# 2010-2011 Analysis Applied Math Seminar

This page contains information about the Analysis and applied Math Seminar at the University of Toronto which was organized in 2010-2011 by Bob Jerrard (rjerrard [at] math) and Amir Moradifam (amir [at] math).

For current 2011-2012 see here.

Of related interest in Toronto (and sometimes cross-listed):

Previous Year's Seminars: 2009-10, 2008-09, 2007-08, 2006-07, 2005-06, 2004-05, 2003-04, 2002-03.

## July 22, López-Gómez, 14:10-15:00 @BA6183

 Julián López-Gómez [] (Universidad Complutense de Madrid) 2010-2011 Analysis Applied Math Seminar Friday July 22 14:10-15:00 BA6183 Title: The theorem of characterization of the maximum principle Abstract: No abstract provided. [ arXiv] 2011_07_22_López-Gómez_Notes 2011_07_22

## July 18, Møller, 16:10-17:00 @BA6183

 Jacob Shach Møller [1] (Aarhus University) 2010-2011 Analysis Applied Math Seminar Monday July 18 16:10-17:00 BA6183 Title: Regularity of Bound States Abstract: In this talk we study regularity properties of bound states of self-adjoint operators. If the associated eigenvalue is isolated the problem is well understood. For eigenvalues embedded into the continuous spectrum one needs more sophisticated methods. A powerful tool are the socalled positive commutator techniques, which in this context goes back to Froese and Herbst who studied decay properties of eigenfunctions of many-body Schrödinger operators. Recently, Cataneo, Graf and Hunziker derived regularity results that feed into a general scheme of Hunziker and Sigal for studying stability of eigenvalues under perturbations. While these results are well suited to deal with many-body Schrödinger operators, they come up short in other contexts, e.g. some massless models of quantum field theory. We will explain the mechanisms behind the proofs, discuss the limitations in the work of Cataneo, Graf and Hunziker, and finally explain how to overcome these limitations. arXiv 2011_07_18_Møller_Notes 2011_07_18

## July 15, Lacave, 15:10-16:00 @BA6183

 Christophe Lacave [2] (Ecole Polytechnique, Paris) 2010-2011 Analysis Applied Math Seminar Friday July 15 15:10-16:00 BA6183 Title: Self-similar asymptotics of solutions to the Navier-Stokes system in two dimensional exterior domain Abstract: . arXiv 2011_07_15_Lacave_Notes 2011_07_15

## July 15, Kuksin, 14:10-15:00 @BA6183

 Sergei Kuksin [3] (Ecole Polytechnique, Paris) 2010-2011 Analysis Applied Math Seminar Friday July 15 14:10-15:00 BA6183 Title: Damped-driven Hamiltonian PDE Abstract: . arXiv 2011_07_15_Kuksin_Notes 2011_07_15

## July 10, Nestoridis, 14:10-15:00 @BA6183

 Vassili Nestoridis [] (University of Athens, Greece) 2010-2011 Analysis Applied Math Seminar Friday July 10 14:10-15:00 BA6183 Title: Extensions of the disc algebra Abstract: We determine the set A˜(D) of uniform limits of polynomials on the closed unit disc Dˉ with respect to the chordal metric χ on C∪{∞}. We study properties of the elements of A˜(D), as well as topological properties of the class A˜(D) endowed with its natural metric topology. More generally, we examine analogous questions replacing C∪{∞} by an arbitrary metrizable compactification of C and replacing the closed unit disc Dˉ by other compact sets (Mergelyan’s theorem). The set of universal Taylor series in the sense of Luh and Chui and Parnes contained in A˜(D) is Gδ and dense in A˜(D). Are there any such universal series in A˜(D)∖A(D), where A(D) denotes the classical disc algebra? [ arXiv] 2011_07_10_Nestoridis_Notes 2011_07_10

