How should you read your TA evaluations?  How do I read them?  I'd like to talk about this and I'd also like to review the first half of the seminar, talking about questions like "What was the main point of the Week ? meeting?".

Suppose f is an n-variate polynomial with integer coefficients and degree d. Many natural computational problems involving the real zero set Z of f are algorithmically difficult. For instance, no known algorithm for computing the number of connected components of Z has complexity polynomial in n+d. Furthermore, no known general algorithm for deciding whether f has root over the p-adic has sub-exponential complexity. So it is worthwhile to seek families of polynomials where these questions are tractable.

Assuming f has n+2 monomial terms, and its exponent vectors do not all lie on an affine hyperplane, we prove that the isotopy type of Z can be determined in time polynomial in log d, for any fixed n. (This family of polynomials --- polynomials supported on circuits ---is highly non-trivial, since it already includes Weierstrass normal forms and several important examples from semi-definite programming.) We also show that a 1998 refinement of the abc-Conjecture (by Baker) implies that our algorithm is polynomial in n as well. Furthermore, the original abc-Conjecture implies that p-adic rational roots for f can be detected in the complexity class NP.

     These results were obtained in collaboration with Kaitlyn Phillipson and Daqing Wan.

A fundamental observation in Katz-Sarnak's study of the zero spacing of L-functions is that Frobenius conjugacy classes in suitable families of varieties over finite fields approximate infinite random matrix statistics. For example, the normalized Frobenius conjugacy classes of smooth plane curves of degree d over F_q approach the Gaussian symplectic ensemble as we take first q to infinity, then d to infinity. In this talk, we explain a sideways version of this result where the limits in d and q are exchanged, and give a Hodge theoretic analog in characteristic zero. 

In this talk, I  will explain the notion of a simple Toeplitz sequence (in the sense of Liu-Qu) and of the subshift associated to it. A description of the elements in the subshift will be given and some basic properties of the subshift will be discussed. 

Raphael Zentner (University of Regensburg): Irreducible SL(2,C)-representations of integer homology 3-spheres
We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein and Wang (which builds on the geometrization theorem of 3-manifolds), it follows that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C).

Zhouli Xu (MIT): TBA
TBA