Abstract List

Ivana Alexandrova, University at Albany SUNY
Semi-Classical-Fourier-Integral-Operator-Valued Pseudodifferential Operators and Scattering in a Strong Magnetic Field
Abstract: We analyze the microlocal structure of the semi-classical scattering amplitude for Schrodinger operators with a strong magnetic and a strong electric fields at non-trapping energies. For this purpose we develop a framework and establish some of the properties of semi-classical-Fourier-integral-operator-valued pseudodifferential operators and prove that the scattering amplitude in this setting is such an operator.

Nadya Askaripour, University of Toronto Mississauga
Approximation by Meromorphic k-differentials
Abstract: Runge theorem is one of the main theorems in complex approximation: we can approximate a holomorphic function on a compact subset of the complex plane with meromorphic functions on the complex plane. In this talk I will explain a generalization of Runge theorem for k-differentials (or automorphic forms) which I proved in a submitted work. There will be an introduction about automorphic forms and the techniques applied. 

Marius Beceanu
Spectral Multipliers and Estimates
Abstract: This talk pertains to some recently posted results about spectral multipliers for -\Delta+V, where V is a scalar potential in an optimal or almost optimal class of potentials. The results are used to prove new estimates for some well-known partial differential equations. All results are in three space dimensions. This is joint work with Gong Chen and, separately, Michael Goldberg.

Jens Christensen, Colgate University
Self-adjoint Differential Operators on Bergman Spaces
Abstract: This talk is concerned with self-adjoint first-order differential operators on Bergman spaces on the unit ball. It turns out that they all arise from the holomorphic discrete series representation of $SU(n,1)$. We also discuss uncertainty principles for such differential operators. This is joint work with Christopher Deng who recently graduated from Colgate University.

Terence Harris, Cornell University
A Euclidean Fourier Analytic Approach to Vertical Projections in the Heisenberg Group
Abstract: I will give a brief overview of projection theorems for Hausdorff dimension in general, and then I will outline a Fourier analytic approach to vertical projections in the Heisenberg group, which answers a question of Fässler and Hovila on almost sure Hausdorff dimension distortion of sets under vertical projections. This approach uses the Euclidean Fourier transform, Basset's integral formula, and modified Bessel functions of the second kind.

Jiho Hong, Korea Advanced Institute of Science and Technology (KAIST)
Inequalities on Laplacian Eigenvalues in Two Dimensions by Holomorphic Coordinate Transformations
Abstract: In this talk, we address the resonance of vibration in two dimensions. In particular, we derive inequalities on two types of eigenvalues: the first Steklov--Dirichlet eigenvalues for eccentric annuli and membrane eigenvalues for simply connected domains with smooth boundaries. We also present numerical computation schemes based on the inequalities. This talk is based on a joint work with Mikyoung Lim and Dong-Hwi Seo and a joint work with Mikyoung Lim.

Jesse Hulse, Syracuse University
The Desingularization of the Cauchy Kernel in Bounded Convex Domains
Abstract: One physical application of the Cauchy integral formula appears in fluid dynamics. However, the Cauchy kernel becomes numerically unstable near the boundary of the given domain. We will present a new technique to desingularize the Cauchy kernel for bounded convex domains which is inspired by Fokas' celebrated Unified Transform method for convex polygons. Time permitting, I will show one application. This is joint work with L. Lanzani (Syracuse U.), S. Lewellyn Smith (UCSD) and E. Luca (U. College London).

Phanuel Mariano, Union College
Improved Upper Bounds for the Hot Spots Constant of Lipschitz Domains
Abstract: In this talk we discuss the Hot Spots constant for bounded smooth domains that was recently introduced by S. Steinerberger as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We use probabilistic techniques to derive a general formula for a dimension-dependent upper bound that can be tailored to any specific class of bounded Lipschitz domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in $\mathbb{R}^{d}$ for both small and asymptotically large $d$ that significantly improve upon the existing results. This is joint work with Hugo Panzo and Jing Wang.

Gabriel Prajitura, SUNY Brockport
Some Open Problems in Linear Dynamics
Abstract: We will discuss some open problems in linear dynamics that we consider important for the future of the field. The selection is rather personal. This is joint work with Gabriela Ileana Sebe from Polytechnic University of Bucharest & The Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy.

Mikolaj Sierzega, Cornell University
On optimal Harnack bounds for a fractional heat equation
Abstract: In this talk I will present a recent sharp result concerning Harnack bounds for nonnegative solutions of a fractional heat flow with a half-laplacian on the real line. While solutions of the standard heat flow enjoy only lower bounds, the non-local setting permits both lower and upper bounds. Time permitting, I will touch on the possible implications that these bounds may have on the problem of extending the classical Li-Yau estimate associated with the standard heat flow to non-local settings.

