Georgia Institute of Technology College of Sciences

The "Convexity Conjecture" by Talagrand asks (very roughly) whether one can "create convexity" in constant steps regardless of the dimension of the ambient space. Talagrand also suggested a discrete version of the Convexity Conjecture and called it "my lifetime favorite problem," offering $1,000 prize for its solution. We introduce a reformulation of the discrete Convexity Conjecture using the new notion of "k-thresholds," which is an extension of the traditional notion of thresholds, introduced by Talagrand. Some ongoing work on understanding k-thresholds, along with a (vague) connection between the Kahn-Kalai Conjecture and the discrete Convexity Conjecture, will also be discussed. Joint work with Michel Talagrand.

4-manifold topology is characterized by unexpected differences between the smooth and topological categories. For instance, it is the only dimension where there can exist infinitely many manifolds $Y_i$ which are homeomorphic to but not diffeomorphic to $X$. A natural question: how does one construct examples of this phenomenon? In this talk, we focus on the method of reverse engineering, which allows for the construction of “small” exotic 4-manifolds. Surprisingly, symplectic geometry is the main ingredient that makes this approach work! We survey the known results related to reverse engineering, and try to pinpoint an error in a paper of Akhmedov-Park, which claimed the existence of an exotic $S^2 \times S^2$.

In this talk I will discuss a recent result that establishes an unconditional proportion of non-vanishing at the central point $s=1/2$ for cubic Hecke $L$-functions over the Eisenstein quadratic number field. This result comes almost 25 years after Soundararajan’s (2000) breakthrough result for the positive proportion of non-vanishing for primitive quadratic Dirichlet $L$-functions over the rational numbers.

In this talk I will explain why number theorists care about non-vanishing for $L$-functions, and why the non-vanishing problem for cubic L-functions has starkly different features to the quadratic case. This is a joint work with A. De Faveri (Stanford), C. David (Concordia), and J. Stucky (Georgia Tech).

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.