# 2011-2012 Analysis Applied Math Seminar

This page contains information about the Analysis and applied Math Seminar at the University of Toronto. The seminar meets regularly on Fridays, 1:10-2pm, 6183 Bahen Center.

The current (interim) organizers are Bob Jerrard (rjerrard [at] math) and Amir Moradifam (amir [at] math).

If you will be speaking, details on how to use the equipment in BA 6183 can be found here: BA6183_Video_Instructions.

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

Previous Year's Seminars: 2010-11, 2009-10, 2008-09, 2007-08, 2006-07, 2005-06, 2004-05, 2003-04, 2002-03.

## April 13, Choksi, 13:10-14:00 @BA6183

 Rustum Choksi (McGill University) 2011-2012 Analysis Applied Math Seminar Friday April 13 13:10-14:00 BA6183 Title: Global Minimization and the Energy Landscape for a Variational Problem with Long-Range Interactions Abstract: Energy-driven pattern formation induced by competing short and long-range interactions is common in many physical systems. A nonlocal perturbation (of Coulombic-type) to the well-known Ginzburg-Landau/Cahn-Hilliard free energy gives rise to a mathematical paradigm with a rich and complex energy landscape. In the first half of this talk, we discuss rigorous asymptotic results concerning global minimizers. In the second part, we discuss a few hybrid numerical methods for accessing ground states. [ arXiv] 2012_4_13_Choksi_Notes 2012_4_13

## March 23, Deegan, 13:10-14:00 @BA6183

 Robert Deegan (University of Bristol) 2011-2012 Analysis Applied Math Seminar Friday March 23 13:10-14:00 BA6183 Title: Wet drop impact Abstract: When a fast moving drop collides with a layer of fluid it a produces a splash, a spray of secondary droplets. There is a bewildering variety of splash morphologies and droplet distributions which manifest as the system parameters (droplet size and speed, layer depth, fluid properties) are varied. Despite this complexity, a splash begins with the formation of a sheet-like jet. There are at least two varieties of jets: the large and slow lamella jet and the small and quick ejecta jet. In this talk I will present our progress towards understanding the simplest of splashes, the so-called crown splash, which results from the disintegration of the lamella. I will also discuss our experimental results on the ejecta jet and the role of the surrounding gas on its evolution. [ arXiv] 2012_3_23_Deegan_Notes 2012_3_23

## March 16, Lan, 13:10-14:00 @BA6183

 Kunquan Lan (Ryerson University) 2011-2012 Analysis Applied Math Seminar Friday March 16 13:10-14:00 BA6183 Title: Nonzero positive solutions of systems of elliptic boundary value problems Abstract: In this talk, I shall present some results on existence of nonzero positive (classic) solutions of systems of second order elliptic boundary value problems under some sublinear conditions involving the principle eigenvalues of the corresponding linear systems. Some results on eigenvalue problems of such elliptic systems are derived and generalize some previous results on the eigenvalue problems of systems of Laplacian elliptic equations. [ arXiv] 2012_3_16_Lan_Notes 2012_3_16

## March 2, Francis, 13:10-14:00 @BA6183

 Bruce Francis (University of Toronto) 2011-2012 Analysis Applied Math Seminar Friday March 2 13:10-14:00 BA6183 Title: Pursuit Laws for Mobile Robots Abstract: This talk describes some recent research in control theory, in particular, the control of autonomous robots. The talk has three parts: 1) A brief motivation for the research and some experimental results. 2) The rendezvous problem with limited vision: Design a motion control law so that a group of mobile robots gather at a single location even though their on-board cameras are near-sighted. The proposed control law leads to coupled differential equations with non-Lipschitz right-hand sides. 3) Infinite chains of robots. In studying the formation of a very large number of robots, one approach is instead to model an infinite number of robots. The relevant question is what mathematical framework to take so that the infinite-chain model correctly describes the behaviour of the large-but-finite chain model. Studies to date take the state space to be the Hilbert space of square-summable sequences. The advantage is that there is a rich Fourier theory available if the formation is spatially invariant. But this Hilbert space formulation leads to anomalous behaviour. For example, an infinite chain of vehicles when displaced will return to their starting points even though the vehicles do not have global sensing capability and therefore could not in reality do so. This talk proposes a different mathematical framework and describes the progress made so far. The problem turns out to be related to some Tauberian theory of Diaconis and Stein. This is joint work with Avraham Feintuch, Math Department, Ben Gurion University of the Negev. [ arXiv] 2012_3_2_Francis_Notes 2012_3_2

## February 17, Athavale, 13:10-14:00 @BA6183

 Prashant Athavale (University of Toronto) 2011-2012 Analysis Applied Math Seminar Friday February 17 13:10-14:00 BA6183 Title: Integro-differential equations and multiscale representations Abstract: In this talk we will discuss various aniosotropic PDEs methods for applications to image processing. We will then discuss integro-differential equations inspired from (BV, L^2) and (BV, L^1) decompositions. Although, the original motivation came from a variational approach, the resulting IDEs can be extended using standard techniques from PDE-based image processing. We use filtering, edge preserving and tangential smoothing to yield a family of modified IDE models with applications to image denoising and image deblurring problems. [ arXiv] 2012_02_17_Athavale_Notes 2012_02_17

## February 10, Zhou, 13:10-14:00 @BA6183

 Gang Zhou (ETH-Zurich) 2011-2012 Analysis Applied Math Seminar Friday February 10 13:10-14:00 BA6183 Title: On Singularity Formation Under Mean Curvature Flow Abstract: In this talk I present our recent works, jointly with D.Knopf and I.M.Sigal, on singularity formation under mean curvature flow. By methods borrowed from dispersive equations and mathematical physics we present a very different way of studying it, and moreover obtain asymptotic of singularity formation on asymmetric surfaces for the first time. After reviewing known results I will compare our approaches to the old ones. Some key elements will be discussed. A few problems, which might be tackled by our techniques, will be formulated. [ arXiv] 2012_2_10_Zhou_Notes 2012_2_10

## February 3, Wang, 13:10-14:00 @BA6183

 Yun Wang (McMaster University) 2011-2012 Analysis Applied Math Seminar Friday February 3 13:10-14:00 BA6183 Title: Critical Sobolev Inequalities and Navier-Stokes Equations Abstract: In this talk, some critical Sobolev inequalities are introduced. These inequalities are generalizations of Brezis-Gallouet-Wainger inequality. We apply such inequalities to the two-dimensional non-homogeneous incompressible Navier-Stokes problem and prove global existence of strong solutions. [ arXiv] 2012_2_3_Wang_Notes 2012_2_3

## January 27, Berlyand, 13:10-14:00 @BA6183

 Leonid Berlyand (Pennsylvania State University) 2011-2012 Analysis Applied Math Seminar Friday January 27 13:10-14:00 BA6183 Title: Flux norm approach to finite-dimensional homogenization approximation with non-separated scales and high contrast Abstract: Classical homogenization theory deals with mathematical models of strongly inhomogeneous media described by PDEs with rapidly oscillating coefficients of the form $A(x/\epsilon)$, $\epsilon \to 0$. The goal is to approximate this problem by a homogenized (simpler) PDE with slowly varying coefficients that do not depend on the small parameter $\epsilon$. The original problem has two scales: fine $O(\epsilon)$ and coarse $O(1)$, whereas the homogenized problem has only a coarse scale. The homogenization of PDEs with periodic or random ergodic coefficients and well-separated scales is well understood. In a joint work with H. Owhadi (ARMA 2010) we consider the most general case of arbitrary $L^\infty$ coefficients, which may contain infinitely many scales that are not necessarily well-separated. Specifically, we study scalar and vectorial divergence-form elliptic PDEs with such coefficients. We establish two finite-dimensional homogenization approximations that generalize the {\it correctors} in classical homogenization. We introduce a flux norm and establish the error estimate in this norm with an explicit and {\it optimal} error constant {\it independent of the contrast} and regularity of the coefficients. A proper generalization of the notion of a cell problem in classical homogenization is the key issue in our consideration. Next we discuss most recent results (L. Zhang, Owhadi) on localized multiscale basis that allows for numerical implementation of our theoretical results and work in progress (with Owhadi and Zhang) on compactness of the solution space and new corrector results in classical periodic homogenization problem. [{{{arxiv}}} arXiv] 2012_1_27_Berlyand_Notes 2012_1_27

## January 20, Serea, 13:10-14:00 @BA6183

 Oana Silvia Serea (Universite de Perpignan) 2011-2012 Analysis Applied Math Seminar Friday January 20 13:10-14:00 BA6183 Title: CANCELLED Abstract: CANCELLED [ arXiv] 2012_1_20_Serea_Notes 2012_1_20

## December 02, Krivodonova, 13:10-14:00 @BA6183

 Lilia Krivodonova (University of Waterloo) 2011-2012 Analysis Applied Math Seminar Friday December 02 13:10-14:00 BA6183 Title: Developments in High-Order Discontinuous Galerkin Methods for Hyperbolic Conservation Laws Abstract: A variety of physical phenomena in fluid mechanics, groundwaterflow, electromagnetics and other areas can be described by hyperbolicconservation laws. The discontinuous Galerkin methods (DGM) have become very popular in recent years due to their ability to accurately capture discontinuities often present in such problems. Their low dispersion and dissipation errors make them very suitable for long time computations. We describe our approach to computing high-order accurate solutions for time dependent problems on structured and unstructured meshes. We review several aspects of our recent work including a connection between high accuracy of the DGM methods, their restrictive CFL condition and the classical Pade approximants. Applications include examples from compressible fluid dynamics and electromagnetics [ arXiv] 2011_12_02_Krivodonova_Notes 2011_12_02

## November 25, Hoell, 13:10-14:00 @BA6183

 Nicholas Hoell (University of Toronto) 2011-2012 Analysis Applied Math Seminar Friday November 25 13:10-14:00 BA6183 Title: Inverting the Attenuated X-Ray Transform Abstract: In this talk we present methods for analytically inverting the attenuated ray transform in 2-dimensional settings. The method is based of a study of the transport equation generating the integral curves over which the unknown function is averaged. This problem first arose in the medical imaging modality SPECT and has recently been useful in the unique determination of interior permittivity and permeability parameters of a conductive body from external measurements. [ arXiv] 2011_11_25_Hoell_Notes 2011_11_25

## November 18, Maggi, 13:10-14:00 @BA6183

 Francesco Maggi (Universita di Firenze) 2011-2012 Analysis Applied Math Seminar Friday November 18 13:10-14:00 BA6183 Title: Sharp Stability Estimates in Geometric Variational Problems Abstract: Optimal stability estimates for isoperimetric and Plateau type problems are presented, together with some improvable results and open problems. [ arXiv] 2011_11_18_Maggi_Notes 2011_11_18

## November 11, Westrich, 13:10-14:00 @BA6183

 Matthias Westrich (McGill University) 2011-2012 Analysis Applied Math Seminar Friday November 11 13:10-14:00 BA6183 Title: Regularity of Eigenstates in Regular Mourre Theory Abstract: We discuss an abstract method to prove that eigenstates, associated with possibly embedded eignvalues, of a self-adjoint operator $H$ are in the domain of the k'th power of a conjugate operator $A$. Conjugate means here that $A$ and $H$ have a positive commutator locally near the relevant eigenvalue in the sense of Mourre. The only requirement is $C^{k+1} (A )$ regularity of $H$. Regarding integer $k$, our result is optimal. Under a natural boundedness assumption on the multiple commutators, we prove that $e^{i\theta A}$ (the eigenstate) is analytic in a strip around the real axis. Natural applications are 'dilation analytic' systems satisfying a Mourre estimate, where our result can be viewed as an abstract version of a theorem due to Balsev and Combes (1971). As a new application we discuss the massive Spin-Boson model. [ arXiv] 2011_11_11_Westrich_Notes 2011_11_11

