The class of sequential quadratic programming (SQP) methods solve a nonlinearly constrained optimization problem by solving a sequence of related quadratic programming (QP) subproblems.  Each subproblem involves the minimization of a quadratic model of the Lagrangian function subject to the linearized constraints. In contrast to the quasi-Newton approach, which maintains a positive-definite approximation of the Hessian of the Lagrangian, second-derivative SQP methods use the exact Hessian of the Lagrangian. In this context, we will discuss a dynamic convexification strategy with two main features. First, the method makes minimal matrix modifications while ensuring that the iterates of the QP subproblem are bounded. Second, the solution of the convexified QP is a descent direction for a merit function that is used to force convergence from any starting point. This talk will focus on the dynamic convexification of a class of primal-dual SQP methods.  Extensive numerical results will be presented.

Weighted Hurwitz numbers were introduced by Harnad and Guay-Paquet as objects covering a wide class of Hurwitz numbers of various types. A particularly strong property of Hurwitz numbers is that they are governed by the celebrated topological recursion (TR) of Chekhov--Eynard--Orantin: a universal algorithm that allows computation of them recursively with respect to their topology. The program of understanding how TR can be used to compute different types of Hurwitz numbers was carried out over the last two decades by considering each case separately, and finally, the general case of rationally-weighted Hurwitz numbers was recently proved by Bychkov--Dunin-Barkowski--Kazarian--Shadrin. 

We will discuss a more general case of weighted $b$-Hurwitz numbers that arise naturally in the context of symmetric functions theory and matrix models. We show that their generating function satisfies the so-called $W$-constraints - certain explicit differential equations arising from representations of $W$-algebras. We will focus on a transition from an algebraic/geometric background to a combinatorial one, which turned out to be crucial in our work. Our result gives a new explanation of the remarkable enumerative properties of Hurwitz numbers following from TR, and extends it to the $b$-deformed case. This is joint work with Nitin Chidambaram and Kento Osuga.

For every surface S, the pure mapping class group G_S acts on the (SL_2)-character variety Ch_S of a fundamental group P of S. The character variety Ch_S is a scheme over the ring of integers. Classically this action on the real points Ch_S(R) of the character variety has been studied in the context of the Teichmuller theory and SL(2,R)-representations of P. 

In a seminal work, Goldman studied this action on a subset of Ch_S(R) which comes from SU(2)-representations of P. In this case, Goldman showed that if S is of genus g>1 and zero punctures, then the action of G_S is ergodic. Previte and Xia studied this question from topological point of view, and when g>0, proved that the orbit closure is as large as algebraically possible. 

Bourgain, Gamburd, and Sarnak studied this action on the F_p-points Ch_S(F_p) of the character variety with boundary trace equal to -2 where S is a puncture torus. They conjectured that in this case, this action has only two orbits, where one of the orbits has only one point. Recently, this conjecture was proved for large enough primes by Chen. When S is an n-punctured sphere, the finite orbits of this action on Ch_S(C) are connected to the algebraic solutions of Painleve differential equations. 

I will report on my joint work with Natallie Tamam in this area.

Hyperkahler manifolds are one of the main classes of manifolds appearing in Berger’s classification of holonomy groups of Riemannian manifolds. It is known that for any non-constant map f from a hyperkahler manifold of dimension 2n, the generic fibers of f are either finite or abelian varieties of dimension n. The latter are Lagrangian fibrations. I will discuss some open problems and some results concerning Lagrangian fibrations on hyperkahler manifolds.

In the upcoming Spring Quarter 2024 I will teach MATH262A 'Random Structures and Statistical Inference:', which is a topics course at the intersection of combinatorial statistics, algorithms and probabilistic combinatorics. The goal of this informal lecture is to give a glimpse into the kind of questions we intend to cover in this course. To this end we shall review the 'hidden clique' problem, which is a simple prototypical example with a surprisingly rich and interesting structure behind.

Inspired by the work of Hayes and Sale showing wreath products of two sofic groups are sofic, we define a notion of soficity for actions of countable discrete groups on countable discrete sets. We shall prove that, if the action $\alpha$ of G on X is sofic, G is sofic, and H is sofic, then the generalized wreath product H $\wr_\alpha$ G is sofic. We shall demonstrate several examples of sofic actions, including actions of sofic groups with locally finite stabilizers, all actions of amenable groups, and all actions of LERF groups. This talk is based on joint work with Srivatsav Kunnawalkam Elayavalli and Gregory Patchell.

This talk discusses computationally feasible algorithms to uncover low-dimensional structures in the Wasserstein space. This line of research is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in $\mathbb{R}^n$, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. One of our algorithms, LOT Wassmap, leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. Experiments demonstrate that LOT Wassmap attains correct embeddings, and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.

This talk is based on joint work with Alex Cloninger, Keaton Hamm, Varun Khurana, Matthew Thorpe, and Bernhard Schmitzer.

Evolutionary dynamics permeates life and life-like systems. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to make sense of many phenomena in the life sciences. How likely is a single mutant to take over a population of individuals? What is the speed of evolution, if things have to get worse before they can get better (aka, fitness valley crossing)? Can cooperation, hierarchical relationships between individuals, spatial interactions, or randomness influence the speed or direction of evolution? Applications to biomedicine will be discussed.

In this talk I'll present a brief history of the National Parks System and discuss some of its merits and drawbacks. I will also compare and contrast it with the National Forest Service and Bureau of Land Management. Since notions of "merit" are inherently subjective, to balance the discussion audience members are encouraged to contribute their views on this subject. Perspectives from those who have had experience with public land management systems in other countries are especially welcome. To close, we'll talk about some of the ways you can get out there and enjoy your public lands!

Compact hyper-Kähler manifolds and their Lagrangian fibrations are higher-dimensional generalizations of K3 surfaces and their elliptic fibrations. I will present a recent exploration of the geometry of isotrivial Lagrangian fibrations conducted with Y. Kim and R. Laza. We show that the smooth fiber of such a fibration is isogenous to the power of an elliptic curve and present a trichotomy arising from the Kodaira dimension of the minimal Galois cover of the base which trivializes monodromy. We are motivated in part by the search for new deformation types of hyper-Kähler manifolds and the boundedness problem.