Abstract: Thermodynamics was born in the 19th century in quest of a way to quantify efficiency of steam engines at the dawn of the industrial age. In the time since, thermodynamics has impacted virtually all other areas in science, from chemistry and biology to the physics of black holes, and yet, progress beyond the classical quasi-static limit towards finite-time thermodynamic transitions has been slow; finite-time is of essence for non-vanishing power generation. In recent years a deeper understanding of non-equilibrium processes has been achieved based on stochastic models with degrees of freedom (state variables) that are subject to Brownian excitation that models heat baths. Within this framework we will explain energy transduction, we will give insights on how anisotropy in thermal or chemical potentials can be tapped for power generation in engineered and physical processes, and we will highlight fundamental bounds on the amount of power that can drawn during finite-time thermodynamic transitions.

The talk is based on joint works with Rui Fu (UCI), Olga Movilla (UCI), Amir Taghvaei (UCI) and Yongxin Chen (GaTech). Research funding by AFOSR, ARO and NSF is gratefully acknowledged.

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Join Zoom Meeting: https://usc.zoom.us/j/94973619069?pwd=VnU5bVlMc1pzVTlEYUVaZUYyNSt6UT09

Meeting ID: 949 7361 9069

Passcode: 925028

Abstract: I will discuss the existence and uniqueness of a singular, non-linear, and path dependent ODE. As an application, we use the ODE solution to prove existence of a Radner equilibrium with homogenous power traders in the limited participation setting of Basak and Cuoco (1998). Joint work with G. Leoni and P. Gasoni.

]]>Abstract: Given an arbitrary finite quiver Q, Maulik and Okounkov defined a new Yangian-style quantum group. It is built from the FRT formalism and their construction of R matrices on the cohomology of Nakajima quiver varieties, via the stable envelopes that they also defined. Just as in the case of ordinary Yangians, there is a Lie algebra g_Q inside their new algebra, and the Yangian is a deformation of the current algebra of this Lie algebra.

Outside of extended ADE type, numerous basic features of g_Q have remained mysterious since the outset of the subject, for example, the dimensions of the graded pieces. A conjecture of Okounkov predicts that these dimensions are given by the coefficients of Kac's polynomials, which count isomorphism classes of absolutely indecomposable Q-representations over finite fields. I will explain a recent proof, with Botta, of this conjecture. By proving that the Maulik-Okounkov Lie algebra g_Q is isomorphic to certain BPS Lie algebras, we prove Okounkov's conjecture, as well as essentially determining the isomorphism class of g_Q, thanks to recent joint work of myself with Hennecart and Schlegel Mejia.

]]>Abstract: Hilbert’s sixth problem asks for a mathematically rigorous justification of the macroscopic laws of statistical physics from the microscopic laws of dynamics. The classical setting of this problem is the justification of Boltzmann’s kinetic equation from Newtonian particle dynamics. This justification has been proven for short times, starting with the work of Lanford in 1975, but its long time justification remains one of the biggest open problems in kinetic theory.

If classical colliding particles are replaced with interacting waves, one formally obtains what is known as “wave kinetic theory”, which is sometimes also called “wave turbulence theory”. This theory of statistical physics for waves has been developed, starting in the late 1920s, for wave systems that arise in various scientific disciplines like many-particle quantum physics, oceanography, climate science, etc. The central mathematical problem there is also the justification of a kinetic equation, known as the wave kinetic equation, starting from the Hamiltonian PDE that governs the corresponding microscopic system. In this talk, we shall describe the state of the art of this problem, leading to a most recent joint work with Yu Deng (USC), in which we give the first instance of a long time justification of a nonlinear (particle or wave) collisional kinetic limit.

]]>Abstract: In this work, we study the event occurrences of individuals interacting in a network. To characterize the dynamic interactions among the individuals, we propose a group network Hawkes process (GNHP) model whose network structure is observed and fixed. In particular, we introduce a latent group structure among individuals to account for the heterogeneous user-specific characteristics. A maximum likelihood approach is proposed to simultaneously cluster individuals in the network and estimate model parameters. A fast EM algorithm is subsequently developed by utilizing the branching representation of the proposed GNHP model. Theoretical properties of the resulting estimators of group memberships and model parameters are investigated under both settings when the number of latent groups G is over-specified or correctly specified. A data-driven criterion that can consistently identify the true G under mild conditions is derived. Extensive simulation studies and an application to a data set collected from Sina Weibo are used to illustrate the effectiveness of the proposed methodology.

]]>Abstract: In searching for a way of evaluating students and assigning a grade, while also motivating students to learn from their mistakes, and making grading assessments easier for me, I settled on a mastery-based grading system. I will discuss the evolution of the grading system I currently use, the pros and cons, as well as issues I am hoping to find a solution for.

]]>Abstract: Mirror symmetry predicts equivalence between Fukaya categories of Coulomb branches (A-side) and derived categories of coherent sheaves of Coulomb branches (B-side). In the talk, I will focus on how this relation works in the decategorified setting. It will explicitly show how to identify a cohomology space associated with the A-side with the equivariant K-theory on the B-side. Based on joint work with A. Smirnov, and with M. Aganagic, Y. Li, V. Shende, P.Zhou.

]]>Abstract: The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking questions about topological vector bundles. But, in general, there are many non-equivalent vector bundles with the same K-theory class. Bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest CW complexes.

Building on work of Hu, we use Weiss calculus and a little chromatic homotopy theory to translate vector bundle enumeration questions to tractable stable homotopy theory computations. We compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. This is joint work with Hood Chatham and Yang Hu.

]]>There will be a projector screen in KAP 414 for those who would like to gather and watch the talk.

Note the speaker will not be present in KAP 414.

Title: TBA

Abstract: TBA

Abstract: Between 1905 and 1910 the idea of the random walk, now ever-present in applied math, was invented simultaneously and independently by multiple people in multiple countries for completely different purposes, from mosquito control to physics to finance to winning a theological argument (really!) I’ll tell some part of this story and also gesture at ways that random walks (or Markov processes, named after the theological arguer) underlie current thinking about artificial intelligence. This talk will be non-technical but should have something new to offer even if you already know what a Markov process is.

]]>Abstract: I spent a portion of 2023 working with a team at DeepMind on the “cap set problem” – how large can a subset of (Z/3Z)n be which contains no three terms which sum to zero? (I will explain, for those not familiar with this problem, something about the role it plays in combinatorics, its history, and why number theorists care about it a lot.) By now, there are many examples of machine learning mechanisms being used to help generate interesting mathematical knowledge, and especially interesting examples. This project used a novel protocol; instead of searching directly for large cap sets, we used LLMs trained on code to search the space of short programs for those which, when executed, output large capsets. One advantage is that a program is much more human-readable than a large collection of vectors over Z/3Z, bringing us closer to the not-very-well-defined-but-important goal of “interpretable machine learning.” I’ll talk about what succeeded in this project (more than I expected!) what didn’t, and what role I can imagine this approach to the math-ML interface playing in near-future mathematical practice.

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