In their 1960 paper, M. Dupius, R. Mazo, and R. Onsager suggested that it might be of interest to consider cases when the contribution of the boundary to the spectral asymptotics is comparable to the contribution of the interiot. I discuss a simple toy problem for the Dirichlet Laplacian in a sequence of domains where it is the case.

The twisted Gan--Gross--Prasad conjectures consider the restriction of representations from GL_n to a unitary group over a quadratic extension E/F. In this talk, I will explain the adaptation of the relative trace formula comparison used in previous work on the global GGP conjecture for unitary groups, to this twisted version. In particular, I will discuss the fundamental lemma that arises, which can be used to obtain the global twisted GGP conjecture (under some local assumptions) in the case that everything is unramified, and how it can be reduced to the Jacquet--Rallis fundamental lemma.

The FOXO family of transcription factors have several important roles in multicellular organisms: they are required for proper development of different tissues, maintain homeostasis in response to diverse cellular stresses, function as tumor suppressors, and have an evolutionarily conserved role in prolonging lifespan. Consistent with their role in diverse cellular processes, FOXO proteins are activated by several different stimuli, leading to the promotion of many different downstream programs often with opposing outcomes. How FOXO protein activation can lead to stimulus-dependent transcriptional outcomes is not known, though several mechanisms have been proposed. Possible mechanisms include differences in FOXO post-translational modifications, binding partners, and the dynamics of FOXO activation. Here, I will describe the current evidence in the literature supporting these mechanisms, and our investigation into the role of dynamic patterns of activation of the FOXO transcription factors. Specifically, I set out to determine whether FOXO responds to different stresses with different temporal patterns of activation. I have shown that FOXO responds to oxidative stress in a sustained, bimodal pattern, while it responds to serum starvation in a stochastic pulsatile pattern. I also found that in MCF7 breast cancer cells, both patterns are controlled by the activity of the primary negative regulator of FOXO, Akt.

In condensed matter physics, when two or more sheets of graphene are twisted by certain angles, a.k.a. magic angles, the resulting material becomes superconducting. The mathematics behind this is a blend of basic representation theory, Bloch-Floquet theory, Jacobi theta functions and holomorphic line bundles. In this talk, I will compare a chiral multilayer graphene (TMG) model with the chiral twisted bilayer graphene (TBG) model studied by Tarnopolsky--Kruchkov--Vishwanath and Becker--Embree--Wittsten--Zworski to show that magic angles of TMG are the same with magic angles of TBG. I will also present a band separation due to the interlayer tunneling by setting up a Grushin problem. Finally, I will present a construction of a holomorphic line bundle with Chern number $-n$. The high Chern number band attracts many attentions in physics for its role in both integer, fractional quantum Hall effect and fractional Chern insulators.

https://arizona.zoom.us/j/85705568785

For a complex projective manifold X, a vanishing cycle is a topological sphere on a smooth hyperplane section that is contracted to a point as the hyperplane section deforms and becomes tangent to X. What is a vanishing cycle on a singular hyperplane section? We will try to answer the question in the case when X is a smooth cubic hypersurface of the complex projective four-space. In particular, we compactify the parameter space of vanishing cycles on smooth hyperplane sections and interpret the boundary points by the Hilbert scheme of X and singularities on cubic surfaces.

We present a numerical method to simulate thick elastic curves that accounts for self-contact and container constraints under large deformations (the motivating model is DNA packing). The base model includes bending, torsion, and inextensibility. A minimizing movements, descent scheme is proposed for computing energy minimizers, under the non-convex inextensibility, self-contact, and container constraints (if the container is non-convex). At each pseudo time-step of the scheme, the constraints are linearized, which yields a convex minimization problem (at every time-step) with affine equality and inequality constraints. First order conditions are established for the descent scheme at each time-step, under reasonable assumptions on the admissible set. Furthermore, under a mild time-step restriction, we prove energy decrease for the descent scheme, and show that all constraints are satisfied to second order in the time-step, regardless of the total number of time-steps taken.

