A central theme of large-scale convex optimization is the search for ``optimal methods.’’ These are the algorithms whose convergence guarantees match complexity theoretic lower bounds for a given problem class. Standard optimal methods are notoriously unintuitive. I will begin by describing a new transparent optimal method for minimizing smooth convex functions that is rooted in elementary cutting plane ideas.

Despite the successes of convex techniques, recent years have seen a resurgence of interest in nonconvex and nonsmooth optimization. In such settings, it is essential to exploit problem structure to make progress. One typical example of favorable structure occurs when minimizing a composition of a finite convex function with a smooth map. In the second part of the talk, I will discuss various aspects of this problem class, focusing on both worst-case and average case guarantees. The phase retrieval problem will illustrate the algorithms and theory.

This is joint work work with D. Davis (Cornell), M. Fazel (Washington), A.S. Lewis (Cornell), C. Paquette (Lehigh), and S. Roy (Washington).

We present several results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we show a sharp upper bound for the $(n-1)$-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems in ${\mathbb R}^n$. We also show certain more recent results concerning semilinear equations (e.g. Navier-Stokes equations) and equations with non-analytic coefficients. The results on the size of nodal sets are connected to quantitative unique continuation, i.e., on the estimate of the order of vanishing of solutions of PDEs at a point. The results on unique continuation are joint with Ignatova and Camliyurt.

We study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model. In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with $m$ random vertices selected with probabilities proportional to their current degrees. (Constant $m$ is the only parameter of the model.) We prove that if $m \ge 1{,}260$, then asymptotically almost surely there exists a perfect matching. Moreover, we show that there exists a Hamiltonian cycle asymptotically almost surely, provided that $m \ge 29{,}500$. One difficulty in the analysis comes from the fact that vertices establish connections only with vertices that are ``older'' (i.e.~are created earlier in the process). However, the main obstacle arises from the fact that edges in the preferential attachment model are not generated independently. In view of that, we also consider a simpler setting---sometimes called uniform attachment---in which vertices are added one by one and each vertex connects to $m$ older vertices selected uniformly at random and independently of all other choices. We first investigate the existence of perfect matchings and Hamiltonian cycles in the uniform attachment model, and then extend the argument to the preferential attachment version.

The mirror symmetry conjecture (inspired by physics) has spurred a
lot of development in symplectic geometry. In the last few years, a
wave of modern homotopy theory has also entered the symplectic
landscape, and begun to present new questions about the structure of
symplectic manifolds. In this talk, we’ll explain a basic invariant
in symplectic geometry (the Fukaya category) and, as time allows,
give a survey of new inroads being opened through Lagrangian
cobordisms, derived geometry, and deformation theory.

Symplectic reductions from compact Hamiltonian Lie group actions on symplectic manifolds are important examples in the study of symplectic topology and mirror symmetry. In late 90s, Givental introduced an equivariant Gromov-Witten theory and used it to prove the mirror conjecture under the semi-positive assumption. During the past ten years, several groups of people have been working hard in generalizing the theory using symplectic vortex equations, but unfortunately, the corresponding moduli spaces suffer serious defect in compactness for higher genus case. In my talk, I will explain my ongoing project with Bohui Chen and Bai-Ling Wang in defining a new Gromov-Witten type of invariants for the equivariant cohomology of the ambient space. Using it, we also construct a quantum Kirwan morphism for a symplectic reduction.