# Set theory and the continuum hypothesis

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This talk is about two prominent axioms of set theory which were introduced independently from one another in the late 80's/early 90's by Foreman-Magidor-Shelah and Woodin and which both decide the size of the continuum in the same way. Answering a long standing question, in a 2021 Annals paper D. Asperó and the speaker showed that these two axioms of set theory are compatible, in fact one implies the other. Both axioms are so-called forcing axioms which are also exploited in topology, algebra, the theory of operator algebras, and elsewhere. I am going to provide a soft hand, accessible introduction to our result.

# Seeing Through Space-Time

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The inverse problem we address is whether we can determine the structure of a region in space-time by measuring point light sources coming from the region. We can also observe gravitational waves since the LIGO detection in 2015. We will also consider inverse problems for nonlinear hyperbolic equations, including Einstein's equations, involving active measurements.

# Piecewise Polynomials by Neural Networks and Finite Elements

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Polynomials and piecewise polynomials are most commonly used function classes in analysis and applications. In this talk, I will use these function classes to motivate and to present some remarkable properties of the function classes given by (both shallow and deep) neural networks. In particular, I will present some recent results on the connections between finite element and neural network functions and comparative studies of their approximation properties and applications to numerical solution of partial differential equations of high order and/or in high dimensions.

Meeting ID: 930 9003 8527

Passcode: 215830

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https://uci.zoom.us/j/93090038527?pwd=cGJaT0g2U1JKUHEvT1FuS1RaSldsdz09

# Quasiperiodicity and quasiintegrability

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We will discuss the critical almost Mathieu operator: Azbel/Hofstadter/Harper model of an electron on the square lattice in a magnetic field. When the commensurability parameter between the lattice and the magnetic field is irrational, the spectrum of the model is a zero-measure Cantor set and its Hausdorff dimension is not larger than 1/2. We will emphasize the significance of the two-dimensionality of the problem, which was used in recent work of the speaker with S. Jitomirskaya. We will also discuss some similarities with integrable two-dimensional statistical models: the Ising model and the dimer problem.

# Spectrum and curvature

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The classical de Rham-Hodge theory implies that each cohomology

class of a compact manifold is uniquely represented by a harmonic

form, signifying the important role of Laplacian in geometry. The talk aims

to explain some results relating curvature to the spectrum of Laplacian. We

plan to start by a brief overview for the case of bounded Euclidean domains

and compact manifolds, highlighting some of the fundamental contributions by

Peter Li and others. We then shift our focus to the case of complete

Riemannian manifolds. In particular, it includes our recent joint work with

Ovidiu Munteanu concerning the bottom spectrum of 3-manifolds with scalar

curvature bounded below.

# Laplace eigenfunctions and frequency of solutions to elliptic PDEs

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A classical idea in the study of eigenfunctions of the Laplace-Beltrami operator is that they behave like polynomials of degree corresponding to the eigenvalue. We will discuss several properties of eigenfunctions which confirm this idea, including the Bernstein and Remez inequalities. As a corollary, we will formulate a local version of the celebrated Courant theorem on the number of nodal domains of eigenfunctions. The proofs of the inequalities rely of the frequency function of solution to elliptic PDEs. In the talk, we will also review some striking properties of this frequency function.

# Playing Games with Entanglement Assistance

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There are many cooperative games which can be won with higher probability if the players are able to access quantum resources. In fact for some games, the players can have very small probability of winning with classical strategies, but the game can be won with probability one with quantum assistance.

The theory of these games has recently been used to solve the Connes Embedding Problem, which had been open since the 1970's, and has been used to show that the mathematical models for describing quantum correlations are all different.

In this talk we introduce these ideas and focus on the family of synchronous games. For synchronous games there is an algebra whose representation theory determines whether or not they can be won with probability one.

This talk will be accessible to anyone with a basic knowledge of operators on a Hilbert space.

# The Polynomial Method and its Algorithmic Aspects

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I will show how basic properties of polynomials can be used

in the study of problems in Combinatorics, Additive Number Theory,

Combinatorial Geometry and Graph Theory, describing recent and

less recent results, problems and algorithmic challenges.

# On a fluid-poroelastic structure interaction problem motivated by the design of a bioartificial pancreas

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The work reported here has been motivated by the design of lab-grown organs, such as a bioartificial pancreas. The design of lab-grown organs relies on using biocompatible materials, typically poroelastic hydrogels, to generate scaffolds to support seeded cells of different organs. Additionally, to prevent the patient's own immune cells from attacking the transplanted organ, the hydrogel containing seeded cells is encapsulated between two semi-permeable, nano-pore size membranes/plates and connected to the patient's vascular system via a tube (anastomosis graft). The semi-permeable membranes are designed to prevent the patient's own immune cells from attacking the transplant, while permitting oxygen and nutrients carrying blood plasma (Newtonian fluid) to reach the cells for long-term cell viability. A key challenge is to design a hydrogel with ``roadways'' for blood plasma to carry oxygen and nutrients to the transplanted cells.

We present a complex, multi-scale model, and a first well-posedness result in the area of fluid-poroelastic structure interaction (FPSI) with multi-layered structures modeling organ encapsulation. We show global existence of a weak solution to a FPSI problem between the flow of an incompressible, viscous fluid, modeled by the time-dependent Stokes equations, and a multi-layered poroelastic medium consisting of a thin poroelastic plate and a thick poroelastic medium modeled by a Biot model. Numerical simulations of the underlying problem showing optimal design of a bioartificial pancreas, will be presented. This is a joint work with bioengineer Shuvo Roy (UCSF), and mathematicians Yifan Wang (UCI), Lorena Bociu (NCSU), Boris Muha (University of Zagreb), and Justin Webster (University of Maryland, Baltimore County).