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3:00 pm
Abhik Pal - University of California San Diego
Sheaf Cohomology of the Supergrassmannian and the Representation Theory of $\mathfrak{gl}(m|n)$
Advancement to Candidacy
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2:00 pm
Sylvester Zhang - University of Minnesota
Schubert calculus and the boson-fermion correspondence
Combinatorics Seminar (Math 269)
APM 7218
AbstractOriginally appearing in string theory, the Boson-Fermion correspondence has found connection to symmetric functions, through its application by the Kyoto school for deriving soliton solutions of the KP equations. In this framework, the space of Young diagrams is conceived as the Fermionic Fock space, while the ring of symmetric functions serves as the Bosonic Fock space. Then the (second part of) BF correspondence asserts that the map sending a partition to its Schur function forms an isomorphism as H-modules, with H being the Heisenberg algebra. In this talk, we give a generalization of this correspondence into the context of Schubert calculus, wherein the space of infinite permutations plays the role of the Fermionic space, and the ring of back-stable symmetric functions represents the Bosonic space.
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11:00 am
Philip Easo - Caltech
The critical percolation probability is local
Math 288 Probability Seminar
Halkin Room APM 6402
AbstractAround 2008, Schramm conjectured that the critical percolation probability $p_c$ of a transitive graph is entirely determined by the local geometry of the graph, subject to the global constraint that $p_c<1$. Previous works had verified the conjecture in various special cases, including nonamenable graphs of high girth (Benjamini, Nachmias and Peres 2012); Cayley graphs of abelian groups (Martineau and Tassion 2013); nonunimodular graphs (Hutchcroft 2017 and 2018); graphs of uniform exponential growth (Hutchcroft 2018); and graphs of (automatically uniform) polynomial growth (Contreras, Martineau and Tassion 2022). In this talk I will describe joint work with Hutchcroft (https://arxiv.org/abs/2310.
10983) in which we resolve this conjecture.
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4:00 pm
Prof. Padmini Rangamani - UC San Diego
The mathematics of cellular mechanotransduction
2024 Murray and Adylin Rosenblatt Endowed Lectures in Applied Mathematics
Natural Sciences Building Auditorium
Please register here:
https://forms.gle/g1XRxXwsUCkdR8YK8 AbstractCellular function often integrates biochemical and mechanical cues in what is known as mechanotransduction. Mechanotransduction is closely tied to cell shape during development, disease, and wound healing. In this talk, I will showcase how mathematical models have helped shed light on some fundamental problems in this area of research including how cell shape can alter biochemical signaling and how cell mechanics can alter cell shape. Throughout, I will highlight the challenges and opportunities for integrating mathematical models with experimental measurements.
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5:10 pm
Prof. Lisa Fauci - Tulane University
Insights from biofluidmechanics: A tale of tails
2024 Murray and Adylin Rosenblatt Endowed Lectures in Applied Mathematics
Natural Sciences Building Auditorium
Please register here:
https://forms.gle/g1XRxXwsUCkdR8YK8 AbstractThe motion of undulating or rotating elastic `tails’ in a fluid environment is a common element in many biological and engineered systems. At the microscale, we will consider models of the journey of extremely long and flexible insect flagella through narrow and tortuous female reproductive tracts, and the penetration of mucosal tissue by helical flagella of bacteria. At the macroscale, we will probe the neuromechanics and fluid dynamics of the lamprey, the most primitive vertebrate and, hence, a model organism. Using a closed-loop model that couples neural signaling, muscle mechanics, fluid dynamics and sensory feedback, we examine the hypothesis that amplified proprioceptive feedback could restore effective locomotion in lampreys with spinal injuries.
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11:00 am
Prof. Brian Hall - University of Notre Dame
Heat flow on polynomials with connections to random matrices and random polynomials
Math 243, Functional Analysis
APM 7218 and Zoom (meeting ID: 94246284235)
AbstractIt is an old result of Polya and Benz the backward heat flow preserves the set of polynomials with all real roots. Recent results have shown a surprising connection between the evolution of real roots under the backward heat flow and the notion of “free convolution” in free probability. Free convolution, in turn, is the operation that allows one to compute the eigenvalue distribution for sums of independent Hermitian matrices in terms of the individual eigenvalue distributions.
The story gets even more interesting when one considers polynomials with complex roots. Recent work of mine with Ho indicates that under the heat flow, the complex roots of high-degree polynomials should evolve in straight lines with constant speed. This behavior also connects to random matrix theory and free probability. I will present some conjectures as well as recent rigorous results with Ho, Jalowy, and Kabluchko.
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11:00 am
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4:00 pm
Paul K. Newton - University of Southern California
Control of evolutionary mean field games and tumor cell population models
UCSD Mathematics Colloquium/MathBio Seminar
AbstractMean field games are played by populations of competing agents who derive their update rules by comparing their own state variable with that of the mean field. After a brief introduction to several areas where they have been used recently, we will focus on models of competing tumor cell populations based on the replicator dynamics mean field evolutionary game with prisoner’s dilemma payoff matrix. We use optimal and adaptive control theory on both deterministic and stochastic versions of these models to design multi-drug chemotherapy schedules that suppress the competitive release of resistant cell populations (to avoid chemo-resistance) by maximizing the Shannon diversity of the competing subpopulations. The models can be extended to networks where spatial connectivity can influence optimal chemotherapy scheduling.