Georgia Institute of Technology College of Sciences

The general linear groups GL_n(A) can be defined for any ring A, and Quillen's definition of K-theory of A takes these groups as its starting point. If A is commutative, one may define symplectic K-theory in a very similar fashion, but starting with the symplectic groups Sp_{2n}(A), the subgroup of GL_{2n}(A) preserving a non-degenerate skew-symmetric bilinear form. The result is a sequence of groups denoted KSp_i(A) for i = 0, 1, .... For the ring of integers, there is an interesting action of the absolute Galois group of Q on the groups KSp_i(Z), arising from the moduli space of polarized abelian varieties. In joint work with T. Feng and A. Venkatesh we study this action, which turns out to be an interesting extension between a trivial representation and a cyclotomic representation.

In the real world, the historical performance of a stock may have impacts on its dynamics and this suggests us to consider models with delays. We consider a portfolio optimization problem of Merton’s type in which the risky asset is described by a stochastic delay model. We derive the Hamilton-Jacobi-Bellman (HJB) equation, which turns out to be a nonlinear degenerate partial differential equation of the elliptic type. Despite the challenge caused by the nonlinearity and the degeneration, we establish the existence result and the verification results.

We study the convergence rate of the least squares estimator (LSE) in a regression model with possibly heavy-tailed errors. Despite its importance in practical applications, theoretical understanding of this problem has been limited. We first show that from a worst-case perspective, the convergence rate of the LSE in a general non-parametric regression model is given by the maximum of the Gaussian regression rate and the noise rate induced by the errors. In the more difficult statistical model where the errors only have a second moment, we further show that the sizes of the 'localized envelopes' of the model give a sharp interpolation for the convergence rate of the LSE between the worst-case rate and the (optimal) parametric rate. These results indicate both certain positive and negative aspects of the LSE as an estimation procedure in a heavy-tailed regression setting. The key technical innovation is a new multiplier inequality that sharply controls the size of the multiplier empirical process associated with the LSE, which also finds applications in shape-restricted and sparse linear regression problems.

A special class of dynamical systems that we will focus on are substitutions. This class of systems provides a variety of ergodic theoretic behavior and is connected to self-similar interval exchange transformations. During this talk we will explore rigidity sequences for these systems. A sequence $\left( n_m \right)$ is a rigidity sequence for the dynamical system $(X,T,\mu)$ if $\mu(T^{n_m}A\cap A)\rightarrow \mu(A)$ for all positive measure sets $A$. We will discuss the structure of rigidity sequences for substitutions that are rank-one and substitutions that have constant length. This is joint work with Jon Fickenscher.

Four dimensions is unique in many ways. For example $n$-dimensional Euclidean space has a unique smooth structure if and only if $n$ is not equal to four. In other words, there is only one way to understand smooth functions on $R^n$ if and only if $n$ is not 4. There are many other way that smooth structures on 4-dimensional manifolds behave in surprising ways. In this talk I will discuss this and I will sketch the beautiful interplay of ideas (you got algebra, analysis and topology, a little something for everyone!) that go into proving $R^4$ has more that one smooth structure (actually it has uncountably many different smooth structures but that that would take longer to explain).

The Institute for Defense Analyses - Center for Computing Sciences is a nonprofit research center that works closely with the NSA. Our center has around 60 researchers (roughly 30 mathematicians and 30 computer scientists) that work on interesting and hard problems. The plan for the seminar is to begin with a short mathematics talk on a project that was completed at IDA-CCS and declassified, then tell you a little about what we do, and end with your questions. The math that we will discuss involves symbolic dynamics and automata theory. Specifically we will develop a metric on the space of regular languages using topological entropy. This work was completed during a summer SCAMP at IDA-CCS. SCAMP is a summer program where researchers from academia (professors and students), the national labs, and the intelligence community come to IDA-CCS to work on the agency's hard problems for 11 weeks.

Please check the meeting webpage at http://10to60.math.gatech.edu for program, titles and abstracts.

For a fixed graph $G$, let $\mathcal{L}_G$ denote the family of Lipschitz functions $f:V(G) \rightarrow \mathbb{R}$ such that $0 = \sum_u f(u)$. The \emph{spread} of $G$ is denoted $c(G) := \frac{1}{|V(G)|} \max_{f \in \mathcal{L}_G} \sum_u f(u)^2$ and the subgaussian constant is $e^{\sigma_G^2} := \sup_{t > 0} \max_{f \in \mathcal{L}_G} \left( \frac{1}{|V(G)|} \sum_u e^{t f(u)} \right)^{2/t^2}$. Motivation of these parameters comes from their relationship with the isoperimetric number of a graph (given a number $t$, find a set $W \subset V(G)$ such that $2|W| \geq |V(G)|$ that minimizes $i(G,t) := |\{u : d(u, W) \leq t \}|$). While the connection to the isoperimetric number is interesting, the spread and subgaussian constant have not been any easier to understand. In this talk, we will present results that describe the functions $f$ achieving the optimal values. As a corollary to these results, we will resolve two conjectures (one false, one true) about these parameters. The conjectures that we resolve are the following. We denote the Cartesian product of $G$ with itself $d$ times as $G^d$. Alon, Boppana, and Spencer proved that the set $\{u: f(u) < k\}$ for extremal function $f$ for the spread of $G^d$ gives a value that is asymptotically close to the isoperimetric number when $d, t$ grow at specific rates and $k=0$; and they conjectured that the value is exactly correct for large $d$ and $k,t$ in ``appropriate ranges.'' The conjecture was proven true for hypercubes by Harper and the discrete torus of even order by Bollob\'{a}s and Leader. Bobkov, Houdr\'{e}, and Tetali constructed a function over a cycle that they conjectured to be optimal for the subgaussian constant, and it was proven correct for cycles of even length by Sammer and Tetali. This work appears in the manuscript https://arxiv.org/abs/1705.09725 .

Associated to a planar cubic graph, there is a closed surface in R^5, as defined by Treumann and Zaslow. R^5 has a canonical geometry, called a contact structure, which is compatible with the surface. The data of how this surface interacts with the geometry recovers interesting data about the graph, notably its chromatic polynomial. This also connects with pseudo-holomorphic curve counts which have boundary on the surface, and by looking at the resulting differential graded algebra coming from symplectic field theory, we obtain new definitions of n-colorings which are strongly non-linear as compared to other known definitions. There are also relationships with SL_2 gauge theory, mathematical physics, symplectic flexibility, and holomorphic contact geometry. During the talk we'll explain the basic ideas behind the various fields above, and why these various concepts connect.