We also give a modification of the scheme that regularizes the inequality constraints, and establish convergence of the regularized solution. We then discretize the regularized problem with a finite element method using Hermite and Lagrange elements. Several numerical experiments are shown to illustrate the method, including an example that exhibits massive amounts of self-contact for a tightly packed curve inside a sphere.  We also demonstrate the effect of parameter choices on packing configurations.

The speaker will be online.

Math, 402 and Zoom   https://arizona.zoom.us/j/85889389967    Password:  applied

In several areas of mathematics, including probability theory, statistics and asymptotic convex geometry, one is interested in high-dimensional objects, such as measures, data or convex bodies.  One common theme is to try to understand what lower-dimensional projections can say about the corresponding high-dimensional objects.   I will describe how this line of inquiry leads to geometric generalizations of some classical results in probability related to tails of random projections, both in commutative and non-commutative settings, and also discuss their relation to some long standing open problems in convex geometry.

Harnessing the demonstrated effectiveness of Bayesian Additive Regression Tree (BART) principles, we develop a fully Bayesian procedure to train an ensemble of small neural networks for regression tasks.  We describe how BART samples from a Bayesian posterior of an ensemble of decision trees, and then adapt this method to neural networks.  Using Markov Chain Monte Carlo, Bayesian Additive Regression Networks (BARN) samples from the space of single hidden layer neural networks that are conditioned on their fit to data.  To create an ensemble of networks, we apply Gibbs' sampling to update each network against the residual target value (i.e. subtracting the effect of the other networks).  We examine the test performance of BARN on several benchmark regression tasks, comparing it to equivalent neuron count single neural networks as well as equivalent tree count BART.  We also use BARN to model an applied problem and compare against the state of the art modeling methods for that specific domain.  BARN provides more consistent and often more accurate results, with a mean root mean square error just 5% higher than the best (or next best) method across 9 datasets with different ``best methods'' in each.  This comes at the cost of significantly greater computation time (minutes vs seconds).  But this may be surmountable with more clever programming, and errors may further shrink with a hyperparameter grid search.

Abstract: TBA

The speaker will be in-person.

https://math.mit.edu/~dunkel/

Math, 501 and Zoom:   https://arizona.zoom.us/j/81337180102 Password:  applied

The stochastic epidemic is a classic example of probabilistic modelling dating back to work of Daniel Bernoulli in 1760, if not earlier. The problem becomes much harder when the model is "spatial" and the disease results in future immunity/death.

Inspired in part by a personal encounter with Covid-19, we consider two models for the spread of infection about a population of diffusing individuals in Euclidean space. The emphasis is upon the case of post-infection immunity. Partial results for the existence (or not) of a pandemic may sometimes be proved via comparisons with branching processes and percolation processes. The principal difficulties lie in the combination of movement and immunity.

The presentation will include summaries of the percolation and contact models, and an outline of their uses in epidemic theory. (Joint work with Zhongyang Li.)

Recently, there has been a growing interest in approximating nonlinear functions and PDEs on tensor manifolds. The reason is simple: tensors can drastically reduce the computational cost of high-dimensional problems when the solution has a low-rank structure. In this talk, I will review recent developments on rank-adaptive algorithms for temporal integration of PDEs on tensor manifolds. Such algorithms combine functional tensor train (FTT) series expansions, operator splitting time integration, and an appropriate criterion to adaptively add or remove tensor modes from the FTT representation of the PDE solution as time integration proceeds. I will also present a new tensor rank reduction method that leverages coordinate flows. The idea is very simple: given a multivariate function, determine a coordinate transformation so that the function in the new coordinate system has a smaller tensor rank. I will restrict the analysis to linear coordinate transformations, which give rise to a new class of functions that we refer to as tensor ridge functions. Numerical applications are presented and discussed for linear and nonlinear advection equations, and for the Fokker-Planck equation.

