Table of Contents

Fall 2025

* August 20th, Wednesday (4:00-5:00pm)

Speaker :
Topic: organizational meeting



* September 10th, Wednesday (4:00-5:00pm)

Speaker : Rohan Sarkar(Binghamton)
Topic: Spectrum of Lévy-Ornstein-Uhlenbeck semigroups on $R^d$

Abstract: We investigate spectral properties of Markov semigroups associated with Ornstein-Uhlenbeck (OU) processes driven by Lévy processes. These semigroups are generated by non-local, non-self-adjoint operators. In the special case where the driving Lévy process is Brownian motion, one recovers the classical diffusion OU semigroup, whose spectral properties have been extensively studied over past few decades. Our main results show that, under suitable conditions on the Lévy process, the spectrum of the Lévy-OU semigroup in the $L^p$-space weighted with the invariant distribution coincides with that of the diffusion OU semigroup. Furthermore, when the drift matrix $B$ is diagonalizable with real eigenvalues, we derive explicit formulas for eigenfunctions and co-eigenfunctions. A key ingredient in our approach is intertwining relationship: we prove that every Lévy-OU semigroup is intertwined with a diffusion OU semigroup, thereby preserving the spectral properties.




* September 17th, Wednesday (4:00-5:00pm)

Speaker : Ziyao Xu (Binghamton)
Topic: A Conservative and Positivity-Preserving Discontinuous Galerkin Method for the Population Balance Equation

Abstract: We develop a conservative, positivity-preserving discontinuous Galerkin (DG) method for the population balance equation (PBE), which models the distribution of particle numbers across particle sizes due to growth, nucleation, aggregation, and breakage. To ensure number conservation in growth and mass conservation in aggregation and breakage, we design a DG scheme that applies standard treatment for growth and nucleation, and introduces a novel discretization for aggregation and breakage. The birth and death terms are discretized in a symmetric double-integral form, evaluated using a common refinement of the integration domain and carefully selected quadrature rules. Beyond conservation, we focus on preserving the positivity of the number density in aggregation-breakage. Since local mass corresponds to the first moment, the classical Zhang-Shu limiter, which preserves the zeroth moment (cell average), is not directly applicable. We address this by proving the positivity of the first moment on each cell and constructing a moment-conserving limiter that enforces nonnegativity across the domain. To our knowledge, this is the first work to develop a positivity-preserving algorithm that conserves a prescribed moment. Numerical results verify the accuracy, conservation, and robustness of the proposed method.



* September 24th, Wednesday (4:00-5:00pm)(Rosh Hashanah)

Speaker : Rosh Hashanah break
Topic:

* October 1st, Wednesday (4:00-5:00pm) (Yom Kippur)

Speaker: Yom Kippur break
Topic:

* October 8th, Wednesday (4:00-5:00pm)

Speaker: Prof. Lixin Shen (Syracuse University)

Topic: Explicit Characterization of the $\ell_p$ Proximity Operator for $0<p<1$

Abstract: The nonconvex $\ell_p$ quasi-norm with $0<p<1$ is a powerful surrogate for sparsity but complicates the evaluation of proximal maps that underpin modern algorithms. In this talk we give an explicit characterization of the scalar proximal operator of $|\cdot|^p$ for all $0<p<1$, including the structure and admissible ranges of global minimizers and conditions ensuring strict, isolated solutions. By applying the Lagrange–Bürmann inversion formula to the stationarity equation, we derive a uniformly convergent series for the larger positive root, yielding an exact and numerically stable formula for the $\ell_p$ proximal map above the classical threshold. We further provide a Mellin–Barnes integral representation and identify the series as a Fox–Wright function, which determines its radius of convergence. Specializations recover the known closed forms for $p=\tfrac12$ and $p=\tfrac23$, and we supply compact hypergeometric expressions for additional rational cases (e.g., $p=\tfrac13$). These results unify scattered formulas into a single framework and enable high-accuracy evaluation of $\ell_p$ proximity operators across the full range $0<p<1$.




