Georgia Institute of Technology College of Sciences

A globally nonnegative polynomial F is called stubborn if no odd power of F is a sum of squares. We develop a new invariant of a singularity of a form (homogeneous polynomial) in 3 variables, which allows us to conclude that if the sum of these invariants over all zeroes of a nonnegative form is large enough, then the form is stubborn. As a consequence, we prove that if an extreme ray of the cone of nonnegative ternary sextics is not a sum of squares, then all of its odd powers are also not sums of squares, and we provide more examples of this phenomenon for ternary forms in higher degree. This is joint work with Khazhgali Kozhasov and Bruce Reznick.

In 1976, Thurston decidedly showed that symplectic geometry and Kähler geometry were strictly distinct by providing the first example of a compact symplectic manifold which is not symplectomorphic to any Kähler manifold. Since this example, first studied by Kodaira, much work has been done in explicating the difference between algebraic manifolds such as affine and projective varieties, complex manifolds such as Stein and Kähler manifolds, and general symplectic manifolds. By building on work first outlined by Seidel, McLean has produced numerous examples of non-affine symplectic manifolds, symplectic manifolds which are not symplectomorphic to any affine variety. McLean approached this problem via analysis of the growth rate of symplectic homology for affine varieties. Every affine variety admits a compactification to a projective variety by a normal crossing divisor. Using this fact, McLean is able to show that the symplectic homology of any affine variety must have a well-controlled growth rate.

We add a bit of subtlety to this already mysterious relationship by providing a particularly interesting example of a non-affine symplectic 4-manifold which admits many normal crossing divisor compactifications. Because of the existence of these nice compactifications, one cannot use growth rate techniques to obstruct our example from being affine and thus cannot apply the work of McLean and Seidel. Our approach to proving this results goes by considering the collection of all symplectic normal crossing divisor compactifications of a particular Liouville manifold  given as a submanifold of the Kodaira-Thurston example . By studying the local geometry of a large collection of symplectic normal crossing divisors, we are able to make several topological conclusions about this collection for  as well as for more general Liouville manifolds  which admit similar compactifications. Our results suggest that a more subtle obstruction must exist for non-affine manifolds. If time permits, we will discuss several structural conclusions one may reach about the collection of divisor compactifications for a more general class of Liouville 4-manifolds.

In enumerative combinatorics, one is often asked to count the number of combinatorial objects.  But the inverse problem is even more interesting: given some numbers, do they have a combinatorial interpretation?  In the main part of the talk I will give a broad survey of this problem, formalize the question in the language of computational complexity, and describe some connections to deep results and open problems in algebraic and probabilistic combinatorics.  In the last part of the talk, I will discuss our recent results on the defect and equality cases of Stanley inequalities for the numbers of bases of matroids and for the numbers of linear extensions of posets (joint work with Swee Hong Chan).  The talk is aimed at the general audience. 

Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network. Consequently, counting motifs in a large network is an important statistical and computational problem. In this talk we will consider the problem of estimating motif densities and fluctuations of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can be Gaussian or non-Gaussian, depending on a notion of regularity of subgraphs with respect to the graphon. Using these results and a novel multiplier bootstrap for graphons, we will construct joint confidence sets for the motif densities. Finally, we will discuss various structure theorems and open questions about degeneracies of the limiting distribution and connections to quasirandom graphs.

Joint work with Anirban Chatterjee, Soham Dan, and Svante Janson

How to achieve the optimal control for general stochastic nonlinear is notoriously difficult, which becomes even more difficult by involving learning and exploration for unknown dynamics in reinforcement learning setting. In this talk, I will present our recent work on exploiting the power of representation in RL to bypass these difficulties. Specifically, we designed practical algorithms for extracting useful representations, with the goal of improving statistical and computational efficiency in exploration vs. exploitation tradeoff and empirical performance in RL. We provide rigorous theoretical analysis of our algorithm, and demonstrate the practical superior performance over the existing state-of-the-art empirical algorithms on several benchmarks. 

Three fifteen-minute talks by local speakers.