I will report on ongoing joint work with Keller VandeBogert and Jerzy Weyman on the total rank of the Tor groups of modules over polynomial rings that arise from representations of Lie algebras. This work is motivated by the problem of understanding lower bounds on the total rank of free complexes with finite length homology and also the problem of computing syzygies of nilpotent orbit closures.

Here are two classical facts about actions of countable group Gamma on topological spaces: 1. Every action of Gamma on a compact space admits an invariant probability measure if and only if Gamma is amenable. 2. If mu is a probability measure on Gamma then every action of Gamma on a compact space always admits a stationary measure, that is, a measure that does not change on average when multiplying by a random element of Gamma drawn from mu. We are interested in how these theorems generalize to actions on non-compact spaces, where measures are required to give compact sets finite mass. For co-compact actions, the first question (about invariant measures) was answered by Kellerhals, Monod, and Rørdam (2013) and is closely related to classical results of Tarski. I will review this and then discuss our recent solution of the problem about stationary measures, joint with Alhalimi, Pan, Tamuz, and Zheng, which also involves a stationary analogue of Tarski's theorem.

We analyze the performance of Belief Propagation Guided Decimation, a physics-inspired message passing algorithm, on the random $k$-XORSAT problem. Specifically, we derive an explicit threshold up to which the algorithm succeeds with a strictly positive probability Ω(1) that we compute explicitly, but beyond which the algorithm with high probability fails to find a satisfying assignment. In addition, we analyze a thought experiment called the decimation process for which we identify a (non-)reconstruction and a condensation phase transition. The main results of the present work confirm physics predictions from [Ricci-Tersenghi and Semerjian: J. Stat. Mech. 2009] that link the phase transitions of the decimation process with the performance of the algorithm, and improve over partial results from a recent article [Yung: Proc. ICALP 2024]. 

We study squared-variable formulations for nonlinear semidefinite programming. We show an equivalence result of second-order stationary points of the nonsymmetric-squared-variable formulations and the nonlinear semidefinite programs. We also show that such an equivalence fails for the local minimizers and second-order stationary points of the symmetric-squared-variable formulations and the nonlinear semidefinite programs, correcting a false understanding in the literature and providing sufficient conditions for such a correspondence to hold.

Starting from the celebrated results of Eells and Sampson, a rich and flourishing literature has developed around equivariant harmonic maps from the universal cover of Riemann surfaces into nonpositively curved target spaces. In particular, such maps are known to be rigid, in the sense that they are unique up to natural equivalence. Unfortunately, this rigidity property fails when the target space has positive curvature, and comparatively little is known in this framework. In this talk, given a closed Riemann surface with strictly negative Euler characteristic and a unitary representation of its fundamental group on a separable complex Hilbert space H which is weakly equivalent to the regular representation, we aim to discuss a lower bound on the Dirichlet energy of equivariant harmonic maps from the universal cover of the surface into the unit sphere S of H, and to give a complete classification of the cases in which the equality is achieved. As a remarkable corollary, we obtain a lower bound on the area of equivariant minimal surfaces in S, and we determine all the representations for which there exists an equivariant, area-minimizing minimal surface in S. The subject matter of this talk is a joint work with Antoine Song (Caltech) and Xingzhe Li (Cornell University).

I will present recent work on Anderson localization for Schrödinger operators generated by hyperbolic transformations. Specifically, we consider subshifts of finite type equipped with an ergodic measure that admits a bounded distortion property. We show that if the Lyapunov exponent is uniformly positive and satisfies a uniform large deviation theorem (LDT) on a compact interval, then the operator exhibits Anderson localization on that interval almost surely. For Hölder continuous potentials with small supremum norms, we establish uniform positivity and a uniform LDT away from an arbitrarily small neighborhood of a finite set. In particular, this yields full spectral localization for such potentials. This talk is based on joint work with A. Avila and D. Damanik.

Topological defects occur in a wide variety of mathematical and physical settings. I will review the origin, structure and dynamics of such defects including a recent realization of nonabelian line defects in a three-dimensional chiral liquid crystal system with the associated entanglement, trivalent junctions and networks, and stable bound states of pairs of defects, thus realizing the notion of topological rigidity envisaged 50 years ago by Poenaru and Toulouse. 

For a complex algebraic variety X embedded inside a smooth variety Y, the local cohomology sheaves of X in Y carry additional structure of a (mixed) Hodge module. In the hypersurface and the local complete intersection (lci) case, this has been widely leveraged to prove various results about higher Du Bois and higher rational singularities, among other things. We investigate these local cohomology sheaves when X is a toric variety (which is typically non-lci) and prove various results about them. A few applications include showing that the local cohomological dimension of a toric variety is NOT a combinatorial invariant, and some new results about the singular cohomology of toric varieties. This is based on joint work with Hyunsuk Kim.