Avy Soffer, Rutgers
The Asymptotic States of Nonlinear Dispersive Equations with Large Initial Data and General Interactions
Abstract: I will describe a new approach to scattering theory, which allows the analysis of interaction terms which are linear and space-time dependent, and nonlinear terms as well. This is based on deriving (exterior) propagation estimates for such equations, which micro-localize the asymptotic states as time goes to infinity. In particular, the free part of the solution concentrates on the propagation set (x=vt), and the localized leftover is characterized in the phase-space as well.
The NLS with radial data in three dimensions is considered, and it is shown that besides the free asymptotic wave, in general, the localized part is smooth, and is localized in the region where |x|^2 is less than t. Furthermore, the localized part has a massive core and possibly a halo which may be a self-similar solution.
This work is joint with Baoping Liu.
This is then followed by new results on the non-radial case and Klein-Gordon equations (Joint works with Xiaoxu Wu).

Mihai Stoiciu, Williams College
The Eigenvalue Distribution for Random Unitary Matrices: An Approach Using Entropy
Abstract: We consider CMV matrices with independent random Verblunsky coefficients. The microscopic eigenvalue distribution of the random unitary matrix depends on the rate of decay of the variance of their Verbunsky coefficients: Poisson for slow decay and “picket fence" (clock) for fast decay. We investigate the transition Poisson-Clock from the perspective of the entropy of the random Verblunsky coefficients.

Khoi Vo, University of California, Riverside
Preliminary Report on Symmetric and Asymmetric Cell Division and Modeling of Interacting Cell Populations in the Colonic Crypt: an Application of PDE in Biological System.
Abstract: Mathematical modeling can be used to describe the behavior of cells within the colonic crypt. The colon is made up of nearly 10 million crypts which are responsible for producing the epithelial cells within the colon. Symmetric and asymmetric stem cells and cycling cells produce the cells within the crypt and when this behavior becomes dysregulated it can lead to the development of colorectal cancer. This model aims to make a simple spatial and time dependent model to describe the behavior of two types of cells within the colon. Both analytic and numerical solutions are presented for a range of initial conditions and time points. The model is then expanded for stochastic analysis to further examine the spatial relationships among the cell types.

Jianxiong Wang, University of Connecticut
Symmetry of Solutions of Higher Order and Fractional Order Semilinear Equations on Hyperbolic Spaces
Abstract: We show that nontrivial solutions to the higher order equations with certain nonlinearity is radially symmetric and non-increasing on bounded domains in the hyperbolic space $\mathbb{H}^n$ as well as on the entire space. Applying Helgason-Fourier analysis on $\mathbb{H}^n$, we developed a moving plane approach for integral equations on $\mathbb{H}^n$. Besides, we established the symmetry of solutions of certain equations with singular terms on Euclidean spaces. Moreover, we obtained symmetry of solutions of some semilinear equations involving fractional order derivatives.

Sijue Wu, University of Michigan
Mathematical Analysis of Water Wave Motion
Abstract: Wave phenomena on water surfaces are one of our most familiar experiences. It can be observed at the beach, on the river, in the ocean, and also in our wash basin. Mathematics provides us tools to understand the wave motion in precise terms. I will explain some of the recent results mathematicians obtained regarding the behavior of the water wave motion, explain the basic tools used leading to these results, and present some open problems.
The first talk will be focused on the local existence results and the second on the long time behavior of the water waves.

Rongwei Yang, University at Albany SUNY
Maxwell's Equations and Yang-Mills Equations in Complex Variables
Abstract: This expository talk first provides a view of Maxwell's equations from the perspective of complex differential forms and Hodge star operator. The electric field and the magnetic field are complex 3-dimensional in this case. We will see that holomorphic functions naturally give rise to nontrivial solutions to Maxwell's equations. The discussion extends to Yang-Mills (YM) equations, where we will take another look at YM-Lagrangian, YM functional as well as the Belavin-Polyakov-Schwartz-Tyupkin instanton solution.
Reference: S. Munshi and R. Yang, Complex solutions to Maxwell's equations}, Complex Analysis and its Synergies (2022) 8:2. https://doi.org/10.1007/s40627-022-00091-6

Ruhan Zhao, SUNY Brockport
Berezin Type Operators and Toeplitz Operators on Bergman Spaces
Abstract: We introduce a class of integral operators called Berezin type operators. It is a generalization of the Berezin transform, and has close relation to the Bergman-Carleson measures. We mainly study the boundedness and the compactness of Berezin type operators from a weighted Bergman space to a weighted Lebesgue space on the unit ball of ${\mathbb C}^n$. We also show that Berezin type operators are closely related to Toeplitz operators. This is a joint work with Gabriel Prajitura and Lifang Zhou.

Gang Zhou, Binghamton University
On the generic singularities formed by mean curvature flow
Abstract: In this talk I will discuss the progress we made in the past few years. By our new techniques, in various cases, we can study a fixed neighborhood of generic singularities formed by mean curvature flow. This includes joint works with Knopf, Sigal and other people.