll end by surveying a few nice open problems in the area.

Based on joint work with Daniel Kane, Shachar Lovett, and Gaurav Mahajan.