This is joint work with Fulvio Gesmundo.īecause of exams and/or travel, Daniel is unable to attend seminars on Oct 11, Oct 18, Nov 15, and Dec 13. Schedule of Talks: March 16, Kevin Tucker (UIC), On the limit of the F-signature function in characteristic zero March 23, Leslie Roberts (Queens University). As a first step in understanding these varieties, we compute a formula for the degrees of Stiefel manifolds using techniques from classical algebraic geometry, representation theory, and combinatorics. This viewpoint is known as algebraic frame theory and still many algebro-geometric properties of these varieties remain unknown, in particular, their degrees. The Stiefel manifold, and many other spaces of finite frames, can be viewed as an algebraic variety embedded in the space of k by n matrices. The Stiefel manifold of orthonormal bases for k-planes in an n-dimensional space goes by a different name in the world of frame theory: the space of Parseval n-frames for a k-dimensional space. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.
Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. I will introduce the class of completely log-concave polynomials in elementary terms, discuss the beautiful real and combinatorial geometry underlying these polynomials, and describe applications to random walks on simplicial complexes. Log-concave polynomials, matroids, and expandersĬomplete log-concavity is a functional property of real multivariate polynomials that translates to strong and useful conditions on its coefficients. We also show that Delta_3 is homotopy equivalent to the 5-sphere, and that Delta_4 has 3-torsion in H_5. \displaystyle are simply connected, for g at least 1. Publications and Preprints A short proof of a conjecture of Matsushita preprint (pdf, arXiv) Functional transcendence of periods and the geometric AndrGrothendieck period conjecture with J. Geometric vertex decomposition and liaison UIC Algebraic Geometry Seminar Midwest Algebraic Geometry Graduate Conference (MAGGC), UIC, August 10-11 2022.
Standard Conjecture D for Matrix Factorizations On the cohomology of the moduli space of parabolic connections