## October 28, Bierri, 13:10-14:00 @BA6183

 Lydia Bierri [1] (University of Michigan) 2011-2012 Analysis Applied Math Seminar Friday October 28 13:10-14:00 BA6183 Title: From the Analysis of Einstein-Maxwell Spacetimes in General Relativity to Gravitational Radiation Abstract: A major goal of mathematical General Relativity (GR) and astrophysics is to precisely describe and finally observe gravitational radiation, one of the predictions of GR. In order to do so, one has to study the null asymptotical limits of the spacetimes for typical sources such as binary neutron stars and binary black hole mergers. D. Christodoulou showed that every gravitational-wave burst has a nonlinear memory, displacing test masses permanently. In joint work with P. Chen and S.-T. Yau we investigated the Einstein-Maxwell (EM) equations in GR and proved that the electromagnetic field contributes at highest order to the memory effect. In this talk, we discuss the null asymptotics for spacetimes solving EM equations, compute the radiated energy and derive limits at null infinity and compare them with the Einstein vacuum (EV) case. The physical insights are based on geometric-analytic investigations of the solution spacetimes. [ arXiv] 2011_10_28_Bierri_Notes 2011_10_28

## October 21, Egli, 13:10-14:00 @BA6183

 Daniel Egli [2] (University of Toronto) 2011-2012 Analysis Applied Math Seminar Friday October 21 13:10-14:00 BA6183 Title: Anderson localization triggered by spin disorder Abstract: The phenomenon of Anderson localization is studied for a class of one-particle Schrödinger operators with random Zeeman interactions. These operators arise as follows: Static spins are placed randomly on the sites of a simple cubic lattice according to a site percolation process with density $x$ and coupled to one another ferromagnetically. Scattering of an electron in a conduction band at these spins is described by a random Zeeman interaction term that originates from indirect exchange. It is shown rigorously that, for positive values of $x$ below the percolation threshold, the spectrum of the one-electron Schrödinger operator near the band edges is dense pure-point, and the corresponding eigenfunctions are exponentially localized. Localization near the band edges persists in a weak external magnetic field, $H$, but disappears gradually, as $H$ is increased. Our results lead us to predict the phenomenon of colossal (negative) magnetoresistance and the existence of a Mott transition, as $H$ and/or $x$ are increased. Our analysis is motivated directly by experimental results concerning the magnetic alloy $\mathsf{Eu}_x\mathsf{Ca}_{1-x}\mathsf{B}_6$. [ arXiv] 2011_10_21_Egli_Notes 2011_10_21

## October 14, Tsugawa, 13:10-14:00 @BA6183

 Kotaro Tsugawa [3] (Nagoya University/University of Toronto) 2011-2012 Analysis Applied Math Seminar Friday October 14 13:10-14:00 BA6183 Title: Local well-posedness of the KdV equation with almost periodic initial data Abstract: We consider the Cauchy problem of the KdV equation. The well-posedness in the Sobolev space of periodic functions has been intensively studied by many people. In this talk, we prove the local well-posedness in an almost periodic function space. The function space contains functions satisfying $f=f_1+f_2+...+f_N$ where $f_j$ is inthe Sobolev space of order $s>-1/2N$ of $a_j$ periodic functions. Note that $f$ is not periodic when the ratio of periods $a_i/a_j$ is irrational. The main tool of the proof is the Fourier restriction norm method introduced by Bourgain. arXiv 2011_10_14_Tsugawa_Notes 2011_10_14

## October 7, Shao, 13:10-14:00 @BA6183

 Arick Shao (University of Toronto) 2011-2012 Analysis Applied Math Seminar Friday October 7 13:10-14:00 BA6183 Title: Breakdown Criteria for Nonvacuum Einstein Equations Abstract: We extend a recent breakdown/continuation result of S. Klainerman and I. Rodnianski for the Einstein-vacuum equations to the Einstein-scalar and the Einstein-Maxwell equations. Roughly, the main theorem states that if an existing local solution of these equations satisfy certain uniform bounds for the second fundamental form, lapse, and matter field, then it can be further extended in time. This can also be reformulated as conditions that must be satisfied when such a solution blows up. In particular, in these nonvacuum settings, we encounter additional difficulties resulting from the nontrivial Ricci curvature and from the coupling between the Einstein and the matter field equations. [ arXiv] 2011_10_7_Shao_Notes 2011_10_7

## September 30, Albin, 13:10-14:00 @BA6183

 Pierre Albin [4] (University of Illinois at Urbana-Champaign) 2011-2012 Analysis Applied Math Seminar Friday September 30 13:10-14:00 BA6183 Title: The signature operator on stratified pseudomanifolds Abstract: The signature operator of a Riemannian metric is an important tool for studying topological questions with analytic machinery. Though well-understood for smooth metrics on compact manifolds, there are many open questions when the metric is allowed to have singularities. I will report on joint work with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza on the signature operator on stratified pseudomanifolds and some of its topological applications. [ arXiv] 2011_09_30_Albin_Notes 2011_09_30

