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Spring Semester 2019

Date / Time Speaker Title Location
26 February 2019
15:15-16:15 		Dr. Michele Coti Zelati
Imperial College London
Event Details

Analysis Seminar

Title Metastability results for the Navier-Stokes equations and related models
Speaker, Affiliation Dr. Michele Coti Zelati, Imperial College London
Date, Time 26 February 2019, 15:15-16:15
Location HG G 43
Abstract We study diffusion and mixing in the incompressible Navier-Stokes equations and related scalar models. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequency. In turn, mixing acts to enhance the dissipative forces, giving rise to what we refer to as enhanced dissipation: this can be understood by the identification of a time-scale faster than the purely diffusive one. This talk is based on two recently obtained results: (1) a general quantitative criterion that links mixing rates (in terms of decay of negative Sobolev norms) to enhanced dissipation time-scales, and (2) a precise identification of the enhanced dissipation time-scale for the Navier-Stokes equations linearized around the Poiseuille flow, along with metastability results and nonlinear transition stability thresholds.
Metastability results for the Navier-Stokes equations and related modelsread_more
HG G 43
5 March 2019
15:15-16:15 		Megan Griffin-Pickering
University of Cambridge
Event Details

Analysis Seminar

Title A Particle Approximation for the Kinetic Isothermal Euler Equation
Speaker, Affiliation Megan Griffin-Pickering, University of Cambridge
Date, Time 5 March 2019, 15:15-16:15
Location HG G 43
Abstract The Kinetic Isothermal Euler equation is a model for plasma. It is the formal limit of the classical Vlasov-Poisson system in the `quasineutral’ limit where the Debye length tends to zero. The Vlasov-Poisson system can itself be derived formally from a system of interacting particles, in the limit as the number of particles tends to infinity. The rigorous justification of this `mean field’ limit remains a major open problem. However, in recent years, researchers have derived the Vlasov-Poisson equation rigorously from various regularised microscopic systems. In this talk, I will present a joint work with Mikaela Iacobelli, in which we give a rigorous derivation of the Kinetic Isothermal Euler equation from a regularised particle system, using a combined mean field and quasineutral limit.
A Particle Approximation for the Kinetic Isothermal Euler Equationread_more
HG G 43
19 March 2019
15:15-16:15 		Yu Wang
Northwestern University
Event Details

Analysis Seminar

Title Title T.B.A.
Speaker, Affiliation Yu Wang, Northwestern University
Date, Time 19 March 2019, 15:15-16:15
Location HG G 43
Title T.B.A. (CANCELLED)
HG G 43
26 March 2019
15:15-16:15 		Dr. Renan Assimos
MPI Leipzig
Event Details

Analysis Seminar

Title The Geometry of the Maximum Principle and a Bernstein Theorem for Codimension Two
Speaker, Affiliation Dr. Renan Assimos, MPI Leipzig
Date, Time 26 March 2019, 15:15-16:15
Location HG G 43
Abstract A cornerstone in the theory of minimal surfaces is Bernstein’s theorem, stating that the only entire minimal graphs in Euclidean 3-space are planes. The effort of many mathematicians lead to several generalizations of this statement. The works of Simons, Bombieri-De Giorgi-Giusti, Moser, Lawson-Osserman and Hildebrandt-Jost-Widman are examples of such results: the first two proving that this theorem is true for minimal hypersurfaces of dimension up to 7 and false for higher dimensions; the third proves the theorem for any minimal hypersurface adding a bounded slope condition; for higher codimensionas, L-O have provided counterexamples even under the extra hypothesis on the slope, while the last work cited gave a stronger condition on the slope to obtain a Bernstein type result. In our work, we present a generalization of Moser’s theorem in codimension 2. More precisely, if f : R^n \to R^2, f(x) = (f1(x),f2(x)) is a smooth map defined everywhere in R^n, M = (x,f(x)) is a minimal submanifold in R^2 and there exists a number \beta_0<+\infty s.t. \Delta_{\beta_0} for all x \in R^p, where \Delta_{\beta_0}:= (det (\delta{\alpha \beta}+\sum f^i_{x_{\alpa}}(x)(f^i_{x_{\beta}}(x)))^{1/2}, then f^i : R^n \to R, i = 1,2 are linear functions on Rn. To prove this theorem we develop general techniques to study the geometry of subsets of a complete Riemannian manifold that contain no image of non-constant harmonic maps. We use this to study regions in a Grassmannian manifold with this property, since the Gauss map of a minimal submanifold is a harmonic map with image into G^+_{p,n}. With these ideas we obtain Bernstein type results.
The Geometry of the Maximum Principle and a Bernstein Theorem for Codimension Tworead_more
HG G 43
2 April 2019
15:15-16:15 		Prof. Dr. Melanie Rupflin
University of Oxford
Event Details

