Date: 09/06/2023
Speaker: Lei Wu (Lehigh University)
Title: Continuum Limit of Ablowitz-Ladik System
Abstract: Ablowitz-Ladik (AL) is a well-known completely integrable equation which acts as discretization of the nonlinear Schrodinger equation (NLS). In this talk, I will focus on the continuum limit of AL to NLS under the low-regularity settings. The proof mainly relies on the compactness argument and the conservation laws. Also, we introduced the localized dispersive estimates and Strichartz estimates.
Date: 09/13/2023
Speaker: Joe Kramer-Miller (Lehigh University)
Title: Recent developments in geometric Iwasawa theory
Abstract: Iwasawa theory is a central topic in modern algebraic number theory. It studies the variation of arithmetic invariants along certain families of number fields (i.e. finite extensions of the rational numbers). Geometric Iwasawa theory aims to study invariants along families of curves (or higher dimensional objects) over finite fields. Most work on geometric Iwasawa theory has only considered arithmetic invariants. However, even the simplest examples show that arithmetic invariants only give a small picture of the rich geometric structures in play. Recently, there has been a great deal of work on geometric Iwasawa theory using finer geometric invariants. In this talk, I will give a brief overview of classical Iwasawa theory for number fields as well as the "arithmetic invariant" approach to geometric Iwasawa theory. Then I will describe recent work and conjectures on geometric invariants.
Date: 09/20/2023
Speaker: Xin Zhou (Cornell University)
Title: Some recent development in minimal surface theory
Abstract: We will present some recent progress on two problems in minimal surface theory posed by S. T. Yau in 1982. In particular, we will discuss the existence of infinitely many closed minimal hypersurfaces in a closed Riemannian manifold and the existence of four closed minimal two-spheres in a Riemannian three-sphere.
Date: 09/27/2023
Speaker: Liangbing Luo (Lehigh University)
Title: Logarithmic Sobolev inequalities on homogeneous spaces
Abstract: The logarithmic Sobolev inequality has been first introduced and studied by L. Gross on a Euclidean space, and since then it found many applications. The fact that the logarithmic Sobolev constant often does not depend on the dimension makes it applicable in infinite-dimensional settings. In this talk, we consider sub-Riemannian manifolds which are homogeneous spaces equipped with a natural sub-Riemannian structure. In such a setting, the corresponding sub-Laplacian is not an elliptic but a hypoelliptic operator. Logarithmic Sobolev inequalities with respect to the hypoelliptic heat kernel measure on such homogeneous spaces are studied. We show that the logarithmic Sobolev constant only depends on the Lie group that acts transitively on such a homogeneous space but the constant is independent of the action of its isotropy group. In some concrete settings, this method will allow us to track the (in)dependence of the logarithmic Sobolev constant on the geometry of the underlying space, especially the dimension-independence of the constant. Several examples will be provided.
Date: 10/04/2023
Speaker: Erdogan Madenci (University of Arizona)
Title: Multiphysics Simulations Using Peridynamics Differential Operator
Abstract: Peridynamics (PD) converts the existing governing field equations from their local to nonlocal form through the PD differential operator while introducing an internal length parameter. The PD differential operator enables differentiation through integration. As a result, the equations become valid everywhere regardless of discontinuities. The lack of an internal length parameter in the classical form of the governing equations is the source of problems when addressing crack initiation and propagation. Although PD is extremely suitable to model the response of structures involving discontinuities, fractures, and other complex phenomena, it is also applicable to other fields, including thermal diffusion, moisture diffusion, electric potential distribution, vacancy diffusion, and neutronic diffusion in either an uncoupled or a coupled manner. This presentation provides an overview of the PD concept, the derivation of the PD differential operator, and multiphysics applications by considering different field equations coupled among themselves or coupled with the deformation field such as hygrothermomechanics, hydromechanics, corrosion and electrodeposition and electromigration.
