Schedule for: 25w5452 - Women in Analysis (WoAN)

Speakers: Nataša Pavlović and Gigliola Staffilani

Abstract: The first part of the talk will focus on the long-time dynamics of the energy spectrum for certain periodic dispersive equations, and particularly the analysis of energy transfer. Since the original work of Bourgain, a connection has been established between the time asymptotic of this transfer and the growth in time of high Sobolev norms of global solutions to the associated evolution equation. Consequently, several works have been generated in the last twenty years to obtain either a polynomial time upper bound for these norms, or to construct initial data that evolve to achieve this growth.

The second part of the talk will focus on emerging connections between dynamics of many body systems and nonlinear PDE. Recently remarkable progress has been achieved in understanding nonlinear dispersive and hyperbolic equations, as well as kinetic and fluid equations. Simultaneously, there has been a lot of progress in understanding the dynamics of many particle systems, that themselves model complex phenomena such as e.g. Bose-Einstein condensation. With fundamental works on derivation of effective equations from quantum many body systems (e.g. nonlinear Schrödinger equation) and effective equations from classical many particle systems (e.g. Vlasov and Boltzmann equations) a channel of communication between mathematical physics and nonlinear PDE communities has opened, contributing to advances in both areas.

We will give a brief review of some of the developments mentioned above.

On the occasion of International Women in Mathematics Day, we will be screening the short film Journeys of Women in Mathematics, created by the IMU Committee for Women in Mathematics, filmed and edited by Micro-Documentaries, and made possible by a grant from the Simons Foundation.

Speakers: Irina Markina and Andreea Nicoara

Abstract: This lecture is meant as an introduction to some of the central ideas in complex analysis in several variables. We will start by characterizing the main notion of convexity, pseudoconvexity, which is the part of convexity that survives biholomorphic mappings. We will proceed to the central PDE in several complex variables, the dbar-Neumann problem. Sup norm estimates are only known in certain cases and the correct hypothesis for (p,q) forms is yet to be understood. The singularities of dbar propagate along complex varieties on the boundary of the domain, which raises interesting questions of integrability. Finally, we will discuss hyperbolicity, which is a way of measuring the failure of the Riemann Mapping Theorem in several complex variables.

Speaker: Fioralba Cakoni, Scattering Group

Abstract: A perplexing question in scattering theory is whether there exist incoming time-harmonic waves, at specific frequencies, that are not scattered by a given inhomogeneity—in other words, whether the inhomogeneity is invisible to such probing waves. We refer to wave numbers corresponding to frequencies at which non-scattering incoming waves exist as non-scattering wave numbers. This question is inherently connected to the inverse scattering problem for inhomogeneous media. Efforts to address it have led to the formulation of the transmission eigenvalue problem, where the wave number serves as the spectral parameter. This is a non-selfadjoint eigenvalue problem with a mathematically challenging structure. The non-scattering wave numbers constitute a subset of the real transmission eigenvalues.

In this presentation, we survey state-of-the-art results on the structure of transmission eigenvalues and non-scattering wave numbers, where the spectral theory of non-selfadjoint operators meets the methods of free boundary regularity.

Speakers: Maya Chhetri and Nsoki Mavinga

Abstract: Linear and Nonlinear Elliptic boundary value problems govern a wide spectrum of complex phenomena, encompassing population dynamics, combustion theory, and chemical reactors. Understanding the qualitative aspects of such PDEs is paramount for gaining deeper insights into these multifaceted processes, and therefore continues to attract attention. One of the classical and influential tools used to study nonlinear partial differential equations is the sub- and supersolutions method. This method requires the construction of two suitable barrier functions, namely, a subsolution and a supersolution. It is, in a sense, elementary, and yet deep results can often be obtained. It yields not only existence results but also provides qualitative information about the solutions.

In this talk, we will survey some classical as well as some recent results. In particular, we discuss results on the existence of minimal and maximal solutions between an ordered pair of sub- and supersolutions for semilinear elliptic equations with reaction terms in the differential equation and on the boundary, especially when the standard monotone iteration method does not apply.

Speakers: M. Cristina Pereyra and Lesley Ward

Abstract: Harmonic analysis revolves around the study of classes of functions and the operators that act on them, with emphasis on the quantitative properties of these objects, including boundedness, integrability, smoothness, and decay estimates. We will survey some recent results and open questions, touching on the parallel and overlapping worlds of dyadic harmonic analysis, multiparameter harmonic analysis, and the setting of spaces of homogeneous type.

For the $t$-Haar multipliers, akin to pseudo-differential operators, necessary and sufficient conditions for their boundedness on $L^p$ spaces are known, and recently, one- and two-weight boundedness properties were resolved. Many questions about them remain to be explored, including sparse domination, compactness properties, commutators, endpoint boundedness, multilinear and multiparameter extensions. For maximal operators, it was shown recently that the dyadic strong maximal function cannot be bounded via sparse domination in its usual form, but is there a variation that works? Also, for which measures $\mu$ is the weighted strong maximal function $M_{s,\mu}$ bounded on the weighted Lebesgue spaces $L^p(\mu)$? A third very active area is that of Kakeya conjectures $-$ the three-dimensional Kakeya conjecture was proved this year $-$ and projections. One can ask for a non-linear analogue of the two-projection theorem: A compact set in the plane that has two distinct projections of zero measure must be purely unrectifiable. These are some of the questions we could explore this week.

Posters:

Presenter: Emily Casey
Title: Rectifiability and anisotropic singular integral operators

Presenter: Cornelia Mihaila
Title: A definition of fractional k-dimensional measure

Presenter:Maricela Ramirez
Title: On the Boundedness of the Cauchy Operator on Chord-Arc Curves

Presenter: Katja Vassilev
Title: One-dimensional wave kinetic theory

Presenter: Luisa Velasco
Title: The Inhomogeneous Six-Wave Kinetic Equation 

Presenter: Julia Wittmann
Title: Diffuse Interface Models for Two-Phase Flows with Phase Transition