## April 08, K. Abou Salem, 14:10-15:00 @BA6180

 Walid K. Abou Salem [] (University of Saskatchewan) 2010-2011 Analysis Applied Math Seminar Friday April 08 14:10-15:00 BA6180 Title: Renormalization group approach to perturbation theory for PDEs Abstract: I discuss the rigorous application of the CGO-renormalization group method to (singular) perturbation theory for nonlinear PDEs. As a paradigm, I consider the concrete example of the nonlinear Schrödinger equation with quadratic nonlinearity in three spatial dimensions. I show how to obtain an approximate solution using the RG method together with an estimate of the difference between the true and approximate solutions. The analysis applies to differential equations where (space-time) resonances are present. Time permitting, I will discuss some open problems. [ arXiv] 2011_04_08_K. Abou Salem_Notes 2011_04_08

{{Seminar

## March 25, Choksi, 14:10-15:00 @BA6180

 Rustum Choksi [] (McGill University) 2010-2011 Analysis Applied Math Seminar Friday March 25 14:10-15:00 BA6180 Title: The Phase Diagram and Small Volume Fraction Limits in Variational Problems with Short and Long-range Interactions Abstract: Energy-driven pattern formation induced by competing short and long-range interactions is common in many physical systems. A simple but rich mathematical paradigm consists of nonlocal perturbations to the well-studied Cahn-Hilliard and isoperimetric problems. This talk will have two parts. In the first, via a combination of numerical and asymptotic analysis, I will address the phase diagram associated with the Cahn-Hilliard-like functional. This part of the talk is joint work with J.F. Williams and M. Maras at SFU. In the second part, I will present a rigorous asymptotic development of the energy in the small volume fraction regime. Using the language of Gamma-convergence, I describe both the leading order behavior --- associated with coarsening of particles, and the next-order behavior --- associated with self-oganiziation of particles. This is joint work with M. Peletier (TU Eindhoven). If time permits, I will also discuss work in progress with N. Le (Columbia) and Peletier which exploits the Gamma-limit structure of the energy to prove convergence of the associated gradient flows. [ arXiv] 2011_03_25_Choksi_Notes 2011_03_25

## March 18, Anco, 14:10-15:00 @BA6180

 Stephen Anco [] (Brock University) 2010-2011 Analysis Applied Math Seminar Friday March 18 14:10-15:00 BA6180 Title: New conserved integrals for fluid flow in multi-dimensions Abstract: Euler's equations are the governing equations of inviscid fluid flow. In this talk, I will give a summary of recent work on classifying conserved integrals (local conservation laws) of Euler's equations in multi-dimensions. For the first half of the talk, a complete classification of conserved integrals is presented for two important cases in all dimensions n>1 : (1) kinematic conservation laws, like entropy, mass, energy, momentum and angular momentum, which depend only on the fluid velocity, pressure, density and entropy (but not their spatial derivatives), in addition to the time and space coordinates; (2) vorticity conservation laws, such as three-dimensional helicity and two-dimensional circulation, which have an essential dependence on the curl of the fluid velocity. As main results, the classification yields a new circulatory entropy in all even dimensions, and also finds that the only other vorticity conserved integrals are enstrophy in all even dimensions and generalized circulation/helicity in all odd dimensions (which are known from work on Hamiltonian Casimirs). For the second half of the talk, new circulatory constants of motion are presented for the case of isentropic fluid flow in all dimensions n>3 . These constants are defined in terms of the fluid velocity and its curl on any closed surface of odd dimension less than n that is transported by the fluid, and reduce to the circulation in dimensions n≤3 . The corresponding local conservation laws are shown to have an equivalent formulation as differential p -forms (with p=1,3,...,2[n/2]−1 ) whose convective Lie derivative along the fluid streamlines is equal to a closed p -form when evaluated for all solutions of Euler's equations. S. Anco and A. Dar, Proc. Roy. Soc. A 466 (2010), 2605--2632. S. Anco and A. Dar, Proc. Roy. Soc. A 465 (2009), 2461--2488. [ arXiv] 2011_03_18_Anco_Notes 2011_03_18

## March 11, Musslimani, 14:10-15:00 @BA6180

 Ziad Musslimani [] (Florida State University) 2010-2011 Analysis Applied Math Seminar Friday March 11 14:10-15:00 BA6180 Title: Numerical study of one-dimensional Bose-Einstein condensates in a random potential Abstract: We present a numerical study of the effect of a disordered potential on a confined one-dimensional Bose-Einstein condensate, in the framework of a mean-field description (Gross-Pitaevskii), using a highly efficient and fast converging numerical scheme. For repulsive interactions, we find that the average spatial extension of the stationary density profile decreases with an increasing disorder strength. The numerical long-time behavior of the condensate will be addressed. [ arXiv] 2011_03_11_Musslimani_Notes 2011_03_11

## March 4, Kempf, 14:10-15:00 @BA6180

 Achim Kempf [4] (University of Waterloo) 2010-2011 Analysis Applied Math Seminar Friday March 4 14:10-15:00 BA6180 Title: Infinitesimal inverse spectral geometry and applications in mathematical physics Abstract: Inverse spectral geometry, i.e., the question of to what extent one can "hear the shape of a drum" is hard because it is highly nonlinear. I show that new insight can be gained by linearizing the problem, namely by perturbing around a given pair of a "drum" and its spectrum. I will also show that inverse spectral geometry naturally arises with key questions of mathematical physics, in particular, quantum gravity and quantum information theory. [ arXiv] 2011_03_4_Kempf_Notes 2011_03_4

## February 18, Ionescu, 14:10-15:00 @BA6180

 Alex Ionescu [5] (Princeton University) 2010-2011 Analysis Applied Math Seminar Friday February 18 14:10-15:00 BA6180 Title: On global well-posedness of defocusing energy-critical nonlinear Schrödinger equations on certain Riemannian manifolds Abstract: I will talk about some work with B. Pausader and G. Staffilani on global regularity of solutions of the defocusing energy-critical NLS in two cases: the hyperbolic space $H3$ and the semiperiodic setting $R \times T3$. arXiv 2011_02_18_Ionescu_Notes 2011_02_18

## February 11, Larsen, 14:10-15:00 @BA6180

 Christopher Larsen [] (Carnegie Mellon University) 2010-2011 Analysis Applied Math Seminar Friday February 11 14:10-15:00 BA6180 Title: TBA Abstract: TBA [ arXiv] 2011_02_11_Larsen_Notes 2011_02_11

## February 04, Selvitella, 14:10-15:00 @BA6180

 Alessandro Selvitella [] (McMaster University) 2010-2011 Analysis Applied Math Seminar Friday February 04 14:10-15:00 BA6180 Title: Some problems concerning a Quasilinear Schrödinger Equation Abstract: In this seminar we will talk about different issues concerning a Quasilinear Schrödinger Equation. In particular we will discuss a joint work with Prof. Louis Jeanjean about uniqueness and nondegeneracy of the ground state. We will also give an outline of what is known, what will be known soon (hopefully...) and what is going to be not known (for a while at least) about the Cauchy Problem for this equation. [ arXiv] 2011_02_04_Selvitella_Notes 2011_02_04

## January 07, Chen, 14:10-15:00 @BA6180

 Thomas Chen [6] (U. Texas (Austin)) 2010-2011 Analysis Applied Math Seminar Friday January 07 14:10-15:00 BA6180 Title: Mean field limits for interacting Bose gases and the Cauchy problem for Gross-Pitaevskii hierarchies Abstract: This talk surveys some recent results, all based on joint work with Natasa Pavlovic, related to the dynamics of Bose gases, and the Cauchy problem for Gross-Pitaevskii (GP) hierarchies. A GP hierarchy is an infinite system of coupled partial differential equations describing an interacting Bose gas in a mean field limit. First, we describe how the quintic nonlinearSchrödinger equation is derived from an N-body Schrödinger system with 3-body interactions and an associated GP hierarchy.Then, the local well-posedness theory for more general GP hierarchies is addressed, for focusing, defocusing, cubic and quintic interactions. In particular, the occurrence of blowup solutions is discussed (joint work with N. Pavlovic and N. Tzirakis). Furthermore, we present new conserved energy functionals which we apply to extend local to global well-posedness. arXiv 2011_01_07_Chen_Notes 2011_01_07

## December 17, El Smaily, 14:10-15:00 @BA6183

 Mohammad El Smaily (Carnegie Mellon University) 2010-2011 Analysis Applied Math Seminar Friday December 17 14:10-15:00 BA6183 Title: The influence of large drift on front propagation Abstract: Pulsating traveling fronts are solutions of heterogeneous reaction-advection-diffusion equations that model some population dynamics. Fixing a unitary direction e, it is a well-known fact that for nonlinearities of KPP type (after Kolmogorov, Petrovsky and Piskunov, f(u)=u(1-u) is a typical homogeneous KPP nonlinearity), there exists a minimal speed c* such that a pulsating traveling front with a speed c in the direction of e exists if and only if c>= c*. In a periodic heterogeneous framework we have the formula of Berestycki, Hamel and Nadirashvili (2005) for the minimal speed of propagation. This formula involves elliptic eigenvalue problems whose coefficients are expressed in terms of the geometry of the domain, the direction of propagation, and the coefficients of reaction, diffusion and advection of our equation. In this talk, I will describe the asymptotic behaviors of the minimal speed of propagation within either a large drift, a mixture of large drift and small reaction, or a mixture of large drift and large diffusion. These large drift limits' are expressed as maxima of certain variational quantities over the family of first integrals' of the advection field. I will give more details about the limit and a necessary and sufficient condition for which the limit is equal to zero in the 2-d case. [ arXiv] 2010_12_17_El Smaily_Notes 2010_12_17

## December 03, Sulem, 14:10-15:00 @BA6183

 Catherine Sulem (University of Toronto) 2010-2011 Analysis Applied Math Seminar Friday December 03 14:10-15:00 BA6183 Title: Interaction between internal and surface waves in a two layers fluid Abstract: We consider a fluid composed of two essentially immiscible layers separated by a sharp interface such as a thermocline or a pycnocline of differential salinity. Internal waves of various types are commonly generated in the world's oceans, and large amplitude, long wavelength nonlinear waves can be produced in the interface and propagate over large distances. In some instances, the visible signature of internal waves on the surface of the ocean is a band of roughness which propagates at the same velocity as the internal wave. Several of the earliest observations are the most striking, consisting of long brightly shining strips of many kilometers in extent and photographed from the Space Shuttle. I will discuss the asymptotic analysis of the coupling between the interface and the free surface of a two layers fluid in a scaling regime chosen to capture these observations, in which the internal mode is treated as a long wavelength nonlinear internal wave, while the surface mode is smaller and taken in a modulational regime. This talk is based on joint work with Walter Craig and Philippe Guyenne. [ arXiv] 2010_12_03_Sulem_Notes 2010_12_03

## November 29, Zworski, 13:00-14:00 @BA1220

 Maciej Zworski (University of California, Berkeley) 2010-2011 Analysis Applied Math Seminar Monday November 29 13:00-14:00 BA1220 Title: Normally hyperbolic trapped sets and quasinormal modes for black holes Abstract: Normally hyperbolic trapped sets produce resonances which are separated from the unitarity axis, that is have decay rates bounded from below (joint work with J Wunsch). The dynamical set up is structurally stable and appears in the geometry of rotating black holes. The results on resonances can then be used to obtain exponential energy decay for the wave equation in Kerr-DeSitter backgrounds (work of S Dyatlov). [ arXiv] 2010_11_29_Zworski_Notes 2010_11_29

## November 26, Griesemer, 14:10-15:00 @BA6183

 Marcel Griesemer (University of Stuttgart) 2010-2011 Analysis Applied Math Seminar Friday November 26 14:10-15:00 BA6183 Title: Hartree-Fock theory for atoms with closed shells Abstract: In this talk we shall discuss the problem of uniqueness of the Hartree-and Hartree-Fock ground states of atoms. We show, for example, that the Hartree-Fock ground state of a closed shell atom is unique provided the atomic number $Z$ is sufficiently large compared to the number $N$ of electrons. More specifically, a two-electron atom with atomic number $Z\geq 35$ has a unique Hartree-Fock ground state given by two orbitals with opposite spins and identical spatial wave functions. This statement is wrong for some $Z>1$, which exhibits a symmetry breaking. -- Joint work with Fabian Hantsch. [ arXiv] 2010_11_26_Griesemer_Notes 2010_11_26

## November 19, Slepcev, 14:10-15:00 @BA6183

 Dejan Slepcev [7] (Carnegie Mellon University) 2010-2011 Analysis Applied Math Seminar Friday November 19 14:10-15:00 BA6183 Title: Coarsening in energy-driven systems Abstract: Many energy-driven systems exhibit coarsening behavior. Starting from an unstable state a system quickly equilibrates locally, while remaining globally far from the equilibrium. The pattern, that initially forms, slowly coarsens over time --- the characteristic length scales of its features grow. I will discuss examples of the coarsening behavior in models of phase segregation, thin liquid films, and grain-boundary networks. In particular, I will present applications of the approach by Kohn and Otto for obtaining rigorous upper bounds on the rate of coarsening. I will discuss some limitations of the approach and how they may be overcome. [ arXiv] 2010_11_19_Slepcev_Notes 2010_11_19

## November 12, Morgan, 14:10-15:00 @BA6183

 Frank Morgan (Williams College) 2010-2011 Analysis Applied Math Seminar Friday November 12 14:10-15:00 BA6183 Title: Double Bubbles from Euclidean to Gauss Space Abstract: In 1884 Schwarz proved that a single round bubble provides the least-perimeter way to enclose given volume in R^3. Similarly, the familiar double bubble that forms when two soap bubbles come together provides the least-perimeter way to enclose and separate two given volumes of air, although we did not prove this until 2000. In Gauss space, the optimal single bubble is a half-space. What is the optimal double bubble? We also mention some new associated function analytic inequalities. [ arXiv] 2010_11_12_Morgan_Notes 2010_11_12

## November 5, Loss, 12:00-13:00 @BA6180

 Michael Loss (Georgia Institute of Technology) 2010-2011 Analysis Applied Math Seminar Friday November 5 12:00-13:00 BA6180 Title: Localization for the random displacement model Abstract: We prove spectral and dynamical localization for the multi-dimensional random displacement model near the bottom of its spectrum by showing that the approach through multiscale analysis is applicable. In particular, we show that a previously known Lifshitz tail bound can be extended to our setting and prove a new Wegner estimate. A key tool is given by a quantitative form of a property of a related single-site Neumann problem which can be described as "bubbles tend to the corners". This is joint work with Frederic Klopp, Shu Nakamura and Gunter Stolz. [ arXiv] 2010_11_5_Loss_Notes 2010_11_5

## November 5, van Veen, 14:10-15:00 @BA6183

 Lennaert van Veen [8] (University of Ontario Institute of Technology) 2010-2011 Analysis Applied Math Seminar Friday November 5 14:10-15:00 BA6183 Title: Bursting shear flow as cycle-to-cycle homoclinic chaos Abstract: Experiments, simulations and theoretical arguments lend mounting evidence for the "edge state" hypothesis on subcritical transition to shear turbulence. The hypothesis asserts that certain simple, spatially smooth states of fluid motion, such as travelling waves and time-periodic flows, mediate between laminar and turbulent motion. Locally, the stable manifold of an edge state separates laminarizing from bursting flows. The global structure of the separatrix, however, is unknown. In this presentation, we show the existence of a flow homoclinic to a time- periodic edge state in plane Couette turbulence. Through the classical Smale-Birkhoff theorem, this implies a complex global geometry of the separating manifold. In particular, we can expect that any turbulent flow is close to the boundary, and small perturbations can cause it to relaminarize. Also, the homoclinic flow give a preferred route from near-laminar to turbulent flow and back. We study the physical characteristics of this cycle in detail and reveal important differences with the conventional picture of turbulent energy dissipation. Joint work with Genta Kawahara (Osaka University) [ arXiv] 2010_11_5_van Veen_Notes 2010_11_5

## October 29, Holzegel, 14:10-15:00 @BA6183

 Gustav Holzegel (Princeton University) 2010-2011 Analysis Applied Math Seminar Friday October 29 14:10-15:00 BA6183 Title: Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem Abstract: Consider a spacetime which approaches a Schwarzschild solution. We will discuss the following problem: Assuming decay of appropriate norms of the Ricci rotation coefficients and their derivatives, can one prove boundedness/ decay for the curvature components and their derivatives? The talk will give a positive answer to this question and explain some of the difficulties arising from the fact that not all curvature components decay. As an important ingredient, we generalize recent work of Dafermos and Rodnianski regarding decay for the wave equation to the setting of the Bianchi equations. [ arXiv] 2010_10_29_Holzegel_Notes 2010_10_29

## October 22, Jerrard, 14:10-15:00 @BA6183

 Robert L. Jerrard (University of Toronto) 2010-2011 Analysis Applied Math Seminar Friday October 22 14:10-15:00 BA6183 Title: Rigidity of Sobolev isometric embeddings Abstract: It has been known for about 50 years that a $C2$ isometric embedding of the n-dimensional Euclidean unit ball into (n+k)-dimensional Euclidean space enjoys certain rigidity properties if $1\le k < n$. For example, its image cannot be contained in a ball of radius much less than 1. This is easily seen to be false if $k\ge n$, and there are dramatic counterexamples, due to Nash and Kuiper, showing that it is also false if one considers isometric embeddings that are merely $C1$. We consider isometric embedding of the Euclidean $n$-ball into $\mathbb{R}^{n+k}$ that belong to the Sobolev space $W^{2,p}$ for certain choices of $p\le n$. Maps with this regularity may fail to be $C1$. Nonetheless, we show that if $p \ge k+1$ then an isometric embedding is $C^{1,\alpha}$ for some positive $\alpha$, and enjoys rigidity properties similar to those of $C2$ isometric embeddings. These results are believed, but not known, to be optimal. This is joint work with Reza Pakzad. [ arXiv] 2010_10_22_Jerrard_Notes 2010_10_22

## October 15, Ivrii, 14:10-15:00 @BA6183

 Victor Ivrii [9] (Toronto) 2010-2011 Analysis Applied Math Seminar Friday October 15 14:10-15:00 BA6183 Title: 2D- and 3D-Magnetic Schrödinger Operator: Short Loops and Pointwise Spectral Asymptotics Abstract: For Schrödinger operator with a strong magnetic field I am considering asymptotics of $e(x,x,\tau)$ (no spatial averaging) as $\mu\to \infty$, $h\to +0$ where $e(x,y,\tau)$ is the Schwartz kernel of the spectral projector, $h$ and $\mu$ are semiclassical and binding parameters respectively. In pointwise asymptotics not only periodic trajectories but also loops are important. And they are plentiful here as for $k=\pm 1, \pm 2,\pm3,\ldots$ there is a loop after $k$ wingings! The full and detailed analysis: Chapter 16 of My Future Monster Book arXiv 2010_10_15_Ivrii_Notes 2010_10_15

## October 8, Czubak, 14:10-15:00 @BA6183

 Magda Czubak (University of Toronto) 2010-2011 Analysis Applied Math Seminar Friday October 8 14:10-15:00 BA6183 Title: Non-uniqueness of the Navier-Stokes equation in the hyperbolic setting Abstract: We consider the Navier-Stokes equation on the two dimensional hyperbolic space with constant sectional curvature −a2. We prove non-uniqueness of smooth solutions which have finite energy, finite dissipation, and satisfy the global energy inequality. We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation. Finally, as a corollary we show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions. This is joint work with Chi Hin Chan. arXiv 2010_10_8_Czubak_Notes 2010_10_8

