Georgia Institute of Technology College of Sciences

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

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.

The geometry of non-arithmetic hyperbolic manifolds is mysterious in spite of how plentiful they are. McMullen and Reid independently conjectured that such manifolds have only finitely many totally geodesic hyperplanes and their conjecture was recently settled by Bader-Fisher-Miller-Stover in dimensions larger than 3. Their works rely on superrigidity theorems and are not constructive. In this talk, we strengthen their result by proving a quantitative finiteness theorem for non-arithmetic hyperbolic manifolds that arise from a gluing construction of Gromov and Piatetski-Shapiro. Perhaps surprisingly, the proof relies on an effective density theorem for certain periodic orbits. The effective density theorem uses a number of ideas including Margulis functions, a restricted projection theorem, and an effective equidistribution result for measures that are nearly full dimensional. This is joint work with K. W. Ohm.

A classic problem in probability theory and combinatorics is to estimate the probability that the random graph G(n,p) contains no triangles.  This problem can be viewed as a question in "non-linear large deviations".   The asymptotics of the logarithm of this probability (and related lower tail probabilities) are known in two distinct regimes.  When p>> 1/\sqrt{n}, at this level of accuracy the probability matches that of G(n,p) being bipartite; and when p<< 1/\sqrt{n}, Janson's Inequality gives the asymptotics of the log.  I will discuss a new approach to estimating this probability in the "critical regime": when p = \Theta(1/\sqrt{n}).  The approach uses ideas from statistical physics and algorithms and gives information about the typical structure of graphs drawn from the corresponding conditional distribution.  Based on joint work with Matthew Jenssen, Aditya Potukuchi, and Michael Simkin.