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 $\mathscr{H}$ 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 $\mathscr{H}$ 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.

Facilitators: Howy Jordan and Maya Ornstein CHAT+ Disabilities in academia

Apr. 18, 2023 2:30pm (MATH 3…

Lie Theory

ALT Faculty

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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)

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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)

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The Lie group $\mathrm{GL}}_{N$ of invertible (complex) $N\times N$ 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 $G}_{t}={G}_{t}^{(N)$. The process $G}_{t$ can also be thought of as a (complex) matrix-valued geometric Brownian motion. It is, with probability $1$, 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 $t>0$, there is a certain annulus that, with probability $1$, contains all the eigenvalues of $G}_{t}^{(N)$ for all large $N$. This shows that the sequence of empirical laws of eigenvalues for $G}_{t}^{(N)$ 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 $G}_{t$.

This is joint work with Bruce Driver and Vaki Nikitopoulos.

Tightness for eigenvalues of $\mathrm{GL}}_{N$ Brownian Motion