In this talk I will report on some joint research in the past year with J.B. Nation and Ralph Freese on ortholattices and with Melissa Sugimoto, José Gil-Férez and Sid Lodhia on involutive partially ordered monoids. Rather than just present the results I will attempt to describe how the automated theorem prover Prover9/Mace4 was used in some aspects of this research. For ortholattices we prove that the variety generated by the 6-element hexagonal ortholattice has an equational basis given by any equational basis of the pentagon lattice variety. We find a list of 9 finite ortholattices that generate varieties covering and present partial results towards showing that this list of finitely generated covers is exhaustive. For locally integral involutive po-monoids we prove that they are Plonka sums of integral involutive po-monoids and we give a more compact dual description of these sums. These results generalize an earlier structural description of finite commutative idempotent involutive residuated lattices by replacing the lattice order by a partial order, idempotence by local integrality and removing finiteness and commutativity. In the idempotent lattice-ordered case we give equational bases for finitely generated varieties by relating them to varieties of Brouwerian algebras.
Richard Green, Flor Orosz Hunziker and Nat Thiem will give brief introductions to their research and their mentoring style.
ALT Open House
Apr. 18, 2023 3:30pm (MATH 3…
Topology
Guchuan Li (University of Michigan)
X
Chromatic homotopy theory is a powerful tool to study periodic phenomena in stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the fixed points of Lubin--Tate theories E_h. The homotopy groups of these fixed points are periodic and computed via homotopy fixed points spectral sequences. In this talk, we prove that at the prime 2, for all heights h and all finite subgroups G of the Morava stabilizer group, the G-homotopy fixed point spectral sequence of E_h collapses after the N(h,G)-page and admits a horizontal vanishing line of filtration N(h,G).
Our proof uses new equivariant techniques developed by Hill-Hopkins-Ravenel in their solution of the Kervaire invariant one problem. As an application, we give a computation of E_2^{hG_{48}} based on this vanishing result.
This is joint work with Zhipeng Duan and XiaoLin Danny Shi.
Vanishing results in Chromatic homotopy theory at the prime 2 Sponsored by the Meyer Fund
Apr. 18, 2023 3:35pm (MATH 2…
Probability
Todd Kemp (University of California San Diego)
X
The Lie group of invertible (complex) matrices is a Riemannian manifold (with its tangent spaces equipped with the metric inherited from the usual Hilbert--Schmidt inner product on matrices), and therefore has a standard Brownian motion . The process can also be thought of as a (complex) matrix-valued geometric Brownian motion. It is, with probability , never a normal matrix, and so the study of the evolution of its eigenvalues is quite complicated. It is yet unknown whether the empirical measure of its eigenvalues (at a fixed time) converges (as $N\to\ifnty$) to some measure on the complex plane. A putative limit measure, called the Brown measure (yes, the Brownian Brown measure) was computed by Driver, Hall, and me in 2022.
As a partial result towards the desired limit theorem, in this talk I will present a tightness result: for each fixed time , there is a certain annulus that, with probability , contains all the eigenvalues of for all large . This shows that the sequence of empirical laws of eigenvalues for forms a tight sequence of random measures. Along the way to the proof, I will also show a ``Dyson Brownian motion'' style interacting particle system that describes the dynamic evolution of the eigenvalues of .
This is joint work with Bruce Driver and Vaki Nikitopoulos.
Tightness for eigenvalues of Brownian Motion
Apr. 18, 2023 11:15pm (MATH …
Diversity
Facilitators: Howy Jordan and Maya Ornstein CHAT+ Disabilities in academia
