Georgia Institute of Technology College of Sciences

For a fixed integer n, consider the nerve L_n of the topological poset of orthogonal decompositions of complex n-space into proper orthogonal subspaces. The space L_n has an action by the unitary group U(n), and we study the fixed points for subgroups of U(n). Given a prime p, we determine the relatively small class of p-toral subgroups of U(n) which have potentially non-empty fixed points. Note that p-toral groups are a Lie analogue of finite p-groups, thus if we are interested in the U(n)-space L_n at a fixed prime p, only the p-toral subgroups of U(n) play a significant role. The space L_n is strongly related to the K-theory analogues of the symmetric powers of spheres and the Weiss tower for the functor that assigns to a vector space V the classifying space BU(V). Our results are a step toward a K-theory analogue of the Whitehead conjecture as part of the program of Arone-Dwyer-Lesh. This is joint work with J.Bergner, R.Joachimi, K.Lesh, K.Wickelgren.

Abstract: (joint work with Peter Scholze) The proetale topology is a Grothendieck topology that is closely related to the etale topology, yet better suited for certain "infinite" constructions, typically encountered in l-adic cohomology. I will first explain the basic definitions, with ample motivation, and then discuss applications. In particular, we will see why locally constant sheaves in this topology yield a fundamental group that is rich enough to detect all l-adic local systems through its representation theory (which fails for the groups constructed in SGA on the simplest non-normal varieties, such as nodal curves).

There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P^3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron-Severi groups contain specified lattices. These constructions, inspired by arithmetic considerations, also involve some fun geometry and combinatorics. This is joint work with Manjul Bhargava and Abhinav Kumar.

Kronecker coefficients lie at the intersection of representation theory, algebraic combinatorics and, most recently, complexity theory. They count the multiplicities of irreducible representations in the tensor product of two other irreducible representations of the symmetric group. While their study was initiated almost 75 years, remarkably little is known about them. One of the major problems of algebraic combinatorics is to find an explicit positive combinatorial formula for these coefficients. Recently, this problem found a new meaning in the field of Geometric Complexity Theory, initiated by Mulmuley and Sohoni, where certain conjectures on the complexity of computing and deciding positivity of Kronecker coefficients are part of a program to prove the "P vs NP" problem. In this talk we will give an overview of this topic and we will describe several problems with some results on different aspects of the Kronecker coefficients. We will explore Saxl conjecture stating that the tensor square of certain irreducible representation of S_n contains every irreducible representation, and present a criterion for determining when a Kronecker coefficient is nonzero. In a more combinatorial direction, we will show how to prove certain unimodality results using Kronecker coefficients, including the classical Sylvester's theorem on the unimodality of q-binomial coefficients (as polynomials in q). We will also present some results on complexity in light of Mulmuley's conjectures. The presented results are based on joint work with Igor Pak and Ernesto Vallejo.

In this talk I will present a new notion of Ricci curvature that applies to finite Markov chains and weighted graphs. It is defined using tools from optimal transport in terms of convexity properties of the Boltzmann entropy functional on the space of probability measures over the graph. I will also discuss consequences of lower curvature bounds in terms of functional inequalities. E.g. we will see that a positive lower bound implies a modified logarithmic Sobolev inequality. This is joint work with Jan Maas.

We conclude the proof of the linearization theorem for fibered holomorphic maps by showing that the iteration scheme we proposed converges. If time allows, we will comment on related work by Mario Ponce and generalizations of the theorem for fibered holomorphic maps in higher dimensions.

Hyperbolic 3-manifolds is a great class of 3-dimensional geometric objects with interesting topology, a rich source of examples (practially one for every knot that you can draw), with arithmetically interesting volumes expressed in terms of dialogarithms of algebraic numbers, and with computer software that allows to manipulate them. Tired of abstract existential mathematics? Interested in concrete 3-dimensional topology and geometry? Or maybe Quantum Topology? Come and listen!

We'll prove the simplest case of Hirzebruch's signature theorem, which relates the first Pontryagin number of a smooth 4-manifold to the signature of its intersection form. If time permits, we'll discuss the more general case of 4k-manifolds. The result is relevant to Prof. Margalit's ongoing course on characteristic classes of surface bundles.

