Analysis seminar

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Spring Semester 2014

Date / Time

Speaker

Title

Location

25 February 2014
15:15-16:15

Daniele Valtorta
EPF Lausanne

Event Details

Analysis Seminar

Title	Quantitative stratification and applications to harmonic functions
Speaker, Affiliation	Daniele Valtorta, EPF Lausanne
Date, Time	25 February 2014, 15:15-16:15
Location	HG G 43
Abstract	Given a harmonic function defined on the unit ball of $\mathbb{R}^n$, we discuss techniques to obtain effective volume estimates on the tubular neighborhood of its critical sets. We use the quantitative stratification technique recently introduced by Jeff Cheeger and Aaron Naber. It is based on approximate symmetries of the function at different scales. Studying how these approximate symmetries interact with each other, we obtain the effective volume estimates. These results are described in a preprint available on the arXiv.

Quantitative stratification and applications to harmonic functionsread_more

HG G 43

11 March 2014
14:15-15:15

Alessio Figalli
The University of Texas at Austin

Event Details

Analysis Seminar

Title	Stability results for the semisum of sets in R^n
Speaker, Affiliation	Alessio Figalli, The University of Texas at Austin
Date, Time	11 March 2014, 14:15-15:15
Location	HG G 43
Abstract	Given a Borel A in R^n of positive measure, one can consider its semisum S=(A+A)/2. It is clear that S contains A, and it is not difficult to prove that they have the same measure if and only if A is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of S is close to the one of A, is A close to his convex hull? More generally, one may consider the semisum of two different sets A and B, in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When n=1, one can approximate a set with finite unions of intervals to translate the problem to the integers Z. In this discrete setting the question becomes a well-studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk I will review some results in the one-dimensional discrete setting and describe how to answer to the problem in arbitrary dimension.

Stability results for the semisum of sets in R^nread_more

HG G 43

11 March 2014
16:00-17:00

Manuel Ritoré
University of Granada

Event Details

Analysis Seminar

Title	Large isoperimetric regions in the product of a compact manifold with Euclidean space
Speaker, Affiliation	Manuel Ritoré, University of Granada
Date, Time	11 March 2014, 16:00-17:00
Location	HG G 43
Abstract	We prove that isoperimetric regions of large volume in the Riemannian product $M \times \mathbb{R}^k$, where $M$ is a compact manifold without boundary, are tubular neighborhoods of $M \times \{p\}$, $p\in \mathbb{R}^k$. This is joint work with Efstratios Vernadakis (http://arxiv.org/abs/1312.1581).

Large isoperimetric regions in the product of a compact manifold with Euclidean spaceread_more

HG G 43

25 March 2014
15:15-16:15

Greg Galloway
University of Miami

Event Details

Analysis Seminar

Title	On the geometry and topology of initial data sets in General Relativity
Speaker, Affiliation	Greg Galloway, University of Miami
Date, Time	25 March 2014, 15:15-16:15
Location	HG G 43
Abstract	We discuss some results concerning the geometry and topology of asymptotically flat initial data sets in three and higher dimensions, with and without horizons. More specifically, we explore the relationship between the topology of such initial data sets and the occurrence of marginally outer trapped surfaces in the initial data. We shall discuss the rationale for this and present relevant background material. This involves work with several collaborators, L. Andersson, M. Dahl, M. Eichmair and D. Pollack.

On the geometry and topology of initial data sets in General Relativityread_more

HG G 43

15 April 2014
15:15-16:15

Thomas Duyckaerts
Université Paris XIII

Event Details

Analysis Seminar

Title	Profiles for nonradial, bounded solutions of the energy-critical wave equation.
Speaker, Affiliation	Thomas Duyckaerts, Université Paris XIII
Date, Time	15 April 2014, 15:15-16:15
Location	HG G 43
Abstract	The energy-critical focusing wave equation admits solitary wave solutions travelling at a constant speed lesser than the speed of light. It is conjectured that any bounded solution of the equation can be written asymptotically as a finite sum of rescaled solitary waves and a dispersive part. In this talk, I will prove a weaker statement, namely that any bounded solution converges in some weak sense, along a sequence of time and after rescaling, to a solitary wave. The proof uses a new compactness-rigidity argument which works for many other dispersive equations. This is based on a joint work with Carlos Kenig and Frank Merle.