* October 30th, Thursday(Special date) (4:00-5:00pm)

Speaker: Zengyan Zhang (Penn State)
Topic: Geometric local parameterization for solving Hele-Shaw problems with surface tension



Abstract: With broad applications in biology, physics, and material science, including tumor growth and fluid interface dynamics, the Hele-Shaw problem with surface tension provides a canonical model for studying the dynamics of evolving interfaces. Solving such problems requires precise treatment of sharp boundaries. However, constructing a global parameterization for complicated surfaces and explicitly tracking boundary motion is challenging. In this work, we present a geometric local parameterization approach for efficiently solving the two-dimensional Hele-Shaw problems, where the boundary is identified only from randomly sampled data. Through convergence and error analysis, as well as numerical experiments, we demonstrate the capability and effectiveness of our approach in resolving complex interface evolution.



November 5th, Wednesday (4:00-5:00pm)

Speaker: Yuanyuan Pan (Syracuse University)
Topic: On the Spectral Geometry and Small-Time Mass of Anderson Models on Planar Domains



Abstract: We consider the Anderson Hamiltonian (AH) and the parabolic Anderson model (PAM) with white noise and Dirichlet boundary condition on a bounded planar domain $D\subset\mathbb R^2$. We compute the small-$t$ asymptotics of the AH's exponential trace up to order $O(\log t)$, and of the PAM's mass up to order $O(t\log t)$. Applications of our main result include the following:

(i) If the boundary $\partial D$ is sufficiently regular, then $D$'s area and $\partial D$'s length can both be recovered almost surely from a single observation of the AH's eigenvalues.

(ii) If $D$ is simply connected and $\partial D$ is fractal, then $\partial D$'s Minkowski dimension (if it exists) can be recovered almost surely from the PAM's small-$t$ asymptotics.

(iii) The variance of the white noise can be recovered almost surely from a single observation of the AH's eigenvalues.




* Room 309, November 20th, Thursday (4:00-5:00pm) (Special room and time)

Speaker: Brian Kirby(Binghamton University)
Topic: Compactifying the Manifold given by the Schwartzchild Metric

Abstract: Consider the metric in $\mathbb{R}^4$ given by $ds^2=f(r)dt^2 - 1/f(r)dr^2 - r^2dg^2$, where $g$ is the standard Riemannian metric in $\mathbb{R}^2$, $f(r) = \phi(r)(r - r_0)$, where $\phi$ is a continuous, differentiable, positive function on $\mathbb{R}$. We will construct the Penrose diagram (the compactified manifold) for the given metric via coordinate changes and compactification. We will then discuss extensions to topological Penrose Diagrams and metric functions with an arbitrary number of roots, if time permits.



* November 26th, Wednesday (4:00-5:00pm) (Thanksgiving Break)

Speaker: Thanksgiving Break
Topic:

* December 3rd, Wednesday (4:00-5:00pm)

Speaker: Job interview
Topic:

Abstract:




Spring 2025

* January 22nd, Wednesday (4-5pm)

Speaker : organizational meeting
Topic: organizational meeting

Abstract: organizational meeting



* January 29th, Wednesday (4-5pm)

Speaker :
Topic: job interview

Abstract:




* March 19th, Wednesday (4-5pm)

Speaker: Pierre Yves Gaudreau Lamarre (Syracuse)
Topic: From critical signal detection to spectral geometry.



Abstract: In this talk, we discuss a remarkable connection between two seemingly unrelated problems in probability/statistics and analysis, namely: detecting low-rank perturbations of random matrices, and recovering information about a differential operator's domain from its spectral asymptotics.

We will then discuss recent works that show how this connection can be exploited to prove new results regarding so-called “critical” perturbations/signals. That is, signals that are right at the threshold for detectability using spectral techniques.

This talk will feature discussions of various joint works with Promit Ghosal, Wilson Li, Yuchen Liao, and Mykhaylo Shkolnikov.




* March 26th, Wednesday (4-5pm)

Speaker: Alper Gunes (Princeton)
Topic: Joint moments of characteristic polynomials of random matrices



Abstract: Joint moments of characteristic polynomials of unitary random matrices and their derivatives have gained attention over the last 25 years, partly due to their conjectured relation to the Riemann zeta function. In this talk, we will consider the asymptotics of these moments in the most general setting allowing for derivatives of arbitrary order, generalising previous work that considered only the first derivative. Along the way, we will examine how exchangeable arrays and integrable systems play a crucial role in understanding the statistics of a class of infinite Hermitian random matrices. Based on joint work with Assiotis, Keating and Wei.




