GAGA Seminar
3:30 pm Tuesday, February 28
TUC 246
Dr. Efton Park, TCU
Diagonalization of Matrices of Continuous Functions
Abstract: One of the most important theorems in linear algebra is the spectral theorem: every complex matrix that commutes with its adjoint is unitarily equivalent to a diagonal matrix. Recently, I have become interested in the question of extending the spectral theorem to matrices whose entries are not complex numbers, but are instead continuous complex-valued functions on some compact Hausdorff space. Looking at this more general problem involves some interesting twists and turns and involves topology in a quite nontrivial way. I will discuss what is known about this problem and what is not. I will not assume any knowledge beyond linear algebra and topology, so the lectures will be appropriate for graduate students.
Frank Stones Memorial Colloquium
3:30 pm Friday, March 2
TUC 246
Dr. Tim Perutz, University of Texas at Austin
Arithmetic Geometry from Symplectic Topology
Abstract:The Fukaya category of a symplectic manifold is an intricate algebraic invariant that encodes refined intersection data about its Lagrangian submanifolds. Kontsevich's homological mirror symmetry conjecture proposed that certain Fukaya categories can be understood in terms of sheaves on "mirror" algebraic varieties. In the talk I'll try to convince you that, in spite of their abstract formulation, these ideas have concrete content. Thus the Fukaya category of the 2-torus encodes multiplication rules for theta-functions associated with a mirror elliptic curve (the Tate curve). I'll outline an integral refinement of mirror symmetry for the 2-torus (joint work with Y. Lekili) which makes a connection between symplectic topology and arithmetic geometry. We show, for instance, that the mirror to the once-punctured 2-torus is the arithmetic curve y2+ xy = x3.