Georgia Institute of Technology College of Sciences

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.

On a closed, simply-connected, symplectic 4-manifold, the Dehn–Seidel twists on Lagrangian spheres and their products provide all known examples of non-trivial elements in the symplectic mapping class group. However, little is known in general about the relations that may hold among Dehn–Seidel twists. 

I will discuss the following result: on a symplectic K3 surface, the squared Dehn--Seidel twists on Lagrangian spheres with distinct fundamental classes are algebraically independent in the abelianization of the (smoothly-trivial) symplectic mapping class group. In a particular case, this establishes an abelianized form of a Conjecture by Seidel and Thomas on the faithfulness of certain Braid group representations in the symplectic mapping class group of K3 surfaces. The proof makes use of Seiberg--Witten gauge theory for families of symplectic 4-manifolds.

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.

Hydrodynamic bores are front-type traveling wave solutions to the two-layer free boundary Euler equations in two dimensions. We  prove that there exists a family of bores that starts at trivial laminar flow where the interface is flat and continues until the interface develops a vertical tangent. This type of behavior was first observed over 45 years ago in computations of internal gravity waves and gravity water waves with vorticity via numerical continuation. Despite considerable progress over the past decade in constructing global families of water waves that potentially overturn, a rigorous proof that overturning definitively occurs has been stubbornly elusive.  

Gravity currents arise when a heavier fluid intrudes into a region of lighter fluid. Typical examples are muddy water flowing into a cleaner body of water and haboobs (dust storms). We give the first rigorous proof of a conjecture of von Kármán on the structure of gravity currents near the rigid boundary. 

This is joint work with Ming Chen (Pittsburgh) and Miles Wheeler (Bath)

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 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).

Consider an elliptic curve $E$ defined over a number field $K$.  The set of $K$-points of $E$ is a finitely generated abelian group $E(K)$ whose rank is an important invariant. It is an open and difficult problem to determine which ranks occur for elliptic curves over a fixed number field $K$. We will discuss recent work which shows that there are infinitely many elliptic curves over $K$ of rank $r$ for each integer $0\leq r \leq 4$.   Our curves will be found in some explicit families.   We will use a result of Kai, which generalizes work of Green, Tao and Ziegler to number fields, to carefully choose our curves in the families.

Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{R}^+$ is $\mu_\alpha$-relatively dense (where $d\mu_\alpha(x) \approx x^{2\alpha+1}\, dx$) for $\alpha > -1/2$, and $\operatorname{supp} \mathcal{F}_\alpha(f) \subset [R,R+1]$, then we show $$\|f\|_{L^2_\alpha(\mathbb{R}^+)} \lesssim \|f\|_{L^2_\alpha(E)},$$ for all $f\in L^2_\alpha(\mathbb{R}^+)$, where the constants in $\lesssim$ do not depend on $R > 0$. Previous results on PLS theorems for the Fourier-Bessel transform by Ghobber and Jaming (2012) provide bounds that depend on $R$. In contrast, our techniques yield bounds that are independent of $R$, offering a new perspective on such results. This result is applied to derive decay rates of radial solutions of the damped wave equation. This is joint work with Ben Jaye.   

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 $S^2 \times S^2$.

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.

We characterize global minimizers below the so-called first critical field of the inhomogeneous version of the Ginzburg-Landau energy functional in a three-dimensional setting. Minimizers of this functional describe the behavior of type-II superconductors exposed to an external magnetic field, which is characterized by the presence of codimension 2 singularities called vortices where superconductivity is locally suppressed. We will talk about how to adapt the results from the standard Ginzburg-Landau theory into an inhomogeneous framework and present results from a recent work in collaboration with Carlos Roman (Pontificia Universidad Catolica de Chile).

Transport maps serve as a powerful tool to transfer information from source to target measures. However, this transfer of information is possible only if the transport map is sufficiently regular, which is often difficult to show. I will explain how taking the source measure to be an infinite-dimensional measure, and building transport maps based on stochastic processes, solves some of these challenges both in the continuous and discrete settings.