seminars:anal
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seminars:anal [2025/09/09 13:32] xxu |
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| - | * **August 27th, Wednesday ** (4:00-5:00pm)\\ \\ | ||
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| - | * **September 3rd, Wednesday ** (4:00-5:00pm)\\ \\ | ||
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| * **October 1st, Wednesday ** (4:00-5:00pm) (Yom Kippur)\\ \\ | * **October 1st, Wednesday ** (4:00-5:00pm) (Yom Kippur)\\ \\ | ||
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| * **October 8th, Wednesday ** (4:00-5:00pm)\\ \\ | * **October 8th, Wednesday ** (4:00-5:00pm)\\ \\ | ||
| **//Speaker//**: Prof. Lixin Shen (Syracuse University) \\ \\ | **//Speaker//**: Prof. Lixin Shen (Syracuse University) \\ \\ | ||
| - | **//Topic//**: TBD | + | **//Topic//**: Explicit Characterization of the $\ell_p$ Proximity Operator for $0<p<1$ |
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| - | **//Abstract//**: TBD | + | **//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 $0<p<1$. |
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| - | * **October 15th, Wednesday **(4:00-5:00pm) \\ \\ | ||
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| - | * **October 29th, Wednesday ** (4:00-5:00pm) \\ \\ | + | * **October 30th, Thursday(Special date) ** (4:00-5:00pm) \\ \\ |
| - | **//Speaker//**: \\ | + | **//Speaker//**: Zengyan Zhang (Penn State) \\ |
| - | **//Topic//**: | + | **//Topic//**: Geometric local parameterization for solving Hele-Shaw problems with surface tension |
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| - | **//Abstract//**: | + | **//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. |
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| **November 5th, Wednesday ** (4:00-5:00pm) \\ \\ | **November 5th, Wednesday ** (4:00-5:00pm) \\ \\ | ||
| **//Speaker//**: Yuanyuan Pan (Syracuse University) \\ | **//Speaker//**: Yuanyuan Pan (Syracuse University) \\ | ||
| - | **//Topic//**: TBD | + | **//Topic//**: On the Spectral Geometry and Small-Time Mass of Anderson Models on Planar Domains |
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| - | **//Abstract//**: | + | **//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: | ||
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| + | (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. | ||
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| + | (iii) The variance of the white noise can be recovered almost surely from a single observation of the AH's eigenvalues. | ||
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| - | * **November 12th, Wednesday ** (4:00-5:00pm)\\ \\ | ||
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| - | + | * **Room 309, November 20th, Thursday ** (4:00-5:00pm) (Special room and time)\\ \\ | |
| - | * **November 19th, Wednesday ** (4:00-5:00pm) \\ \\ | + | **//Speaker//**: Brian Kirby(Binghamton University)\\ |
| - | **//Speaker//**: \\ | + | **//Topic//**: Compactifying the Manifold given by the Schwartzchild Metric |
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| - | **//Abstract//**: | + | **//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. |
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| * **December 3rd, Wednesday ** (4:00-5:00pm)\\ \\ | * **December 3rd, Wednesday ** (4:00-5:00pm)\\ \\ | ||
| - | **//Speaker//**: \\ | + | **//Speaker//**: Job interview \\ |
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| - | ====Spring 2025==== | + | ====Spring 2026==== |
| - | * **January 22nd, Wednesday ** (4-5pm)\\ \\ **//Speaker //**: organizational meeting \\ **//Topic//**: organizational meeting | + | * **January 21st, Wednesday ** (4-5pm)\\ \\ |
| + | **//Speaker //**: TBD \\ | ||
| + | **//Topic//**: TBD | ||
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| - | <WRAP box 80%> **//Abstract//**: organizational meeting | + | <WRAP box 80%> **//Abstract//**: |
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| - | * **January 29th, Wednesday ** (4-5pm)\\ | + | * **January 28th, Wednesday ** (4-5pm)\\ |
| + | \\ **//Speaker //**: Chad Nelson (Binghamton University) | ||
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| + | * **February 4th, Wednesday ** (4-5pm)\\ | ||
| + | \\ **//Speaker //**: TBD | ||
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| + | * **February 11th, Wednesday ** (4-5pm)\\ | ||
| \\ **//Speaker //**: | \\ **//Speaker //**: | ||
| - | \\ **//Topic//**: job interview | + | \\ **//Topic//**: |
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| - | * **March 19th, Wednesday ** (4-5pm) \\ \\ | + | * **February 18th, Wednesday ** (4-5pm)\\ |
| - | **//Speaker//**: Pierre Yves Gaudreau Lamarre (Syracuse) \\ | + | \\ **//Speaker //**: |
| - | **//Topic//**: From critical signal detection to spectral geometry. | + | \\ **//Topic//**: |
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| + | <WRAP box 80%> **//Abstract//**: | ||
