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For Zoom URL please contact Laura Keller

Spring Semester 2021

Date / Time Speaker Title Location
23 February 2021
15:15-16:15
Matt Rosenzweig
MIT
Event Details

Analysis Seminar

Title Scaling-Critical Mean-Field Theory of Point Vortices and Other Coulomb Systems
Speaker, Affiliation Matt Rosenzweig, MIT
Date, Time 23 February 2021, 15:15-16:15
Location Online via Zoom
Abstract We consider classical Helmholtz-Kirchoff point vortices, a model example of a system with Coulomb interactions. In the mean-field regime where the magnitudes of the vortex circulations are inversely proportional to the number of vortices, which is very large, we expect the evolution to effectively be described by the vorticity formulation of the two-dimensional incompressible Euler equation. We will present a result on this approximation problem when the limiting vorticity is only in L^∞, a scaling-critical function space for the well-posedness of the equation. We will also discuss the generalization of our result to higher-dimensional Coulomb systems and systems where multiplicative noise of transport-type is added to the dynamics. Time permitting, we will discuss some of the goals and challenges of going beyond mean-field theory.
Scaling-Critical Mean-Field Theory of Point Vortices and Other Coulomb Systemsread_more
Online via Zoom
2 March 2021
15:15-16:15
Prof. Dr. Filomena Pacella
University "La Sapienza" of Rome
Event Details

Analysis Seminar

Title Constant mean curvature surfaces and overdetermined problems in cones
Speaker, Affiliation Prof. Dr. Filomena Pacella, University "La Sapienza" of Rome
Date, Time 2 March 2021, 15:15-16:15
Location Online via Zoom
Abstract We present some recent results about the characterization of constant mean curvature surfaces with boundary in cones. A parallel question is to determine the domains inside a cone which admit solution for a partial overdetermined problem. The relation between the two problems will be discussed as well as the role of the convexity of the cone and the construction of counterexamples.
Constant mean curvature surfaces and overdetermined problems in cones read_more
Online via Zoom
23 March 2021
15:15-16:15
Dr. Martin Lesourd
Harvard University
Event Details

Analysis Seminar

Title The Positive Mass Theorem for Manifolds with Arbitrary Ends
Speaker, Affiliation Dr. Martin Lesourd, Harvard University
Date, Time 23 March 2021, 15:15-16:15
Location Online via Zoom
Abstract The Positive Mass Theorem is a classic result of Differential Geometry and Geometric Analysis. Since its original proof in 1979 by Schoen-Yau, it has been generalized and sharpened in a variety of ways. Here, using recent advances in the study of spaces with Scalar Curvature bounded below (in particular the stability of hypersurfaces with prescribed mean curvature), we generalize the theorem one step further to cover a wider class of spaces. This settles a conjecture of Schoen-Yau 1988 and gives a new proof of another conjecture of Schoen-Yau 1988 on the structure of locally conformally flat manifolds with nonnegative scalar curvature. (The other proof coming from combining L.-Unger-Yau 2020 and Chodosh-Li 2020.) This is joint work with Unger-Yau https://arxiv.org/abs/2103.02744
The Positive Mass Theorem for Manifolds with Arbitrary Endsread_more
Online via Zoom
20 April 2021
15:15-16:15
Itamar Oliveira
Cornell University
Event Details

Analysis Seminar

Title The Fourier Extension problem through a time-frequency perspective
Speaker, Affiliation Itamar Oliveira, Cornell University
Date, Time 20 April 2021, 15:15-16:15
Location Online via Zoom
Abstract An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps L^{2+\frac{2}{d}}([0,1]^{d}) to L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1}) for every \varepsilon>0. It has been fully solved only for d=1 and there are many partial results in higher dimensions regarding the range of (p,q) for which L^{p}([0,1]^{d}) is mapped to L^{q}(\mathbb{R}^{d+1}). In this talk, we will take an alternative route to this problem: one can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all g of the form g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d}). Time permitting, we will also address multilinear versions of the statement above and get similar results, in which we will need only one of the many functions involved in each problem to be of such kind to obtain the desired conjectured bounds. This is joint work with Camil Muscalu.
The Fourier Extension problem through a time-frequency perspectiveread_more
Online via Zoom
4 May 2021
15:15-16:15
Dr. Ruben Jakob
Technion
Event Details

Analysis Seminar

Title Three different types of Willmore flows and their analytic properties
Speaker, Affiliation Dr. Ruben Jakob, Technion
Date, Time 4 May 2021, 15:15-16:15
Location Online via Zoom
Assets Abstractfile_download
Three different types of Willmore flows and their analytic propertiesread_more
Online via Zoom
11 May 2021
15:15-16:15
Dr. Joao Pedro Goncalves Ramos
ETH Zurich, Switzerland
Event Details

