Georgia Institute of Technology College of Sciences

The Kauffman bracket skein module S(M) of a 3-manifold M classifies polynomial invariants of links in M satisfying Kauffman bracket skein relations. Witten conjectured that the skein module (over a field, with generic A) is finite dimensional for any closed 3-manifold M. This conjecture was proved by Gunningham, Jordan, and Safronov, however their work does not lead to an explicit computation of S(M).
In fact, S(M) has been computed for a few specific families of closed 3-manifolds so far. We introduce a novel method of computing these skein modules for certain rational homology spheres. (This is joint work with R. Detcherry and E. Kalfagianni.)

I will discuss some problems in geometric topology, and relate them to graph-theoretic properties. I will give some open problems, and answer questions of Kalai, Belolipetski, Gromov and others.

We study quantizations of SL_n-character varieties, which appears as moduli spaces for many geometric structures. Our main goal is to establish the existence of several quantum trace maps. In the classical limit, they reduce to the Fock-Goncharov trace maps, which are coordinate charts on moduli spaces of SL_n-local systems used in higher Teichmuller theory. In the quantized theory, the algebras are replaced with non-commutative deformations. The domains of the quantum trace maps are the SL_n-skein algebra and the reduced skein algebra, and the codomains are quantum tori, which are non-commutative analogs of Laurent polynomial algebras. In this talk, I will review the classical theory and sketch the definition of the quantum trace maps.

We consider the problem of estimating the factors of a rank-1 matrix with i.i.d. Gaussian, rank-1 measurements that are nonlinearly transformed and corrupted by noise. Considering two prototypical choices for the nonlinearity, we study the convergence properties of a natural alternating update rule for this nonconvex optimization problem starting from a random initialization. We show sharp convergence guarantees for a sample-split version of the algorithm by deriving a deterministic recursion that is accurate even in high-dimensional problems. Our sharp, non-asymptotic analysis also exposes several other fine-grained properties of this problem, including how the nonlinearity and noise level affect convergence behavior.

On a technical level, our results are enabled by showing that the empirical error recursion can be predicted by our deterministic sequence within fluctuations of the order n−1/2 when each iteration is run with n observations. Our technique leverages leave-one-out tools originating in the literature on high-dimensional M–estimation and provides an avenue for sharply analyzing higher-order iterative algorithms from a random initialization in other high-dimensional optimization problems with random data.