Analysis Seminar

Title Singularities of Teichmueller harmonic map flow
Speaker, Affiliation Prof. Dr. Melanie Rupflin, University of Oxford
Date, Time 2 April 2019, 15:15-16:15
Location HG G 43
Abstract We discuss singularities and the asymptotic behaviour of Teichmueller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to singular behaviour of the metric component.
Singularities of Teichmueller harmonic map flowread_more
HG G 43
16 April 2019
15:15-16:15 		Prof. Dr. Joaquim Serra
ETH Zurich, Switzerland
Event Details

Analysis Seminar

Title On stable solutions to elliptic semilinear PDE
Speaker, Affiliation Prof. Dr. Joaquim Serra, ETH Zurich, Switzerland
Date, Time 16 April 2019, 15:15-16:15
Location HG G 43
Abstract It is a classical fact that there exist singular stable energy solutions to elliptic semilinear PDE in dimension 10. For few concrete nonlinearities it was known that stable solutions must be smooth up to dimension 9. This is a PDE analogue of the conjectured smoothness of stable minimal surfaces up to dimension 7. In a forthcoming work with Cabre, Figalli, and Ros-Oton we establish the boundedness (and hence smoothness) up to dimension 9 of stable solutions for any nonlinearity. We also prove a priori energy bounds for stable solutions in any dimension. We will explain these results, the open questions they settle, as well as their motivations and some classical consequences of them.
On stable solutions to elliptic semilinear PDEread_more
HG G 43
16 April 2019
16:30-17:30 		Prof. Dr. Peter Topping
University of Warwick
Event Details

Analysis Seminar

Title Pyramid Ricci flows in higher dimensions
Speaker, Affiliation Prof. Dr. Peter Topping, University of Warwick
Date, Time 16 April 2019, 16:30-17:30
Location HG G 43
Abstract In this talk I will take a look at the problem of running the Ricci flow in more general situations than when working on closed manifolds, or when starting with a metric of bounded curvature. In general this is apparently not possible, as I will explain, but there are ways around these obstructions, and one of them is via the notion of “pyramid Ricci flow”. I will explain what these are, give some applications, and give details of some forthcoming new work on the topic with Andrew McLeod.
Pyramid Ricci flows in higher dimensionsread_more
HG G 43
30 April 2019
15:15-16:15 		Dr. Yi Zhang
Universität Bonn
Event Details

Analysis Seminar

Title Sobolev regularity of infinity Laplace equations in the plane and its application
Speaker, Affiliation Dr. Yi Zhang, Universität Bonn
Date, Time 30 April 2019, 15:15-16:15
Location HG G 43
Abstract The infinity Laplacian, introduced by Aronsson in 1960's, is a highly degenerate nonlinear second elliptic partial differential operator. The interior regularity of infinity Laplace equations is one of the main questions in this field. In 2005 C^1_{\loc}-regularity of planar infinity harmonic functions is proved by Savin, and later in 2008 by Evans and Savin the C_{\loc}^{1,\,\alpha}-regularity. The everywhere differentiability is proved by Evans and Smart in 2011. In this talk, I present my joint work with Prof. Koch and Prof. Zhou showing that |Du|^2\in W^{1,\,2}_{\loc}-regularity for infinity Laplace equations with some other sharp regularity results, together with my resent joint work with Prof. Zhou which gives a new understanding of C^1_{\loc}-regularity of planar infinity harmonic functions by Savin
Sobolev regularity of infinity Laplace equations in the plane and its applicationread_more
HG G 43
7 May 2019
15:15-16:15 		Dr. Shane Cooper
Durham University
Event Details

Analysis Seminar

Title A homogenisation theory for a general class of high-contrast problems: asymptotics with error estimates
Speaker, Affiliation Dr. Shane Cooper , Durham University
Date, Time 7 May 2019, 15:15-16:15
Location HG G 43
Abstract In this talk, we present a framework to study the asymptotic behaviour of (a large class of) periodic non-uniformly elliptic systems with respect to a (small period) parameter. We determine, under very few readily verifiable assumptions, the leading- order approximation of the solution and derive error estimates, uniform in right-hand- side. Spectral asymptotics with error estimates directly follow.
A homogenisation theory for a general class of high-contrast problems: asymptotics with error estimatesread_more
HG G 43
28 May 2019
15:15-16:15 		Prof. Dr. Armin Schikorra
University of Pittsburgh
Event Details

Analysis Seminar

Title Thresholds for Measuring Degree in fractional Sobolev Spaces
Speaker, Affiliation Prof. Dr. Armin Schikorra, University of Pittsburgh
Date, Time 28 May 2019, 15:15-16:15
Location HG G 43
Abstract The degree of a map between two spheres of same dimension can be estimated by Sobolev norm of said map (of the right class). In this talk I will discuss to what extend this is possible for the Hopf degree as well – and why the estimate we have is "analytically optimal" but probably not "topologically optimal". Joint work with J. Van Schaftingen.
Thresholds for Measuring Degree in fractional Sobolev Spacesread_more
HG G 43

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