## September 23, Galvao-Sousa, 13:10-14:00 @BA6183

 Bernardo Galvao-Sousa [5] (University of Toronto) 2011-2012 Analysis Applied Math Seminar Friday September 23 13:10-14:00 BA6183 Title: Thin films for the Ginzburg-Landau model Abstract: I will present recent results in collaboration with Stan Alama and Lia Bronsard on thin film London limits of the Ginzburg--Landau model. We obtain $\Gamma$--convergence results for the first and second critical fields under particular asymptotic ratios between the magnitude of the parallel applied magnetic field and the thickness of the film. For the first critical field, we study the optimal density of vortices via an obstacle problem for some examples to illustrate how the geometry of the domain will affect the position of vortices. [ arXiv] 2011_09_23_Galvao-Sousa_Notes 2011_09_23

## September 16, Sutherland, 14:10-15:00 @BA6180

 Scott Sutherland [6] (SUNY Stonybrook) 2011-2012 Analysis Applied Math Seminar Friday September 16 14:10-15:00 BA6180 Title: Bounds on the cost of root finding Abstract: We discuss a path-lifting method for finding an approximate zero of f(z)=0, where f is a complex polynomial, and show that the number of function evaluations required to locate a solution depends only on the geometry of the polynomial, not on the degree. [ arXiv] 2011_09_16_Sutherland_Notes 2011_09_16

## September 16, Seis, 13:10-14:00 @BA6183

 Christian Seis [7] (University of Toronto) 2011-2012 Analysis Applied Math Seminar Friday September 16 13:10-14:00 BA6183 Title: Rayleigh-Bénard convection: Bounds on the Nusselt number Abstract: We consider Rayleigh--Bénard convection as modelled by the Boussinesq equations in the infinite-Prandtl-number limit. We are interested in the scaling of the average upward heat transport, the Nusselt number $Nu$, in terms of the non-dimensionalized temperature forcing, the Rayleigh number $Ra$. Experiments, asymptotics and heuristics suggest that $Nu \sim Ra^{1/3}$. This work is mostly inspired by two earlier rigorous work on upper bounds of $Nu$ in terms of $Ra$: 1.) The work of Constantin and Doering establishing $Nu \lesssim Ra^{1/3} \ln^{2/3}Ra$ with help of a (logarithmically failing) maximal regularity estimate in $L^{\infty}$ on the level of the Stokes equation. 2.) The work of Doering, Reznikoff and Otto establishing $Nu\lesssim Ra^{1/3}\ln^{1/3}Ra$ with help of the background field method. We present two results: 1.) The background field method can be slightly modified to yield $Nu\lesssim Ra^{1/3}\ln^{1/15}Ra$ --- which is optimal for the background flield method. 2.) The estimates behind the background field method can be combined with the maximal regularity in $L^{\infty}$ to yield $Nu\lesssim Ra^{1/3}\ln^{1/3}\ln Ra$ --- an estimate that is only a double logarithm away from the supposed optimal scaling. This is joint work with Felix Otto. [ arXiv] 2011_09_16_Seis_Notes 2011_09_16

## September 09, Zworski, 13:10-14:00 @BA6183

 Maciej Zworski [8] (University of California at Berkeley) 2011-2012 Analysis Applied Math Seminar Friday September 09 13:10-14:00 BA6183 Title: A semiclassical proof of Quillen's theorem Abstract: In this expository talk I give a semiclassical interpretation of Quillen's original proof of his 1968 theorem on decomposition of positive complex bi-homogeneous forms into sums of Hermitian squares. The result can be interpreted as a hermitian analogue of Hilbert's 17th problem and it was rediscovered by Catlin and D'Angelo in 1996. Both proofs are related to the Fourier–Bros–Iagolnitzer transform (FBI) transform, quantum harmonic oscillator and the calculus of Toeplitz operators. [ arXiv] 2011_09_09_Zworski_Notes 2011_09_09