The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history, from early works of Sylvester, Satge, Selmer etc. and, up to recent work of Alpoge-Bhargava-Shnidman. A conjecture attributed to Sylvester predicts that the primes >2 in the residue classes 2,5 mod 9 are not sums of two rational cubes, while those in the residue classes 4,7 or 8 mod 9 are. Primes which are 1 mod 9 may or may not be sums of two rational cube sums. We rephrase the problem in terms of elliptic curves, and use certain integral binary cubic forms to prove unconditionally that there are infinitely many primes in each of the residue classes 1 mod 9 and 8 mod 9 that are expressible as sums of two rational cubes. More generally, we prove that every non-zero residue class a (mod q), for any prime q, contains infinitely many primes which are sums of two rational cubes. Among other results, we show that corresponding to any positive integer n, there are infinitely many imaginary quadratic fields in which n is a sum of two cubes. These results represent joint work with Somnath Jha and Dipramit Majumdar. The starting point of this work was an accidental encounter in earlier work with Dipramit Majumdar when we classified all cubic cyclic extensions of Q.

We show that the amalgamated free product of weakly exact von Neumann algebras is weakly exact. This is done by using a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize weakly exact von Neumann algebras.

Erdős and Pósa proved in 1965 that every graph contains either $k$ vertex-disjoint cycles or a set of at most $O(k \log k)$ vertices intersecting every cycle. Such an approximate duality does not hold for odd cycles due to certain projective-planar grids, as pointed out by Lovász and Schrijver, and Reed showed in 1999 that these grids are the only obstructions to this duality. In this talk, we generalize these results by characterizing the obstructions in group-labelled graphs. Specializing to the group $\mathbb{Z}/m\mathbb{Z}$ gives a characterization of when cycles of length $\ell \bmod m$ satisfy this approximate duality, resolving a problem of Dejter and Neumann-Lara from 1988. We discuss other applications and analogous results for $A$-paths.

Based on joint work with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum.

Many statistical learning problems, where one aims to recover an underlying low-dimensional signal, are based on optimization, e.g., the linear programming approach for recovering a sparse vector. Existing work often either overlooked the high computational cost in solving the optimization problem, or required case-specific algorithm and analysis -- especially for nonconvex problems. This talk addresses the above two issues from a unified perspective of conditioning. In particular, we show that once the sample size exceeds the intrinsic dimension of the signal, (1) a broad range of convex problems and a set of key nonsmooth nonconvex problems are well-conditioned, (2) well-conditioning, in turn, inspires new algorithms and ensures the efficiency of off-the-shelf optimization methods.

A half-translation surface is a collection of polygons on the plane with side identifications by translations or half-turns in such a way that the resulting topological surface is closed and orientable. We also assume that the total Euclidean area of the polygons is finite. Two half-translations are equivalent if a sequence of cut-and-paste operations takes one to the other. From the view of complex geometry, an equivalent definition is a Riemann surface endowed with a meromorphic quadratic differential with poles of order at most one.

A stratum of half-translation surfaces consists of those with prescribed cone angles at the vertices of the polygons. Strata are, in general, not connected. A natural flow, the diagonal or Teichmüller flow, acts on stratum components.

In this talk, we investigate some topological properties of stratum components. We show that the (orbifold) fundamental group of such a component is “detected” by the diagonal flow in that every loop is homotopic to a concatenation of closed geodesics (coned to a base-point). Using this result, we show that the Lyapunov spectrum of the homological action of the diagonal flow is simple, thus establishing the Kontsevich–Zorich conjecture for quadratic differentials.

This is a joint work with Mark Bell, Vincent Delecroix, Vaibhav Gadre, and Saul Schleimer.

The sequence-structure-function relationship for proteins is well established, but are there corresponding relationships at larger scales, from organelles to specialized subcellular structures? My research seeks to address how molecular organization influences the shape of cellular membranes. In this talk I will discuss our recent progress towards developing approaches to enable the incorporation of biological complexity in models of cellular membrane mechanics. This includes a new simulation engine called Mem3DG which uses concepts from discrete differential geometry to model the coupled mechochemical feedback of in-plane membrane components interacting with membrane geometry. I will show biological problems we have been able to address and give a perspective on possible directions for future mathematical and computational development.

A graph $X$ is defined inductively to be $(a_0, . . . , a_{n−1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius 1 around $v$ is an $(a_1, . . . , a_{n−1})$-regular graph. A family $F$ of graphs is said to be an expander family if there is a uniform lower bound on the Cheeger constant of all the graphs in $F$. 

After briefly (re)introducing Coxeter groups and their geometries, we will describe how they can be used to construct very regular polytopes, which in turn can yield highly regular graphs. We will then use the super-approximation machinery, whenever the Coxeter group is hyperbolic, to obtain the expansion of these family of graphs. As a result, we obtain interesting infinite families of highly regular expander graphs, some of which are related to the exceptional groups. 

The talk is based on work joint with Conder, Lubotzky and Schillewaert.

A game of Nim is traditionally played with heaps of objects called counters. In this game, two players take turns choosing a heap and then removing any positive number of counters from that heap. Play continues until the final counter is removed, at which point the player responsible for this is declared the winner. In search of a general winning strategy for Nim, we encounter and prove several core principles of combinatorial game theory.