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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.

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.

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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.

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.

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.

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.

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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.

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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.

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).