* April 2nd, Wednesday (4-5pm)

Speaker: Zhihan Wang (Cornell)
Topic: Shape of Mean Curvature Flow near and Passing Through a Non-degenerate Singularity

Abstract: A central question in geometric flow is to understand how the geometry and topology change after passing through singularities. I will explain how the local dynamics influence the shape of a mean curvature flow, the negative gradient flow of area functional, near a singularity, and how the geometry and topology of the flow change after passing through a singularity with generic dynamics. This talk is based on the joint work with Ao Sun and Jinxin Xue.



* April 9th, Wednesday (4-5pm)

Speaker: Yanfei Wang (Johns Hopkins University)
Topic: Weyl law improvement on products of Zoll manifolds



Abstract: Iosevich and Wyman have proved that the remainder term in classical Weyl law can be improved from $O(\lambda^{d-1})$ to $o(\lambda^{d-1})$ in the case of product manifold by using a famous result of Duistermaat and Guillemin. They also showed that we could have polynomial improvement in the special case of Cartesian product of round spheres by reducing the problem to the study of the distribution of weighted integer lattice points. In this paper, we show that we can extend this result to the case of Cartesian product of Zoll manifolds by investigating the eigenvalue clusters of Zoll manifold and reducing the problem to the study of the distribution of weighted integer lattice points too.




* April 16th, Wednesday, 2:20-3:20pm, WH 329 (Special time and room)

Speaker: Merrick Chang (Binghamton)
Topic: ABD Exam

Abstract:




* April 16th, Wednesday (4-5pm)

Speaker: Mikołaj Sierżęga (Cornell University/ University of Warsaw)
Topic: Li-Yau-Type Bounds for the Fractional Heat Equation

Abstract: Differential Harnack bounds are a key analytical device that bridge partial differential equations of the elliptic or parabolic type with Harnack bounds, which provide pointwise estimates on the local variability of solutions. A prime example is the famous Li-Yau inequality, which applies to positive solutions of the classical heat equation.

The growing interest in the theory and applications of nonlocal diffusion models naturally raises questions about analogues of Li-Yau-type inequalities in the nonlocal setting. However, despite many parallels between local and nonlocal diffusion models, even the model case of fractional heat flow presents both conceptual and technical challenges.

In my talk, I will discuss recent progress on optimal differential Harnack bounds for fractional heat flow. In particular, I will show how the structural properties of these estimates offer new insights into classical results for the standard heat equation.




* April 30th, Wednesday (4-5pm)

Speaker: Chad Nelson (Binghamton)
Topic: ABD Exam: Pseudodifferential Operators and Hodge Theory on Compact Manifolds

Abstract: The goal of Hodge theory is to relate the de Rham cohomology of a compact manifold, which is essentially a topological object, with precise information regarding the differentiation of differential forms on the manifold. One elegant way to do this is to employ pseudodifferential operators. These are operators that generalize the notion of a differential operator, motivated by the Fourier transform.

First, we will develop the theory of pseudodifferential operators on Euclidean space. This involves, for example, proving properties regarding the taking of adjoints, of composing two operators, etc. We will prove the existence of a pseudo-inverse, or a parametrix, for elliptic differential operators. Next, we will translate this theory from Euclidean space to compact manifolds. We will then give a precise description of the de Rham cohomology (and more!) using the parametrix construction for elliptic operators on the manifold.

No prior knowledge about differential equations or cohomology will be assumed.




* May 7th, Wednesday (4-5pm)

Speaker: Marius Beceanu (Albany)
Topic: Uniform decay estimates for Hamiltonians with first and second-order perturbations



Abstract: I will present new results regarding the uniform decay of solutions to Schroedinger and wave equations, whose Hamiltonian $H=-\Delta+iA \cdot \nabla + V$ contains a magnetic potential (a first-order perturbation) or where the Laplacian is replaced by the Laplace-Beltrami operator on a more general manifold (second-order perturbations).




