I will give an example of a moduli space of curves that looks like it should be connected, but it has two components. Those components will be recognized by a developed by Fried and Serre. We use this example to illustrate much theory about these spaces, including understanding them as covers of the -line:
their degrees as covers, their genuses, and whether they have fine moduli.
An efficient device for making this computation is the sh(ift)-incidence matrix.
I will explain this using a 2-page handout. This is a prelude to seeing the relation between this lift invariant and the . For a curve rather than an abelian variety, this is a perfect pairing . This pairing first appeared to explain a phenomena on modular curves. This talk is a prelude to showing the Weil pairing in this famous place gets incorporated into this theory, making it available for generalization beyond any previous use of it.
Conventional wavelets in two dimensions use the operations of integer translation and dilation by an expansive integer matrix to compress images for more efficient storage and transmission. This talk will discuss wavelets built by replacing the group of integer translations by one of the 17 wallpaper symmetry groups. We will focus on wavelet sets, which can be used to decompose the wavelet representation into irreducibles.
Wavelet Sets for Crystallographic Groups
May. 03, 2018 3pm (MATH 350)
Probability
Kyle Luh (Harvard)
X
Despite recent advances in the graph isomorphism problem, the current algorithms still only run in quasipolynomial time. It is known, however, that for graphs with adjacency matrices that have simple spectrum (meaning no repeated eigenvalues) there is a polynomial-time algorithm. The natural question is whether the majority of graphs and matrices fall in this category. It was conjectured by Babai that Erd\H{o}s-R\enyi random graphs with edge probability 1/2 have simple spectrum with high-probability. This conjecture was recently verified by Tao and Vu for all constant p. We extend their result to allow p to depend on the number of vertices (p \geq n^{-1+\delta} for any \delta>0). In particular, we show that a large class of sparse random matrices have simple spectrum with high probability. Along with the proof of this result, we will survey several recurring tools from random matrix theory. This talk is based on joint work with Van Vu.
Sparse Random Matrices have Simple Spectrum
May. 03, 2018 11pm (MATH 350)
PDE/Analysis
Zachary Grey (University of Colorado, Boulder, Aerospace Engineering) A Riemmanian View on Active Subspaces: Active Manifold-Geodesics Sponsored by the Meyer Fund
Components of Moduli spaces and the Weil Pairing