PDF of Schedule and Abstract (updated 04/25)
Title: The Dirichlet Problem in Rough Domains
Abstract:
In this talk, we will review the Dirichlet problem for second-order, linear, divergence form elliptic operators. We will discuss some classical and recent well-posedness results for domains that satisfy very weak regularity conditions.
Title: The Quantitative Geometry of Geodesics
Abstract:
The goal of quantitative geometry is to provide effective versions of known existence theorems for geometric objects. For example, following Serre's proof of the existence of infinitely many geodesics connecting any two points on a closed Riemannian manifold, one may attempt to prove a length bound for these geodesics. In this talk, we will provide a survey of current quantitative theorems concerning geodesics and explore how such results can be proven. In particular, we discuss recent work on the existence of "short" geodesics that meet a given submanifold orthogonally.
Title: Radio Graceful Labelings of Graphs
Abstract:
Radio graceful labeling of graphs originates from the Channel Assignment Problem, seeking to assign frequencies to transmitters in a way that avoids interference between the transmitters. Through graph homomorphisms, radio graceful and circular radio graceful labelings are closely related to two special families of graphs, namely, distance graphs and circulant graphs which have been studied by many. In this talk, we investigate similar properties for graphs with a radio graceful labeling, providing upper bounds for diameter, clique size, and number of edges. We extend these results to graphs with a circular radio graceful labeling.
Title: Multiscale Hybrid and Stochastic Modeling of Complex Biological Systems
Abstract:
Computational biology plays a critical role in understanding the dynamic behavior of biological systems across scales. While both discrete and continuous modeling approaches have led to significant advances in representing subcellular processes, physiological dynamics, and developmental biology, traditional single-scale models often fall short in capturing the intricate interactions that span multiple biological levels. In this talk, I will present a novel 3D hybrid modeling framework designed to bridge these gaps by integrating multiple scales of biological complexity. The framework incorporates: (1) a discrete stochastic Subcellular Element Model to capture individual cell mechanics and behavior; (2) continuum partial differential equations to describe the reaction, diffusion, and advection of extracellular biomolecules; and (3) a stochastic decision-making model for simulating cell lineage transitions. Through case studies involving intestinal crypt organization, embryonic development, and epidermal tissue regeneration, I will demonstrate how this multiscale modeling approach yields deeper mechanistic insights and predictive capabilities for complex biological phenomena.
Title: Hybrid Models for Inverse Problems in Scientific Imaging
Abstract:
Imaging plays a vital role in basic science, but extracting scientific insights from costly experiments is often challenging. Quantities of interest usually cannot be measured directly, and therefore, the problem is mathematically formulated as an inverse problem, associating the available measurements (input) to the quantities of interest (solution) via a forward model, a mathematical model of the measurement process. In practical terms, it is rarely feasible to take enough measurements to uniquely determine the solution, resulting in an under-determined, ill-posed problem with a potentially infinite number of solutions. A fruitful approach to deal with this ambiguity is to introduce a regularization function that encodes prior knowledge and allows to constrain the solution to satisfy desirable properties. In this talk we review some of the machine learning (ML) tools developed for regularizing inverse problems and discuss new techniques that combine data-driven models and simulations to enable quantitative scientific image reconstruction.
Title: Linear and Nonlinear Dispersive Equations in Domains with a Boundary
Abstract:
A plethora of physical phenomena are modeled by partial differential equations (PDEs). In the case of waves in water, optical fibers, Bose-Einstein condensates, or other media, one encounters the phenomenon of dispersion, namely waves of different frequencies propagating at different speeds. The associated PDEs are referred to as dispersive, and their analysis has been at the center of interest within the broader PDE/harmonic analysis/nonlinear waves communities during the past fifty years. In this talk, we will emphasize the importance of studying dispersive PDEs in the presence of nonzero boundary conditions. Such conditions are directly motivated by applications that take place in domains with a boundary, either in nature or in the laboratory. The relevant problems are known as initial-boundary value problems and their study turns out to be significantly more involved than the one of the more standard initial value problems, which take place on fully unbounded domains. Fundamental dispersive PDEs like the nonlinear Schrödinger and the Korteweg-de Vries equations will serve as motivating examples in order to guide us through the main steps of the analysis.
Title: Additive Properties of the Hausdorff Metric
Abstract:
For a compact set A ⊂ R^n, the Hausdorff distance to convex hull is defined by d(A) := d_H(A, conv(A)), where d_H is the Hausdorff metric. A notable conjecture in this area is the Dyn-Farkhi conjecture, which posits that d^2 is subadditive on compact sets in R^n. In 2018, Fradelizi, Madiman, Marsiglietti and Zvavitch disproved the Dyn-Farkhi conjecture when n ≥ 3, but the question remained open in R^2. Recent work has found that the Dyn-Farkhi conjecture is an affirmative in R^2. In this talk, I will outline the proof that d^2 is subadditive on compact sets in R^2.
