For a self-symmetric tracial von Neumann algebra $A$, we study rescalings of $A^{*n}*L\mathbb F_r$  for $n\in\mathbb N$ and $r\in (1,\infty]$ and use them to obtain an interpolation $\mathcal F_{s,r}(A)$ for all real numbers $s > $0 and $1 − s < r \leq\infty$. In this talk, I will first review the literature around this topic and explain well-definedness of the family $\mathcal F_{s,r}(A)$. I will discuss our definition of self-symmetry which includes all diffuse abelian tracial von Neumann algebras regardless of separability, and then focus on free products of infinitely many members of the family $\mathcal F_{s,r}(A_i)$. This is joint work with Ken Dykema.

The celebrated Newlander-Nirenberg theorem states that on a smooth manifold, an almost complex structure $J$ is a complex structure if and only if it is integrable, namely, the Nijenhuis tensor $N_J$ vanishes. It was known from Hill and Taylor that if $J$ has Hölder regularity above $C^{1/2}$ then $N_J$ makes sense as a tensor with distributional coefficients. However $N_J$ is undefined for generic $C^{1/2}$ tensor due to the failure of multiplication for $C^{1/2}$ functions and $C^{-1/2}$ distributions.

In the talk, we will explore the integrability condition when $J$ has regularity below $C^{1/2}$. We give a necessary and sufficient condition for $J$ being a complex structure (at least) for $J\in C^{1/3+}$ using Bony's paradifferential calculus. 

This is an in progress work joint with Gennady Uraltsev.

Recently, a class of algorithms combining classical fixed-point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as 10^108 x 10^108. So far, a complete mathematical explanation for this success has proven elusive. The family of methods has not yet been extended to the important case of linear system solves. Our recent work proposes a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution is too large to store as a dense vector.

Jacobi's method is the oldest-known algorithm for the symmetric eigenvalue problem. It is also optimal; depending on the implementation, Jacobi can (1) compute small eigenvalues to higher relative accuracy than any other algorithm and (2) attain the arithmetic/communication complexity lower bounds of matrix multiplication (in both serial and parallel settings). This talk surveys efforts to optimize Jacobi as a one-algorithm case study into recent trends in numerical linear algebra. Based on joint work with James Demmel, Hengrui Luo, and Yifu Wang.

To achieve the greatest possible speed, practitioners regularly implement randomized algorithms for low-rank approximation and least-squares regression with structured dimension reduction maps. This talk outlines a new perspective on structured dimension reduction, based on the injectivity properties of the dimension reduction map. This approach provides sharper bounds for sparse dimension reduction maps, and it leads to exponential improvements for tensor-product dimension reduction. Empirical evidence confirms that these types of structured random matrices offer exemplary performance for a range of synthetic problems and contemporary scientific applications.

Joint work with Chris Camaño, Ethan Epperly, and Raphael Meyer; available at arXiv:2508.21189.

3d mirror symmetry predicts deep relationships between certain algebraic symplectic varieties. One such expectation is an "equivalence" between curve counts in a Higgs branch and those in the corresponding Coulomb branch. When it can be precisely formulated, this equivalence takes the form of an equality (after analytic continuation and change of variables) of meromorphic functions associated to the two branches. Bow varieties provide the largest currently known setting where the appropriate curve counts can be defined and their equivalence precisely formulated. In this talk, I will give an overview of these ideas and discuss my work with Tommaso Botta, in which we prove the duality of curve counts for finite type A bow varieties. Our proof combines geometric, combinatorial, and analytic arguments to eventually reduce to the case of the cotangent bundle of the complete flag variety. Time permitting, I will also discuss ongoing work to incorporate "descendant insertions" into the statements by using Hecke modifications of vector bundles.