Analysis Seminar

Title Uncertainty principles and interpolation formulae in analysis and PDE
Speaker, Affiliation Dr. Joao Pedro Goncalves Ramos, ETH Zurich, Switzerland
Date, Time 11 May 2021, 15:15-16:15
Location Online via Zoom
Abstract The famous Heisenberg uncertainty principles predicts, in one of its versions, that a functions $f$ and its Fourier transform $\widehat{f}$ cannot both be too concentrated in space, otherwise $f \equiv 0.$ This intriguing principle has many classical counterparts of similar flavour, such as the Hardy uncertainty principle and the Amrein-Berthier theorem on annihilating pairs. In recent years, however, several breakthrough results have shed new light onto problems in the uncertainty realm. In particular, we mention first, in the realm of PDEs, a series of papers by Escauriaza-Kenig-Ponce-Vega, which generalised the Hardy uncertainty to a more general Schrödinger equation setting in its sharp form. On the other hand, from a more purely Fourier-analytic perspective, we mention the Radchenko-Viazovska interpolation formula, which provides one with an explicit way to recover an even function $f \in \mathcal{S}(\mathbb{R})$ given its values $f(\sqrt{n}), \widehat{f}(\sqrt{n}), n \ge 0.$ In this talk, we will go through some of these uncertainty results, from the most classical to some of the most recent. Our focus will be more on ideas than on technical details, with a particular emphasis on possible directions, open problems and conjectures in this area.
Uncertainty principles and interpolation formulae in analysis and PDEread_more
Online via Zoom
18 May 2021
15:15-16:15
Prof. Dr. Davi Maximo
University of Pennsylvania
Event Details

Analysis Seminar

Title The Waist Inequality and Positive Scalar Curvature
Speaker, Affiliation Prof. Dr. Davi Maximo, University of Pennsylvania
Date, Time 18 May 2021, 15:15-16:15
Location Online via Zoom
Abstract The topology of three-manifolds with positive scalar curvature has been (mostly) known since the solution of the Poincare conjecture by Perelman. Indeed, they consist of connected sums of spherical space forms and S^2 x S^1's. In spite of this, their "shape" remains unknown and mysterious. Since a lower bound of scalar curvature can be preserved by a codimension two surgery, one may wonder about a description of the shape of such manifolds based on a codimension two data (in this case, 1-dimensional manifolds). In this talk, I will show results from a recent collaboration with Y. Liokumovich elucidating this question for closed three-manifolds.
The Waist Inequality and Positive Scalar Curvature read_more
Online via Zoom
25 May 2021
15:15-16:15
Dr. Hon To Hardy Chan
ETH Zurich, Switzerland
Event Details

Analysis Seminar

Title Local and nonlocal ODEs in the singular fractional Yamabe problem
Speaker, Affiliation Dr. Hon To Hardy Chan, ETH Zurich, Switzerland
Date, Time 25 May 2021, 15:15-16:15
Location Online via Zoom
Abstract In conformal geometry, the Yamabe problem asks for Yamabe metrics, or conformal metrics of constant scalar curvature. In search of singular Yamabe metrics, one is led to the study of the Lane-Emden equation with a Sobolev-subcritical exponent that depends on the dimension of the singularity. The radial profile, which solves a classical ODE, is well-understood.
One could pose the same problem concerning the fractional curvature, a general notion that includes the scalar curvature, the curvatures associated to Paneitz and GJMS operators, as well as those with non-integer order. For the investigation of the corresponding radial profile, we discuss the development of the nonlocal ODE theory. Apart from the localizing Caffarelli-Silvestre extension, we show that nonlocal ODE can also be understood as a coupled infinite system of second order ODEs. Finally, we also mention a simple while surprising transformation that reduces the nonlocal ODE into almost a scalar first order ODE.
Local and nonlocal ODEs in the singular fractional Yamabe problemread_more
Online via Zoom
1 June 2021
15:15-16:15
Prof. Dr. Anna Mazzucato
Penn State
Event Details

Analysis Seminar

Title Transport, mixing, and enhanced dissipation
Speaker, Affiliation Prof. Dr. Anna Mazzucato, Penn State
Date, Time 1 June 2021, 15:15-16:15
Location Online via Zoom
Abstract I will discuss transport of passive scalars by incompressible flows and measures of optimal mixing and stirring. I will present two examples of opposite effects of mixing: one leads to irregular transport and a dramatic, instantaneous loss of regularity for transport equation, the other is enhanced dissipation, which can lead to global existence in non-linear, dissipative systems. In particular, I will show how mixing leads to global existence for the 2D Kuramoto-Sivashisky equation, a model for flame front propagation.
Transport, mixing, and enhanced dissipationread_more
Online via Zoom

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