Fall 2024

* August 21st, Wednesday (3:30-4:30pm)

Speaker :
Topic: organizational meeting



* September 18th, Wednesday (3:30-4:30pm)

Speaker : Ao Sun (Lehigh University)
Topic: Local dynamics and shape of mean curvature flow passing through a singularity

Abstract: A central question in geometric flow is to understand how the geometry and topology change after passing through singularities. I will explain how the local dynamics influence the shape of the flow near a singularity, and how the geometry and topology of the flow will change after passing through a singularity with generic dynamics. This talk is based on joint work with Zhihan Wang and Jinxin Xue



* October 2nd , Wednesday (3:30-4:30pm) (Rosh Hashanah and Fall Break)

Speaker:
Topic:

Abstract:




* October 23rd, Wednesday (3:30-4:30pm)

Speaker: David Renfrew (Binghamton University)
Topic: Universality for roots of derivatives of entire functions

Abstract: We show for a large class of entire functions, $f$, that after proper rescaling, on compact sets, the derivatives of $f$ converge to cosine, in particular their roots become evenly spaced. This proves a conjecture of Farmer and Rhoades [Trans. Amer. Math. Soc., 357(9):3789–3811, 2005] and Farmer [Adv. Math., 411:Paper No. 108781, 14, 2022] for our class of entire functions. A main ingredient of our proof is to show that high derivatives of high degree polynomials behave like Hermite polynomials, which we prove using the techniques from the newly developed field of finite free probability.



* October 30th, Wednesday (3:30-4:30pm)

Speaker: Shukai Du (Syracuse University)
Topic: Forward and inverse computation for radiative transfer via hp-adaptive mesh refinement and machine learning acceleration

Abstract: The forward and inverse problems for radiative transfer are important in many applications, such as climate modeling, optical tomography, and remote sensing. However, these problems are notoriously challenging to compute due to their high dimensionality, significant memory requirements, and the computational expense associated with solving the inverse problem iteratively. To address these challenges, we present recent progress on two approaches. The first approach is hp-adaptive mesh refinement, which has proved effective in efficiently representing solutions where they are smooth with high-order approximations, while also providing the flexibility to resolve local features through adaptive refinements. For the forward problem, we demonstrate that exponential convergence with respect to degrees of freedom (DOFs) can be achieved even when the solution exhibits sharp gradients. For the inverse problem, we introduce a goal-oriented hp-adaptive mesh refinement method that blends the two optimization processes—one for inversion and one for mesh adaptivity—thereby reducing computational cost and memory requirements. The second approach, termed element learning, aims to accelerate finite element-type methods through machine learning. This approach retains the desirable features of finite element methods while substantially reducing training costs. It draws on principles from hybridizable discontinuous Galerkin (HDG) methods, replacing HDG's local solvers with machine learning models. Numerical tests for both approaches are presented to demonstrate their computational efficiency in addressing the forward and inverse computations of radiative transfer.



* November 27th, Wednesday (3:30-4:30pm) (Thanksgiving Break)

Speaker:
Topic:

Abstract:




* December 6th, Friday (3:30-4:30pm)

Speaker:Jacob Shapiro (Princeton University)
Topic: Topological Classification of Insulators: the non-interacting spectrally gapped case

Abstract: An important theme in contemporary condensed matter physics is “topological phases of matter”. This refers to exotic materials which exhibit a number of striking phenomena. E.g., they have a quantized macroscopic observable which is stable under large classes of perturbations and in their bulk they are insulators though they exhibit robust edge currents along their boundaries. To mathematically explain this, we define an appropriate topological space of quantum mechanical Hamiltonians which describe the motion of (single) electrons in an insulator, and calculate its path-connected components. Hamiltonians in the same path-component are said to be topologically “equivalent”. The presentation will be based on joint pre-prints together with Jui-Hui Chung.




Spring 2024

* January 24th, Wednesday (4-5pm)

Speaker : organizational meeting
Topic: organizational meeting

Abstract: organizational meeting



* February 29th, Thursday (Special date) (4-5pm)

Speaker: Alex Iosevich (Rochester)

Topic: Signal recovery, uncertainty principles and Fourier restriction theory



Abstract: We are going to consider functions $f: {\mathbb Z}_N \to {\mathbb C}$ and view them as signals. Suppose that we transmit this signal via its Fourier transform

$$\widehat{f}(m)=\frac{1}{N} \sum_{x=0}^{N-1} e^{-\frac{2 \pi i xm}{N}} f(x),$$

and that the values of $\widehat{f}(m), m \in S$, are lost. Under what circumstances is it possible to recover the original signal? We shall see how this innocent question quickly leads us into the deep dark forest of Fourier analysis.



* March 6th, Wednesday (4-5pm) (Spring Break)

Speaker: