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Autumn Semester 2015

Date / Time Speaker Title Location
6 October 2015
15:15-16:15 		Prof. Dr. Annamaria Montanari
University of Bologna
Event Details

Analysis Seminar

Title Harnack Inequality for a Subelliptic PDE in Nondivergence Form
Speaker, Affiliation Prof. Dr. Annamaria Montanari, University of Bologna
Date, Time 6 October 2015, 15:15-16:15
Location HG G 43
Abstract We consider subelliptic equations in non divergence form of the type (1) Lu = 􏰀aijXjXiu = 0 i≤j where Xj are the Grushin vector fields, and the matrix coefficient is uniformly elliptic. We obtain a scale invariant Harnack’s inequality on the Xj ’s Carnot Carath ́eodory balls for nonnegative solutions under the only assumption that the ratio between the maximum and minimum eigenvalues of the coefficient matrix is bounded. In the seminar we first prove a weighted Aleksandrov Bakelman Pucci estimate, and then we show a critical density estimate, the double ball property and the power decay property. Once this is established, Harnack’s inequality follows directly from the axiomatic theory developed by Di Fazio, Gutierrez and Lanconelli in 2008. Our motivation to study the operator L comes from the geometric theory of several complex variable, where nonlinear second order Partial Differential Equations of “de- generate elliptic”- type appear. In particular, in looking for a characterization property of domains of holomorphy in term of a differential property of the boundary (pseudo- convexity), one has to handle the Levi curvatures equations, which are fully nonlinear equations in non-divergence form. The existence theory for viscosity solutions to such equations is quite well settled down. On the contrary, the problem of the regularity is well understood only in R3 and it is still widely open in higher dimension. This is mainly due to the lack of pointwise estimates for solutions to linear sub-elliptic equations with rough coefficients. These equations in cylindrical coordinates are non divergence pde’s Lu = f, with L structured as in (1).
Harnack Inequality for a Subelliptic PDE in Nondivergence Formread_more
HG G 43
13 October 2015
15:15-16:15 		Dr. Alessandro Carlotto
ETH-ITS
Event Details

Analysis Seminar

Title The finiteness problem for minimal surfaces of bounded index in a 3-manifold
Speaker, Affiliation Dr. Alessandro Carlotto, ETH-ITS
Date, Time 13 October 2015, 15:15-16:15
Location HG G 43
Abstract Given a closed, Riemannian 3-manifold (N,g) without symmetries (more precisely: generic) and a non-negative integer p, can we say something about the number of minimal surfaces it contains whose Morse index is bounded by p? More realistically, can we prove that such number is necessarily finite? This is the classical "generic finiteness" problem, which has a rich history and exhibits interesting subtleties even in its basic counterpart concerning closed geodesics on surfaces. We settle such question when g is a bumpy metric of positive scalar curvature by proving that either finiteness holds or N does contain a copy of RP^3 in its prime decomposition and we discuss the obstructions to any further generalisation of such result. When g is assumed to be strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by White) then the finiteness conclusion is true for any compact 3-manifold without boundary.
The finiteness problem for minimal surfaces of bounded index in a 3-manifoldread_more
HG G 43
10 November 2015
15:15-16:15 		Prof. Dr. Isabelle Gallagher
University Paris VII
Event Details

Analysis Seminar

Title "From molecular dynamics to linear fluid equations"
Speaker, Affiliation Prof. Dr. Isabelle Gallagher, University Paris VII
Date, Time 10 November 2015, 15:15-16:15
Location HG G 43
"From molecular dynamics to linear fluid equations"
HG G 43
1 December 2015
15:15-16:15 		Prof. Dr. Luca Martinazzi
Universität Basel
Event Details

Analysis Seminar

Title The fractional Liouville equation in dimension 1 - Compactness and quantization
Speaker, Affiliation Prof. Dr. Luca Martinazzi, Universität Basel
Date, Time 1 December 2015, 15:15-16:15
Location HG G 43
Abstract In this talk I will introduce the fractional Liouville equation on the circle S^1 and its geometric interpretation in terms of conformal immersions of the unit disk into the complex plane. Using this interpretation we can show that the solutions of the fractional Liouville equation have very precise compactness properties (including quantization) with a clear geometric counterpart. I will also compare these result with analogue ones for the classical Liouville equation in dimension 2, used to prescribe the Gaussian curvature. This is a joint work with Francesca Da Lio and Tristan Riviere.
The fractional Liouville equation in dimension 1 - Compactness and quantizationread_more
HG G 43
8 December 2015
15:15-16:15 		Dr. Thomas Mettler
Institut für Mathematik Goethe Universität Frankfurt
Event Details

Analysis Seminar

Title Extremal conformal structures on projective surfaces
Speaker, Affiliation Dr. Thomas Mettler, Institut für Mathematik Goethe Universität Frankfurt
Date, Time 8 December 2015, 15:15-16:15
Location HG G 43
Abstract Given a prescription of paths on a surface — one for every direction in the tangent space — one might ask if those paths are the geodesics of a Riemannian metric. Generically they are not, hence one might look for a Riemannian metric whose geodesics are ‘as close as possible’ to the prescribed paths. This gives rise to a natural variational problem. In this talk I will explain how the critical points relate to certain weakly conformal maps and minimal mappings.
Extremal conformal structures on projective surfacesread_more
HG G 43
15 December 2015
15:15-16:15 		Jonas Luehrmann
ETH
Event Details

Analysis Seminar

Title Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation
Speaker, Affiliation Jonas Luehrmann, ETH
Date, Time 15 December 2015, 15:15-16:15
Location HG G 43
Abstract We present a proof of global regularity and scattering for arbitrarily large solutions to the energy critical Maxwell-Klein-Gordon equation. The proof is based upon a novel "twisted" Bahouri-Gerard profile decomposition and a concentration compactness/rigidity argument by Kenig-Merle, following the method developed by Krieger-Schlag in the context of energy critical wave maps. This is joint work with Joachim Krieger.
Concentration Compactness for the Critical Maxwell-Klein-Gordon Equationread_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

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