Profiles for nonradial, bounded solutions of the energy-critical wave equation.read_more

HG G 43

20 May 2014
15:15-16:15

Christiane Tretter
Universität Bern

Event Details

Analysis Seminar

Title	Spectral theory of block operator matrices and applications in mathematical physics
Speaker, Affiliation	Christiane Tretter, Universität Bern
Date, Time	20 May 2014, 15:15-16:15
Location	HG G 43
Abstract	Block operator matrices are matrices the entries of which are linear operators. They arise frequently in applications, e.g. when considering coupled systems of differential equations. In this talk we shall investigate the spectral theory of block operator matrices. Questions to be addressed include the location and structure of the spectrum and criteria for block diagonalization; the latter is closely related to the existence of solutions of algebraic Riccati equations. Applications to problems from hydrodynamics, magnetohydrodynamics, and quantum mechanics will be presented.

Spectral theory of block operator matrices and applications in mathematical physicsread_more

HG G 43

27 May 2014
15:15-16:15

Paul Feehan
Mathematics Rutgers University

Event Details

Analysis Seminar

Title	Global existence and convergence of smooth solutions to Yang-Mills gradient flow over compact four-manifolds
Speaker, Affiliation	Paul Feehan, Mathematics Rutgers University
Date, Time	27 May 2014, 15:15-16:15
Location	HG G 43
Abstract	Given a compact Lie group and a principal bundle over a closed Riemannian manifold, the quotient space of connections, modulo the action of the group of gauge transformations, has fundamental significance for algebraic geometry, low-dimensional topology, the classification of smooth four-dimensional manifolds, and high-energy physics. The quotient space of connections is endowed with the Yang-Mills energy functional and Atiyah and Bott (1983) had proposed that its gradient flow with respect to the natural Riemannian metric on the quotient space should prove to be an important tool for understanding the topology of the quotient space via an infinite-dimensional Morse theory. The critical points of the energy functional are gauge-equivalence classes of Yang-Mills connections. However, thus far, smooth solutions to the Yang-Mills gradient flow has only been known to exist for all time and converge to critical points, as time tends to infinity, in relatively few cases, including (1) when the base manifold has dimension two or three (Rade, 1992), (2) when the base manifold is a complex algebraic surface (Donaldson, 1985), and (3) in the presence of rotational symmetry in dimension four (Schlatter, Tahvildar-Zadeh, and Struwe, 1998). Global existence of solutions with up to finitely many possible singularities (caused by the ``bubbling'' phenomenon) was proved independently by Struwe (1994) and Kozono, Maeda, Naito (1995). However, the question of global existence in of smooth solutions over general compact, Riemannian, four-dimensional base manifolds has remained unresolved. In this talk we shall describe our proof of the following result: Given a compact Lie group and a smooth initial connection on a principal bundle over a compact, Riemannian, four-dimensional manifold, there is a unique, smooth solution to the Yang-Mills gradient flow which exists for all time and converges to a smooth Yang-Mills connection on the given principal bundle as time tends to infinity.

Global existence and convergence of smooth solutions to Yang-Mills gradient flow over compact four-manifoldsread_more

HG G 43

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Organizers: Francesca Da Lio, Tom Ilmanen, Thomas Kappeler, Tristan Rivière, Michael Struwe

Archive: SS 24 AS 23 SS 23 AS 22 SS 22 AS 21 SS 21 AS 20 SS 20 AS 19 SS 19 AS 18 SS 18 AS 17 SS 17 AS 16 SS 16 AS 15 SS 15 AS 14 SS 14 AS 13 SS 13 AS 12 SS 12 AS 11 SS 11 AS 10 SS 10 AS 09