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.
***Special Dynamical Systems and Ergodic Theory Seminar***
Furstenberg's proof of Szemeredi's theorem introduced the Correspondence Principle, a general technique for translating a combinatorial problem into a dynamical one. While the original formulation suffices for certain patterns, including arithmetic progressions and some infinite configurations, higher order generalizations have required refinements of these tools. We discuss the new techniques introduced in joint work with Moreira, Richter, and Robertson that are used to show the existence of infinite patterns in large sets of integers.
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.
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.
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.
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.
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.
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.