Date: 10/11/2023
Speaker: Ludovic Sacchelli (Nice)
Title: A discontinuous approach to the stabilization of unobservable systems
Abstract: Control theory in finite dimension is the study of ODEs subject to dynamics that admit time-dependent functions as parameters, known as controls or inputs. Feedback control focuses on the design of input functions not as depending on time per se but rather on the state of the solution of the ODE. In that way, the dynamical system becomes autonomous, which may allow us to create a stable equilibrium. Yet, in many applications it is simply not possible to assume that the state of a dynamical system is known. In general it is only partially measured. A popular approach is then to provide the feedback controller with an online estimate of the state. This requires solving in parallel a sort of inverse problem that will recover the state of the system from the knowledge of the partial measurement. However it may happen that the inverse problem has a unique solution only some of the time, a case we call unobservable. For a long time, the popular approach was to design ad-hoc methods to make sure that a system never became unobservable. Recently, there has been a renewed interest for strategies that can simultaneously manage the two competing goals of estimation and control. I’m going to present some recent advances that rely on the introduction of discrete events and observability monitoring to overcome these challenges.
Date: 10/18/2023
Speaker: Taeho Kim (Lehigh University)
Title: Prediction via Maximum Agreement
Abstract: While Karl Pearson’s correlation coefficient (PCC) assesses the degree of linear association between two variables or paired values of two quantitative features, there exists another measure known as Lin’s Concordance Correlation Coefficient (CCC). The CCC quantifies the degree of agreement between two variables or paired values of two quantitative features. One of the fundamental challenges in statistics, machine learning, and many scientific disciplines is the problem of prediction. This problem involves utilizing the values of a set of features to predict the value of a specific target feature. In this presentation, we will show predictors designed to maximize agreement, as measured by the CCC, between the predictor and the predictand. We will also compare these Maximum Agreement Predictors (MAP) with the widely recognized Least-Squares Predictor (LSP), which minimizes mean-squared prediction error or maximizes the PCC between the predictor and the predictand. Throughout our discussion, we will provide finite and asymptotic properties of these predictors. To illustrate their practical application, we will employ two real datasets: one related to eye data and the other concerning body fat data.
Date: 10/25/2023
Speaker: Jonathan Boretsky (Harvard)
Title: Flag positroids and Bruhat Order
Abstract: Positroids are a rich combinatorial object. We will see how they can help us to simplify a topological problem about the Grassmannian. Positroids are in bijection with a number of sets of objects and we will discuss how a few of these connect to other areas of math. A natural way to build on the theory of positroids is by studying flag positroids. We will see that many results about positroids also hold for flag positroids. However, we will also highlight a result that is specific to flag positroids by explaining a surprising connection between flag positroids and the Bruhat order on the symmetric group. This talk will feature results from joint work with Chris Eur and Lauren Williams.
Date: 11/01/2023
Speaker: Andre Neves (University of Chicago)
Title: Recent progress on Minimal surfaces
Abstract: We will talk about recent progress regarding the existence and asymptotic behavior of minimal surfaces.
Date: 11/08/2023
Speaker: George Daskalopoulos (Brown University)
Title: Analytic questions arising in the theory of best Lipschitz maps.
Abstract: After I give a general exposition, I will describe several open problems arising in Thurston's theory of stretch maps and geodesic laminations. This is joint work with Karen Uhlenbeck.
Date: 11/15/2023
Speaker: Michael Gekhtman (Notre Dame)
Title: Five Glimpses of Cluster Algebras
Abstract: Cluster algebras were introduced by Fomin and Zelevinsky just over 20 years ago and have since found exciting applications in many areas including algebraic geometry, representation theory, integrable systems and theoretical physics. I will use examples to explain the definition of a cluster algebra and then sketch several applications of the theory, including Somos-5 recursion, pentagram map and generalizations of Abel's pentagon identity. (Portions of this presentation will also be of interest to upper-level undergraduate students.)
Date: 11/29/2023
Speaker: Andrew Harder (Lehigh)
Title: Periods and Feynman integrals
Abstract: Amplitudes are important quantities in quantum field theory that allow one to study the behavior of interacting subatomic particles. They can be written as sums of integrals of many variables called Feynman integrals. Physicists have studied these integrals for many decades. It is well known that the values that appear have a lot of structure, and that they are often related to interesting mathematical quantities. More recently, it has been observed that Feynman integrals can be interpreted as periods in algebraic geometry. This opens the door to applying the toolbox of algebraic geometry to describe the structure of Feynman integrals. I'll talk about how one can use this relationship to show that the integrals attached to an infinite class of planar 2-loop Feynman graphs are built from hyperelliptic and algebraic functions.