I will introduce the cluster value problem, and its relation to the Corona problem, in the setting of Banach algebras of analytic functions on unit balls. Then I will present a reduction of the cluster value problem in separable Banach spaces, for the algebras $A_u$ and $H^{\infty}$, to those spaces that are $\ell_1$ sums of a sequence of finite dimensional spaces. This is joint work with William B. Johnson.

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse.

A highlight in the study of quantum physics was the work of Knizhnik, Polyakov and Zamolodchikov (1988), in which they proposed a relation (KPZ relation) between the scaling dimension of a statistical physics model in Euclidean geometry and its counterpart in the random geometry. More recently, Duplantier and Sheffield (2011) used the 2-dim Gaussian free field to construct the Liouville quantum gravity measure on a planar domain, and gave the first mathematically rigorous formulation and proof of the KPZ relation in that setting. Inspired by the work of Duplantier and Sheffield, we apply a similar approach to extend their results and techniques to higher even dimensions R^(2n) for n>=2. This talk mainly focuses on the case of R^4. I will briefly introduce the notion of Gaussian free field (GFF). In our work we adopt a specific 4-dim GFF to construct a random Borel measure on R^4 which formally has the density (with respect to the Lebesgue measure) being the exponential of an instance of the GFF. Further we establish a 4-dim KPZ relation corresponding to this random measure. This work is joint with Dmitry Jakobson (McGill University).

First-order (a.k.a. subgradient) methods in convex optimization are a popular choice when facing extremely large-scale problems, where medium accuracy solutions suffice. The limits of performance of first-order methods can be partially understood under the lens of black box oracle complexity. In this talk I will present some of the limitations of worst-case black box oracle complexity, and I will show two recent extensions of the theory: First, we extend the notion of oracle compexity to the distributional setting, where complexity is measured as the worst average running time of (deterministic) algorithms against a distribution of instances. In this model, the distribution of instances is part of the input to the algorithm, and thus algorithms can potentially exploit this to accelerate their running time. However, we will show that for nonsmooth convex optimization distributional lower bounds coincide to worst-case complexity up to a constant factor, and thus all notions of complexity collapse; we can further extend these lower bounds to prove high running time with high probability (this is joint work with Sebastian Pokutta and Gabor Braun). Second, we extend the worst-case lower bounds for smooth convex optimization to non-Euclidean settings. Our construction mimics the classical proof for the nonsmooth case (based on piecewise-linear functions), but with a local smoothening of the instances. We establish a general lower bound for a wide class of finite dimensional Banach spaces, and then apply the results to \ell^p spaces, for p\in[2,\infty]. A further reduction will allow us to extend the lower bounds to p\in[1,2). As consequences, we prove the near-optimality of the Frank-Wolfe algorithm for the box and the spectral norm ball; and we prove near-optimality of function classes that contain the standard convex relaxation for the sparse recovery problem (this is joint work with Arkadi Nemirovski).

We show that after stabilizations of opposite parity and braid isotopy, any twobraids in the same topological link type cobound embedded annuli. We use this to prove thegeneralized Jones conjecture relating the braid index and algebraic length of closed braidswithin a link type, following a reformulation of the problem by Kawamuro. This is joint workwith Doug Lafountain.

The main paradigm of smoothed analysis on graphs suggests that for any large graph G in a certain class of graphs, perturbing slightly the edges of G at random (usually adding few random edges to G) typically results in a graph having much nicer properties. In this talk we discuss smoothed analysis on trees, or equivalently on connected graphs. A connected graph G on n vertices can be a very bad expander, can have very large diameter, very high mixing time, and possibly has no long paths. The situation changes dramatically when \eps n random edges are added on top of G, the so obtained graph G* has with high probability the following properties: - its edge expansion is at least c/log n; - its diameter is O(log n); - its vertex expansion is at least c/log n; - it has a linearly long path; - its mixing time is O(log^2n) All of the above estimates are asymptotically tight. Joint work with Michael Krivelevich (Tel Aviv) and Wojciech Samotij (Tel Aviv/Cambridge).