## October 1, Moradifam, 14:10-15:00 @BA6183

 Amir Moradifam (University of Toronto) 2010-2011 Analysis Applied Math Seminar Friday October 1 14:10-15:00 BA6183 Title: Regularity of the extremal solution in fourth order nonlinear eigenvalue problems Abstract: We study the regularity of the extremal solution of the semilinear biharmonic equation $\Delta^2 u =\frac{\lambda}{(1-u)^2}$ on a ball $B \subset R^N$, under Navier boundary conditions $u=\Delta u=0$ on $\partial B$, where $\lambda >0$ is a parameter. It is known that there exists a $\lambda^{*}$ such that for $\lambda>\lambda^{*}$ there is no solution while for $\lambda<\lambda^{*}$ there is a branch of minimal solutions. Our main result asserts that the extremal solution $u^{*}$ is regular ($\sup_{B}u^{*}<1$) for $N\leq 8$ and it is singular ($\sup_{B}u^{*}=1$) for $N\geq 9$. Our proof for the singularity of extremal solutions in dimensions $N\geq 9$ is based on certain improved Hardy-Rellich inequalities. [ arXiv] 2010_10_1_Moradifam_Notes 2010_10_1

## September 24, Merigot, 14:10-15:00 @BA6183

 Quentin Merigot (University of Toronto) 2010-2011 Analysis Applied Math Seminar Friday September 24 14:10-15:00 BA6183 Title: Stability of Federer curvature measures Abstract: The curvature measures defined by Federer provide a notion of extrinsic curvature for a large class of compact subsets of the Euclidean space that includes compact convex sets and compact smooth submanifolds. These curvature measures are defined through a tube formula that generalizes those of Steiner and Weyl. In this presentation, we will study the somewhat simpler notion of boundary measure, which can be used to recover the curvature measures through polynomial fitting. The boundary measure of a compact set K with respect to a bounded open set E is the pushforward of the restriction of the Lebesgue measure to E by the projection function onto K. Our main result is the following: the boundary measure of a compact set K depends continuously on K (for the Hausdorff distance); moreover, if one endows the set of measures with a given mass with the Wasserstein distance (from optimal transportation), the dependence is 1/2-Hölder. This stability result will be obtained either by a fine study of the covering numbers of the medial axis of a compact set (also called "ambiguous locus" in Riemanian geometry) or as the consequence of a L^1 stability result for gradient of convex functions. If time permits, we will briefly discuss the algorithmic aspects of computing boundary measures. [ arXiv] 2010_09_24_Merigot_Notes 2010_09_24

## September 10, Frantzikinakis, 15:10-16:00 @BA6183

 Nikos Frantzikinakis [10] (Crete) 2010-2011 Analysis Applied Math Seminar Friday September 10 15:10-16:00 BA6183 Title: Ergodic methods in combinatorics Abstract: I am going to survey some old and new results that showcase the interaction of ergodic theory and combinatorial number theory. A typical such example is Furstenberg's proof of Szemeredi's theorem on arithmetic progressions that was proved using ergodic techniques. Several far reaching extensions of this result became accessible only recently, and their proof uses some rather deep structural theorems in ergodic theory (that exploit the dichotomy between structure and randomness) and some new equidistribution results on nilmanifolds (these are generalizations of compact Abelian Lie groups). [ arXiv] 2010_09_10_Frantzikinakis_Notes 2010_09_10

## September 03, Laul, 14:10-15:00 @BA6183

 Parul Laul [11] (UNC) 2010-2011 Analysis Applied Math Seminar Friday September 03 14:10-15:00 BA6183 Title: Localized energy estimates for wave equations on high dimensional Schwarschild space-times Abstract: Localized energy estimates for the wave equation on Minkowski and (1+3)-dimensional Schwarzschild space-times have had various applications; for example, in the proof of Price’s Law. We discuss a similar localized energy estimate for the homogeneous wave equation $\square_g = 0$ on the $(1 + n)$-dimensional hyperspherical Schwarzschild manifold. arXiv 2010_09_03_Laul_Notes 2010_09_03