Lobachevsky started to work on computing volumes of hyperbolic polytopes long before the first model of the hyperbolic space was found. He discovered an extraordinary formula for the volume of an orthoscheme via a particular function called dilogarithm. We will discuss a generalization of the formula of Lobachevsky to higher dimensions. For reasons I do not fully understand, a mild modification of this formula leads to the proof of a conjecture of Goncharov about the depth of multiple polylogarithms. Moreover, the same construction leads to a functional equation for polylogarithms generalizing known equations of Abel, Kummer, and Goncharov.

***Distinguished Visitor Colloquium***

Resolving a conjecture of Erdos and Turan from the 1930's, in the 1970's Szemerédi showed that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon thereafter, Furstenberg used ergodic theory to give a new proof of this result, leading to the development of combinatorial ergodic theory. These tools have led to uncovering new patterns that occur in any sufficiently large set of integers, but until recently all such patterns have been finite. Based on joint work with Joel Moreira, Florian Richter, and Donald Robertson, we discuss recent developments for infinite patterns, including the resolution of a conjecture of Erdos. 

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.

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

Join Zoom Meeting

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

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.