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-====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 $R^d$
-\\ \\
-<WRAP box 80%>
-**//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.
- \\
-</WRAP>
-\\ \\
- * **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
-\\ \\
-<WRAP box 80%>
-**//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.
- \\
-</WRAP>
-\\ \\
- * **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 $0<p<1$
-\\     \\
-<WRAP box 80%>
-**//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 $0<p<1$.
-\\
-</WRAP>
-\\ \\
-* **October 22nd, Wednesday ** (4:00-5:00pm) \\  \\
-**//Speaker//**:  \\
-**//Topic//**:
-\\     \\
-* **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
-\\     \\
-<WRAP box 80%>
-**//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.
-</WRAP>
-\\ \\
- **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
-\\     \\
-<WRAP box 80%>
-**//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.
- \\
-</WRAP>
-\\ \\
- * **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
-\\     \\
-<WRAP box 80%>
-**//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.
- \\
-</WRAP>
-\\ \\
-* **November 26th, Wednesday ** (4:00-5:00pm) (Thanksgiving Break)\\  \\
-**//Speaker//**:  Thanksgiving Break  \\
-**//Topic//**:
-\\     \\
-* **December 3rd, Wednesday ** (4:00-5:00pm)\\  \\
-**//Speaker//**:  Job interview  \\
-**//Topic//**:
-\\     \\
-<WRAP box 80%> **//Abstract//**:
- \\
-</WRAP>
-\\ \\
-----
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  * **January 28th, Wednesday ** (4-5pm)\\
 \\   **//Speaker //**:   Chad Nelson (Binghamton University)
-\\      **//Topic//**:
+\\      **//Topic//**: Fredholmness of Elliptic Operators on Manifolds with Boundary
 \\ \\
-<WRAP box 80%> **//Abstract//**:
+<WRAP box 80%> **//Abstract//**:  The classical calculus of pseudodifferential operators extends differential operators in a way that is suited to the construction of parametrices (pseudo-inverses) for elliptic operators. A fundamental consequence is that elliptic operators are Fredholm between appropriate Sobolev spaces on compact manifolds.
+On manifolds with boundary, this implication no longer holds. Melrose’s calculus of b-pseudodifferential operators is the analogous class of operators which leads to Fredholm properties for elliptic operators satisfying a certain condition related to the boundary. In this talk, I will compare the classical case and the boundary case, emphasizing the new features introduced by the boundary—most notably the b-stretched product and the indicial operator—and explain how these lead to Fredholmness on weighted b-Sobolev spaces.
  \\
@@ Line 205: / Line 51: @@
  * **February 4th, Wednesday ** (4-5pm)\\
-\\   **//Speaker //**:   TBD
+\\   **//Speaker //**: Emmanuel Adara (Binghamton University)
-\\      **//Topic//**:
+\\      **//Topic//**:  On Methods of Solution to Chemical Master Equation in Biochemical Systems
 \\ \\
-<WRAP box 80%> **//Abstract//**:
+<WRAP box 80%> **//Abstract//**:  In chemical kinetics, accurately modeling the dynamic behavior of chemical systems is essential for predicting reaction outcomes and optimizing processes. However, the challenge known as the “curse of dimensionality” has posed significant difficulties for conventional techniques employed in addressing the chemical master equation (CME). This predicament arises when the state space of the Markov chain expands exponentially with the number of species, rendering the CME computation practically unsolvable.
+In this talk, I will discuss some methods of solving the CME, including Gillespie’s algorithm, the Chemical Langevin Equation, and the Method of Moments, along with an overview of tensor train and machine learning-based methods, which offer promising strategies for gaining insights into complex biological systems.
  \\
@@ Line 216: / Line 64: @@
- * **February 11th, Wednesday ** (4-5pm)\\
-\\   **//Speaker //**:
+ * **Tuesday, February 10 (Joint with Data Science Seminar) ** (12:15-1:15pm)\\
-\\      **//Topic//**:
+\\   **//Speaker //**: Dr. Yizeng Li (Department of Biomedical Engineering at Binghamton University)
+\\      **//Topic//**: Multiphase Continuum Models for Cell Migration.
 \\ \\
-<WRAP box 80%> **//Abstract//**:
+<WRAP box 80%> **//Abstract//**: Cell migration is a fundamental process in physiology and disease, yet it poses challenging problems in multiscale modeling and continuum mechanics. Cell motility arises from the coupling of intracellular transport, active force generation, and evolving geometry. Cytoskeletal dynamics, in particular actin turnover and force production, provides a rich setting for mathematical analysis. In this talk, I will present a mathematical framework for mammalian cell motility based on multiphase continuum models with moving boundaries. The formulation incorporates fluid-structure interaction and active stresses to describe the coupled evolution of cytoskeletal flow and cell shape. The model predicts how migration efficiency depends on actin dynamics and geometric features of the cell. If time permits, I will also present a mechanical-electrical-chemical coupled model for water-driven cell motility induced by polarized membrane ion transport. This second framework highlights how transport processes and force balance together generate directed motion.
