Schedule

In this talk, we examine the asymptotic behavior of Fubini-Study currents and equilibrium metrics associated with sequences of Hermitian metrics on holomorphic line bundles over a fixed compact Kähler manifold. We establish conditions under which these currents, when suitably rescaled, converge weakly. We provide a detailed description of the limit, in doing so we generalize a theorem of Tian. Our justification is based on techniques developed by Demailly. Notably, our results do not require the positivity of the underlying Hermitian metrics, and we discuss the implications of this relaxation.

"A mathematician is a machine for turning coffee into theorems." - Erdös

In this presentation, we intend to discuss new results on the unconditional well-posedness for a pair of periodic nonlinear dispersive equations using an abstract framework introduced by Kishimoto. This framework is based on a normal form reductions argument coupled with a number of crucial multilinear estimates.

Given a compact Riemannian manifold whose Laplace-Beltrami spectrum is simple, one can uniquely recover the Riemannian metric from the local Weyl counting function. In plain language, one can hear the shape of most drums if one knocks at each point on its surface. I'll discuss a short proof of this result and give some examples that illustrate why the spectrum needs to be simple. I'll also give a bit of history about some important results to provide context, including Milnor's isospectral tori and Uhlenbeck's theorem on the genericity of manifolds with simple spectra.

This is joint work with Xing Wang and Yakun Xi (https://arxiv.org/abs/2407.18797).

The complex Monge Ampere operator plays on important role in pluripotential theory and several complex variables. This talk will focus on the Pluricomplex Green Function, the Lempert function, and the relationship between the two. A formula will be presented for the pluricomplex Green function with two poles in the bidisk.

"Food is our common ground, a universal experience." - James Beard 

Abstract: One of the foundational reasons for calculus is to understand macroscopic behavior of functions from its microscopic, or asymptotic behavior. First developed in Euclidean setting and then extended to the setting of smooth manifolds, calculus has been used in various settings in physics and in dynamics. However, when the underlying space in which the function measures certain quantities is not a smooth space, the notions of calculus do not make sense. In this talk we will discuss a few ways of doing calculus in settings that do not have smooth structures.

In many cases, the energy-minimal function h: U->V is a homeomorphism. We study this question for the Dirichlet energy and when U and V are embedded tori in R^3.  For the standard smooth tori U and V, the existence and global invertibility of the minimizers are obtained. Surprisingly, when V is replaced by a non-smooth torus-like target surface, injectivity of the minimizer is lost. The key tools in finding an extremal homeomorphism are the free Lagrangians. We demonstrate these ideas in the case of the Dirichlet energy and a pair of planar rectangles. 

"A mathematician is a machine for turning coffee into theorems." - Erdös

This talk is on the analysis of overdetermined boundary value problems (OBVP). These problems model various physical phenomena and are characterized by the imposing both Dirichlet and Neumann type boundary conditions.

The focus will be on OBVP for second-order, homogeneous, constant complex coefficient, weakly elliptic systems in non-smooth domains, with boundary data in Whitney–Lebesgue spaces with integrability $p\in (1,\infty)$. We will discuss integral representation formulas, jump formulas, characterization of admissible boundary data, and the existence and uniqueness of solutions for OBVP in uniformly rectifiable domains.

This is a joint work with Irina Mitrea (Temple University), Dorina Mitrea, and Marius Mitrea (Baylor University).

Abstract: Over the years I have been both on the receiving and giving hand of the complex process of research grant applications. I have gone through disappointment and elation, and benefitted in many ways from these career propellers and prodders of better communication skills.  In this presentation, I will mostly focus on the NSF funding opportunities for mathematicians, provide insights on the NSF merit review process, and discuss the tenets of successful grant proposals.

Abstract: One of the ways of measuring the energy of a function is to consider the integral of the square of its derivative; this is what is done in order to measure the energy of a satellite in motion. When the motion is in an inviscid medium, the energy is the integral of a power of its "speed". These are examples of local energies, that is, two functions that agree outside of a small region have differing energies only in that region. In some phenomena such local behavior is not applicable. In this talk we will discuss some ways of measuring nonlocal energies by using local energies. 

"A mathematician is a machine for turning coffee into theorems." - Erdös

We prove the existence of viscosity solutions to complex Hessian equations that satisfy a determinant dominant condition.  This viscosity solution is shown to be unique when the right hand is strictly monotone increasing in terms of the solution. This is joint work with Jingrui Cheng.

Given a formally integrable almost complex structure $X$ defined on the closure of a bounded domain $D \subset \mathbb C^n$, and provided that $X$ is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on $\overline D$ that transforms $X$ into the standard complex structure, under certain geometric and regularity assumptions on $D$. In this talk I will present my recent work on the ½ estimate for global Newlander-Nirenberg problem on strongly pseudoconvex domains. The main ingredients in our proof are the construction of Moser-type smoothing operators on bounded Lipschitz domains using Littlewood-Paley theory and a convergence scheme of KAM type. 

The classical Cauchy integral operator is one of the most famous and most studied singular integral operator in mathematics. In this talk, I will be presenting a higher-order analogue of the existing theory for the classical Cauchy operator, in which the salient role of the Cauchy-Riemann operator $\overline{\partial}$ is now played by natural powers of this. A central role will be played by integral representation formulas, jump relations and higher-order Fatou-type theorems. 

This is joint work with Irina Mitrea (Temple University), Dorina Mitrea and Marius Mitrea (Baylor University).