SHORT BIO: Dr. Venturi is a professor of Applied Mathematics in the Baskin School of Engineering at UC Santa Cruz. He received his MS in Mechanical Engineering in 2002, and his PhD in Applied Physics in 2006 from the University of Bologna (Italy). His research activity has been recently focused on the numerical approximation of PDEs on tensor manifolds, including high-dimensional PDEs arising from the discretization of functional differential equations (infinite-dimensional PDEs).

WEBPAGE:  https://venturi.soe.ucsc.edu

Math, 402 and Zoom   https://arizona.zoom.us/j/85889389967  Password:  applied

We study reinforcement learning (RL) in a setting with a network of agents whose states and actions interact in a local manner where the objective is to find policies such that the (discounted) global reward is maximized. A fundamental challenge in this setting is that the state-action space size scales exponentially in the number of agents, rendering the problem intractable for large networks. In this paper, we present our framework that exploits the network structure to conduct reinforcement learning in a scalable manner. The key feature in our framework is that we prove spatial decay properties for the Q function and the policy, meaning their dependence on faraway agents decays when the distance increases. Such spatial decay properties enable approximations by truncating the Q functions and policies to local neighborhoods, hence drastically reducing the dimension and avoiding the exponential blow-up in the number of agents.

The speaker will be in-person.

https://engineering.cmu.edu/directory/bios/qu-guannan.html

Math, 501 and Zoom https://arizona.zoom.us/j/81337180102 Password:  applied

To design and control magnetic confinement fusion reactors, one must compute the geometry of the confined plasma at equilibrium. In this talk, I present a new method for solving this free-boundary problem through integral equations and PDE-constrained optimization. I detail a formulation of the equilibrium equations that couples the Grad-Shafranov equation for the interior magnetic field with an integral equation for the exterior vacuum field. I introduce high order numerical methods for solving the associated integral equation, including a novel application of the Kapur-Rokhlin quadrature rule for singular integrals in axisymmetric magnetic confinement systems. Finally, I frame the coupled equations in the larger PDE-constrained optimization framework and discuss gradient descent methods for the optimization iteration.

Math, 402 and Zoom   https://arizona.zoom.us/j/85889389967  Password:  applied

The discovery in 1998 of a link between the Wasserstein-2 metric, entropy, and the heat equation, by Jordan, Kinderlehrer, and Otto, precipitated the increasing relevance of optimal mass transport in the evolving theory of finite-time thermodynamics, aka stochastic energetics. Specifically, dissipation in finite-time thermodynamic transitions for Langevin models of colloidal particles can be measured in terms of the Wasserstein length of trajectories. This enabling new insight has led to quantifying power and efficiency of thermodynamic cycles that supersede classical quasi-static Carnot engine concepts that alternate their contact between heat baths of different temperatures. Indeed, naturally occurring processes often harvest energy from temperature or chemical gradients, where the enabling mechanism responsible for transduction of energy relies on non-equilibrium steady states and finite-time cycling. Optimal mass transport provides the geometric structure on the manifold of thermodynamic states for studying energy harvesting mechanisms. In this, dissipation and work output can be expressed as path and area integrals, and fundamental limitations on power and eficiency in geometric terms lead to isoperimetric problems. The analysis presented provides guiding principles for building autonomous engines that extract work from thermal or chemical anisotropy in the environment. Based on joint work with Olga Movilla Miangolarra, Amirhossein Taghvaei, Rui Fu, and Yongxin Chen

Bio:  Tryphon T. Georgiou was educated at the National Technical University of Athens, Greece (Diploma 1979) and the University of Florida, Gainesville (PhD. 1983).  He is a Distinguished Professor at the Department of Mechanical and Aerospace Engineering, University of California, Irvine, and Professor Emeritus at the University of Minnesota. He is a Fellow of IEEE, SIAM, IFAC, AAAS and a Foreign Member of the Royal Swedish Academy of Engineering Sciences (IVA).

https://georgiou.eng.uci.edu/    

The speaker will be in-person. 

Math, 501 and Zoom https://arizona.zoom.us/j/81337180102    
Password:  applied   

For details, see https://science.arizona.edu/academics/graduation-convocation