+Biography of the speaker: Yizeng Li is an Assistant Professor in the Department of Biomedical Engineering at Binghamton University. She received MS from Mathematics and PhD from the Department of Mechanical Engineering at the University of Michigan-Ann Arbor. Afterwards, she was a postdoctoral researcher at Johns Hopkins University's Department of Mechanical Engineering and Institute for NanoBioTechnology. Her backgrounds are in theoretical mechanics and applied mathematics with applications to biophysics and mechanobiology. Li develops physiology-based mathematical models for cell motility, polarization, volume regulation, electro-homeostasis, signal transduction, and other biophysics problems. She also combines mathematical models with experimental data to explain non-intuitive cell biology phenomena.
  \\
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- * **February 18th, Wednesday ** (4-5pm)\\
-\\   **//Speaker //**:
-\\      **//Topic//**:
-\\ \\
-<WRAP box 80%> **//Abstract//**:
- \\
-</WRAP>
-\\ \\
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 </WRAP>
-\\ \\
+\\
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 </WRAP>
-\\ \
+\\
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 \\ \\
- \
  * **March 18th, Wednesday ** (4-5pm) \\  \\
- **//Speaker//**:   \\
+ **//Speaker//**: Zheng Sun (University of Alabama, Tuscaloosa)\\
- **//Topic//**:
+ **//Topic//**: On a numerical artifact of solving shallow water equations with a discontinuous bottom
 \\     \\
-<WRAP box 80%> **//Abstract//**:
+<WRAP box 80%> **//Abstract//**: The nonlinear shallow water equations are used to model the free surface flow in rivers and coastal areas for which the horizontal length scale is much greater than the vertical length scale. They have wide applications in oceanic sciences and hydraulic engineering. In this talk, we study a numerical artifact of solving the shallow water equations over a discontinuous riverbed. For various first-order methods, we report that the numerical solution will form a spurious spike in the numerical momentum at the discontinuous point of the bottom. This artifact will cause the convergence to a wrong solution in many test cases. We present a convergence analysis to show that this numerical artifact is caused by the numerical viscosity imposed at the discontinuous point. Motivated by our analysis, we propose a numerical fix which works for the nontransonic problems.
@@ Line 292: / Line 132: @@
 **//Speaker//**:   Yiming Zhao(Syracuse University) \\
 **//Topic//**:  TBD
 \\     \\
 <WRAP box 80%> **//Abstract//**:
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+\\
 </WRAP>
 \\ \\
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  **//Speaker//**:   \\
  **//Topic//**:
 \\     \\
 <WRAP box 80%> **//Abstract//**:
@@ Line 333: / Line 166: @@
-* **April 15th, Wednesday, 4:00-5:00pm  \\  \\
+* **April 15th, Wednesday,** 4:00-5:00pm  \\  \\
- **//Speaker//**:    \\
+ **//Speaker//**:   Brian Kirby(Binghamton University)  \\
-**//Topic//**:
+**//Topic//**:   PhD Thesis Defense
 \\     \\
 <WRAP box 80%> **//Abstract//**:
@@ Line 343: / Line 176: @@
 \\ \\
- * **April 22th, Wednesday ** (4-5pm) ****\\  \\
+ * **April 22th, Wednesday ** (4-5pm) \\  \\
  **//Speaker//**: \\
  **//Topic//**:
@@ Line 360: / Line 193: @@
-* **April 29th, Wednesday ** (4-5pm) \\  \\
+ **April 29th, Wednesday ** (4-5pm) \\  \\
- **//Speaker//**:    \\
+ **//Speaker//**: Yahong Yang (Georgia Institute of Technology)  \\
- **//Topic//**:
+ **//Topic//**: Multiscale Neural Networks for Approximating Green’s Functions and Operators
 \\     \\
-<WRAP box 80%> **//Abstract//**:
+<WRAP box 80%> **//Abstract//**: Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) with broad applications in physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is to learn the associated Green’s function. However, Green’s functions are notoriously difficult to approximate due to their poor regularity, often requiring large neural networks and long training times.
+In this talk, we address these challenges by leveraging multiscale neural networks to learn Green’s functions efficiently. Through theoretical analysis based on multiscale Barron space techniques, together with numerical experiments, we show that the multiscale approach significantly reduces the required network size and accelerates training. We then extend this framework to operator learning, enabling neural networks to efficiently and accurately learn the mapping from coefficient functions to Green’s functions.
@@ Line 375: / Line 210: @@
 * **May 6th, Wednesday ** (4-5pm) \\  \\
- **//Speaker//**:  \\
+ **//Speaker//**: \\
  **//Topic//**:
 \\     \\

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