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| - | <WRAP box 80%> **//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. | + | </WRAP> |
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| - | 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. | + | * **February 25th, Wednesday ** (4-5pm)\\ |
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| + | * **March 4th, Wednesday ** (4-5pm)\\ | ||
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| + | * **March 11th, Wednesday ** (4-5pm)\\ | ||
| + | \\ **//Speaker //**: | ||
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| + | * **March 18th, Wednesday ** (4-5pm) \\ \\ | ||
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| - | * **March 26th, Wednesday ** (4-5pm) \\ \\ | + | * **March 25th, Wednesday ** (4-5pm) \\ \\ |
| - | **//Speaker//**: Alper Gunes (Princeton) \\ | + | **//Speaker//**: \\ |
| - | **//Topic//**: Joint moments of characteristic polynomials of random matrices | + | **//Topic//**: |
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| - | <WRAP box 80%> **//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. | + | <WRAP box 80%> **//Abstract//**: |
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| - | * **April 2nd, Wednesday ** (4-5pm) \\ \\ | + | * **April 1st, Wednesday ** (4-5pm) \\ \\ |
| - | **//Speaker//**: Zhihan Wang (Cornell)\\ **//Topic//**: Shape of Mean Curvature Flow near and Passing Through a Non-degenerate Singularity | + | **//Speaker//**: Spring Break \\ |
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| - | <WRAP box 80%> **//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. | + | <WRAP box 80%> **//Abstract//**: |
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| - | * **April 9th, Wednesday ** (4-5pm) \\ \\ | + | * **April 8th, Wednesday ** (4-5pm) \\ \\ |
| - | **//Speaker//**: Yanfei Wang (Johns Hopkins University)\\ | + | **//Speaker//**: \\ |
| - | **//Topic//**: Weyl law improvement on products of Zoll manifolds | + | **//Topic//**: |
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| - | <WRAP box 80%> **//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. | + | <WRAP box 80%> **//Abstract//**: |
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| - | * **April 16th, Wednesday, 2:20-3:20pm, WH 329** (Special time and room) \\ \\ | + | * **April 15th, Wednesday, 4:00-5:00pm \\ \\ |
| - | **//Speaker//**: Merrick Chang (Binghamton) \\ | + | **//Speaker//**: \\ |
| - | **//Topic//**: ABD Exam | + | **//Topic//**: |
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| - | * **April 16th, Wednesday ** (4-5pm) ****\\ \\ | + | * **April 22th, Wednesday ** (4-5pm) ****\\ \\ |
| - | **//Speaker//**: Mikołaj Sierżęga (Cornell University/ University of Warsaw)\\ | + | **//Speaker//**: \\ |
| - | **//Topic//**: Li-Yau-Type Bounds for the Fractional Heat Equation | + | **//Topic//**: |
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| - | <WRAP box 80%> **//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. | + | <WRAP box 80%> **//Abstract//**: |
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| - | 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. | + | |
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| - | 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. | + | |
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| - | * **April 23rd, Wednesday ** (4-5pm) ****\\ \\ | ||
| - | **//Speaker//**: \\ | ||
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| - | + | * **April 29th, Wednesday ** (4-5pm) \\ \\ | |
| - | * **April 30th, Wednesday ** (4-5pm) \\ \\ | + | **//Speaker//**: \\ |
| - | **//Speaker//**: Chad Nelson (Binghamton) \\ | + | **//Topic//**: |
| - | **//Topic//**: ABD Exam: Pseudodifferential Operators and Hodge Theory on Compact Manifolds | + | |
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| - | <WRAP box 80%> **//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. | + | <WRAP box 80%> **//Abstract//**: |
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| - | 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. | + | |
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| - | No prior knowledge about differential equations or cohomology will be assumed. | + | |
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| - | * **May 7th, Wednesday ** (4-5pm) \\ \\ | + | * **May 6th, Wednesday ** (4-5pm) \\ \\ |
| - | **//Speaker//**: Marius Beceanu (Albany) \\ | + | **//Speaker//**: \\ |
| - | **//Topic//**: Uniform decay estimates for Hamiltonians with first and | + | **//Topic//**: |
| - | second-order perturbations | + | |
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| - | <WRAP box 80%> **//Abstract//**: I will present new results regarding the uniform decay of | + | <WRAP box 80%> **//Abstract//**: |
| - | 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). | + | |
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seminars/anal.1757439155.txt · Last modified: 2025/09/09 13:32 by xxu