Schur-Weyl duality involves the commuting actions of the general linear group and the symmetric group on a tensor space, relating the irreducible representations of these two groups. The idea can be generalised to other groups using the partition algebra and its subalgebras. I will discuss one such generalisation, `Schur-Weyl-Jones duality', as well as a refinement of this used to obtain a combinatorial formula for irreducible characters of the symmetric group. Time permitting, I will discuss an application of this formula towards obtaining new bounds on the expected irreducible character of a wrandom permutation, that is, a random permutation obtained via a word map $w : S_n \times \cdots  \times S_n \rightarrow S_n$.

We study a general class of finite-player stochastic games with mean-field interaction where the linear-quadratic cost functional includes linear operators acting on controls in L^2. We propose a new approach for deriving the Nash equilibrium of these games in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations which can be solved. Moreover, by deriving stability results for the system of Fredholm equations, we obtain the convergence of the finite-player Nash equilibrium to the mean-field equilibrium in the infinite player limit. Our general framework includes examples of stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.

This is joint work with Eduardo Abi Jaber (Ecole Polytechnique) and Eyal Neuman (Imperial College London). The paper is available at https://ssrn.com/abstract=4470883.
 

This talk will be colloquial and geared towards people from other fields. I will talk about smooth mod $p$ representations of $p$-adic Lie groups. In stark contrast to the complex case, these categories typically do not have any (nonzero) projective objects. For reductive groups this is a byproduct of a stronger result on the derived functors of smooth induction. The talk is based on joint work with Peter Schneider.

Cell polarization, in which a uniform distribution of substances becomes asymmetric due to internal or external stimuli, is a fundamental process underlying cell mobility and cell division. Budding yeast provides a good system to study how biochemical signals and mechanical properties coordinate with each other to achieve stable cell polarization and give rise to certain morphological change in a single cell. Recent experimental data suggests yeast budding develops into two trajectories with different bud shapes as mother cells become old. We first developed a 2D model to simulate biochemical signals on a shape-changing cell and investigated strategies for robust yeast mating. Then we extended and coupled this biochemical signaling model with a 3D subcellular element model to take into account cell mechanics, which was applied to investigate how the interaction between biochemical signals and mechanical properties affects the cell polarization and budding initiation. This 3D mechanochemical model was also applied to predict mechanisms underlying different bud shape formation due to cellular aging.

In the early years of the XX century, Weyl initiated study of compact perturbations of pseudo-differential operators. The Weyl-von Neumann theorem asserts that two self-adjoint operators on a complex Hilbert space are unitarily equivalent modulo compact perturbations if and only if their essential spectra coincide. Berg and Sikonia (independently) extended this result to normal operators. New impetus to the subject was given in 1970s by Brown, Douglas, and Fillmore, who replaced single operators with (separable) C*-algebras and realized that compact perturbations can be considered as extensions by the ideal of compact operators. After passing to the quotient (the Calkin algebra, Q) and identifying an extension with a *-homomorhism into Q, analytic methods had been supplemented with methods from algebraic topology, homological algebra, and (most recently) logic.  Around the same time, Shelah proved one of his many influential results, by showing that the assertion `all automorphisms of $\ell_\infty/c_0$ are trivial' is relatively consistent with ZFC. Surprisingly, these two directions of research are intimately connected. 

This talk will be about rigidity of quotient structures and it is  partially based on the preprint Farah, I., Ghasemi, S., Vaccaro, A., and Vignati, A. (2022). Corona rigidity. arXiv preprint arXiv:2201.11618

https://arxiv.org/abs/2201.11618 and some more recent results. 

The 29th SCGAS will be held at the Department of Mathematics of University of California at San Diego on Saturday, April 13, 2024 and Sunday, April 14, 2024. The lectures will be held in Natural Science Building Auditorium (04/13) and Center Hall 105 (04/14) due to the campus event of Triton Day. For directions on how to get to Natural Science Building see map; For the Center Hall, here is a map.

And here are directions from the BW-Del Mar to the UCSD campus.

Registration starts at 10am Saturday morning. The first talk will be at 11:00am and the last talk will finish at 12:30pm on Sunday, to allow for travel.

Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to join our annual seminar. Partial financial support is available.

The Seminar is supported by the NSF and by the School of Physical Sciences at UC San Diego.

Invited Speakers: Guido De Philippis (CIMS), Bruce Kleiner (CIMS), Yi Lai (Stanford), Bill Minicozzi (MIT), Song Sun (Berkeley/Zhejiang Univ.), Guofang Wei (UCSB), Xin Zhou (Cornell)

Registration: Participants are asked to register online: the electronic registration form is now available.