Georgia Institute of Technology College of Sciences

How likely is $G(n;p)$ to have a less-than-typical number of triangles? This is a foundational question in non-linear large deviation theory. When $p >> n^{-1/2}$ or $p >> n^{-1/2}$ the answer is fairly well-understood, with Janson's inequality applying in the former case and regularity- or container-based methods applying in the latter. We study the regime $p = c n^{-1/2}$, with $c>0$ fixed, with the large deviation event having at most $E$ times the expected number of triangles, for a fixed $0 <= E < 1$.

We prove explicit formula for the log-asymptotics of the event in question, for a wide range of pairs $(c,E)$. In particular, we show that for sufficiently small $E$ (including the triangle-free case $E = 0$) there is a phase transition as $c$ increases, in the sense of a non-analytic point in the rate function. On the other hand, if $E > 1/2$, then there is no phase transition.

As corollaries, we obtain analogous results for the $G(n;m)$ model, when $m = C n^{3/2}$. In contrast to the $G(n;p)$ case, we show that a phase transition occurs as $C$ increases for all $E$.

Finally, we show that the probability of $G(n;m)$ being triangle free, where $m = C n^{3/2}$ for a sufficiently small constant $C$, conforms to a Poisson heuristic.

Joint with Matthew Jenssen, Will Perkins, and Aditya Potukuchi.

In their celebrated work, Gordon and Luecke proved that knots in the three-dimensional sphere are determined by their complements. Their result inspired several conjectures seeking to generalize the theorem, including "Cosmetic surgery conjecture" proposed by Gordon and the "Knot complement problem for null-homotopic knots" proposed by Boileau. In this talk, I will discuss applications of tools from Yang—Mills gauge theory to these Dehn surgery problems.

Convex relaxations are of central interest in optimization, and it is typically challenging to determine whether a given convex relaxation will be tight for a given problem. We introduce a topological framework for analyzing  situations in which a constrained optimization problem over a nonconvex set (such as a manifold) has a tight convex relaxation. In particular, we give a criterion for the existence of such a tight convex relaxation in terms of the existence of a continuous function of Lagrange multipliers for the constrained problem maximizing the corresponding Lagrangian. We call such a function a Lagrangian dual section, in reference to the topological notion of a section of a bundle.

As a corollary of this result, we will give new criteria for the exactness of SDP relaxations for Stiefel manifold optimization and inverse eigenvalue problems in terms of linear subspaces of matrices satisfying spectral properties such as being nonsingular. We will also illustrate a homotopy continuation style algorithm with global optimality guarantees with applications to the unbalanced procrustes problem.

In a recent joint work with J. Buzzi, Y. Shi, and J. Yang, given a diffeomorphism preserving a one-dimensional expanding foliation $\mathcal F$ with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with controlled Jacobian and a Gibbs property.

We then prove that for any measure of maximal $\mathcal F$-entropy, its conditional measures on each leaf must be equivalent to the reference measures.

When the reference measures are equivalent to the leafwise Lebesgue measure, we prove that the log-determinant of $f$ must be cohomologous to a constant.

We will consider several applications, including the strong and center foliations of Anosov diffeomorphisms, factor over Anosov diffeomorphisms, and perturbations of the time-one map of geodesic flows on surfaces with negative curvature. We will also discuss several conjectures on the unique ergodicity and (exponential) equidistribution for the strong unstable foliations of Anosov systems. 

Zoom link: https://gatech.zoom.us/j/92005780980?pwd=ptlx7KdBAbHI3DTvv6V7fjFn27LDaE.1

Meeting ID: 920 0578 0980
Passcode: 604975