Title: On the Philosophy of Geometric Harmonic Analysis
Abstract:
This talk is a journey into the five volume series Geometric Harmonic Analysis, joint work with Dorina Mitrea and Marius Mitrea, in which we develop a brand of Harmonic Analysis of definite geometric flavor and we build the necessary machinery capable of dealing with problems involving Partial Differential Equations in very general geometric and analytic settings. The goal is to emphasize the philosophy underpinning this work showing how the PDE, the Geometry, and the Functional and Harmonic Analysis interact with one another and ultimately come together to create a powerful, effective, and forever renewing mechanism for the study of second and higher-order elliptic boundary value problems.
Title: Functional Stochastic Volatility Models and Change Point Detection: An FDA Perspective
Abstract:
This talk presents our recent research at the intersection of Functional Data Analysis (FDA) and Stochastic Differential Equations (SDEs). After a brief introduction to FDA, we will highlight novel contributions that enhance the FDA toolbox for better adaptability to dynamic and stochastic systems. Key applications arise in finance and economics.
We introduce a stochastic volatility model for time series of curves. It is motivated by the dynamics of intraday price curves that exhibit both between days dependence and intraday price evolution. In this context, we study intraday curves, with one curve per trading day. We establish the properties of the model and propose several approaches to its estimation.
Next, we explore applications of these methods in developing testing procedures for detecting the presence of a change point in the intraday volatility pattern. The proposed methods test for changes in shape, magnitude, and overall structure, and are supported by asymptotically correct sizes and consistent estimators.
Title: Structure Identification and Learning for Ultra-High-Dimensional Generalized Partially Linear Spatially Varying Coefficient Models
Abstract:
In this paper, we build a flexible generalized partially linear spatially varying coefficient model (GPLSVCM) in the ultra-high dimension, which accounts for both constant and varying effects of covariates. GPLSVCM is significant for accurately capturing and interpreting spatially varying relationships. It utilizes the bivariate penalized splines over triangulation for the coefficient estimation and adaptive group LASSO techniques for structure identification in ultra-high dimensional data analysis. Under mild assumptions, the proposed procedure is consistent in both variable selection and the separation of varying and constant coefficients with established convergence rates for estimators and asymptotic normality for the constant coefficient estimators. Simulations and an application to the New York City crime dataset validate the theoretical results.
Title: Dynamics of Epidemics on Random Graphs
Abstract:
We discuss the spread of infectious diseases on random graphs through various epidemic models, including SIS, SIR, and SIRS. Our focus is on understanding key dynamical properties such as survival time, phase transitions, and the long-term behavior of infections in these networks. We examine how the underlying graph structure influences the spread, persistence, and extinction of epidemics. This talk is based on joint works with Bhamidi, Lam, Nam, Sly, and Yang.
Title: Virtual Unknotting Numbers of Families of Virtual Torus Knots
Abstract:
A virtual torus knot K sits charmingly in the intersection of the well-understood torus knot and the not-so-well-understood virtual knot, making it an intriguing object to study!
The unknotting number of a knot K is defined unambiguously when K is a classical knot. However, "the" unknotting number of a virtual knot is not as clear to define, since virtual knots have both classical and virtual crossings. We will define virtual unknotting number vu(K) as the minimum number of (classical) crossing changes required to unknot K. Under this definition of virtual unknotting, not all virtual knots can be unknotted. We call those which can be unknotted virtually nullhomotopic.
Working under this same definition of vu(K), Masaharu Ishikawa and Hirokazu Yanagi establish bounds on vu(K) for a particular family of virtual torus knots which they construct, making use of the u-invariant u(K) and P-invariant P(K). We similarly make use of these tools, among others, in pursuit of virtual unknotting information on families of virtual torus knots.
Our findings include families of virtually nullhomotopic virtual torus knots with virtual unknotting number computed or bounded, families of virtual torus knots which are not virtually nullhomotpic, and observations on the relationship between the P-invariant and the size of the virtual torus knot.
Title: Domain Decomposition Methods and Some Applications
Abstract:
Domain decomposition methods (DDMs) reduce large problems into collections of smaller problems and most of these smaller problems can be solved in parallel. These algorithms have been widely used for large-scale simulation on massively parallel computers. In this talk, two classes (overlapping and non-overlapping) of the DDMs will be introduced and their applications to various partial differential equations (PDEs) with different discretization methods will be provided. Some recent applications of the DDMs to the neural network training for solving PDEs will be discussed.
Title: The Contact Cut Graph and a Weinstein L-invariant
Abstract:
The cut complex associated to a surface is a powerful tool in the study of smooth 4-manifolds. In this talk, I will introduce the cut complex and its analogue in the contact and symplectic setting. Adding a restriction to the allowable curves in a cut system, the contact cut graph is a subgraph of the cut complex. I will go through some recent results exploring the structure of the contact cut graph, and use it to define a new invariant of 4-dimensional Weinstein domains. This is based on joint work with Castro, Islambouli, Min, Sakalli, and Starkston.