2019-20 Montreal Analysis Seminar

Seminars are usually held on Mondays or Fridays at Concordia, McGill or Universite de Montreal
To attend a zoom session, and for suggestions, questions etc. please contact Galia Dafni (galia.dafni@concordia.ca), Dmitry Jakobson (dmitry.jakobson@mcgill.ca), Damir Kinzebulatov (damir.kinzebulatov@mat.ulaval.ca) or Iosif Polterovich (iossif@dms.umontreal.ca)

WINTER 2020

Joint seminar with geometric analysis
Friday, January 17, 13:30-14:30, McGill, Burnside Hall, Room 1104
Henrik Matthiesen (University of Chicago)
Handle attachment and the normalised first eigenvalue
Abstract: I will discuss asymptotic lower bounds of the first eigenvalue for two constructions of attaching degenerating handles to a given closed Riemannian surface. One of these constructions is relatively simple but often fails to strictly increase the first eigenvalue normalized by area. Motivated by this negative result, we then give a much more involved construction that always strictly increases the first eigenvalue normalized by area. As a consequence we obtain the existence of a metric that maximizes the first eigenvalue among all unit area metrics on a given closed surface. This is based on joint work with Anna Siffert.

Friday, January 31, 13:30-14:30, Concordia, Library building, Room LB 921-4
Alexey Kokotov (Concordia)
Flat conical Laplacian in the square of the canonical bundle and its regularized determinants
Abstract: We discuss two natural definitions of the determinant of the Dolbeault Laplacian acting in the square of the canonical bundle over a compact Riemann surface equipped with flat conical metric given by the modulus of a holomorphic quadratic differential with simple zeroes. The first one uses the zeta-function of some special self-adjoint extension of the Laplacian (initially defined on smooth sections vanishing near the zeroes of the quadratic differential), the second one is an analog of Eskin-Kontsevich-Zorich (EKZ) regularization of the determinant of the conical Laplacian acting in the trivial bundle. In contrast to the situation of operators acting in the trivial bundle, these two regularizations turn out to be essentially different. Considering the regularized determinant of the Laplacian as a functional on the moduli space of quadratic differentials with simple zeroes on compact Riemann surfaces of a given genus, we derive explicit expressions for this functional for the both regularizations. The expression for the EKZ regularization is closely related to the well-known explicit expressions for the Mumford measure on the moduli space of compact Riemann surfaces.

Friday, February 14, 13:30-14:30, McGill, Burnside Hall, Room 1104
Bradley Siwick (Chemistry and Physics, McGill)
STRUCTURE AND DYNAMICS WITH ULTRAFAST ELECTRON MICROSCOPES … or how to make atomic-level movies of fundamental processes in molecules and materials
Abstract: In this talk I will describe how combining ultrafast lasers and electron microscopes in novel ways makes it possible to directly ‘watch’ the time-evolving structure of condensed matter and the couplings between carrier and lattice degrees of freedom on the fastest timescales open to atomic motion [1-4]. By combining such measurements with complementary (and more conventional) spectroscopic probes we can now develop structure-property relationships for materials under even very far from equilibrium conditions [2]. I will assume no familiarity with ultrafast lasers or electron microscopes and highlight opportunities for theoretical/computational developments that support these experiments. Rigorous modeling of the processes described is currently intractable.
[1] Morrison et al Science 346 (2014) 445
[2] Otto et al, PNAS, 116 (2019) 450
[3] Stern et al, Phys. Rev. B 97 (2018) 165416
[4] Rene de Cotret et al, Phys. Rev. B 100 (2019) 214115

Joint with Symplectic Geometry Seminar
Friday, February 28, 13:40-14:40, UQAM, Pavillon President Kennedy, Salle PK-5115
Egor Shelukhin (Universite de Montreal)
Smith theory in Floer homology, persistence, and dynamics
Abstract: Recent years have seen a renewed interest in a classical topological inequality that originated in the work of P. A. Smith, extended to the framework of Floer homology. We describe such inequalities in the setting of persistence modules obtained from Floer homology, and their recent applications to questions in Hamiltonian dynamics. In particular, we show that for a class of symplectic manifolds including complex projective spaces, a Hamiltonian diffeomorphism with more fixed points, counted suitably, than the dimension of the ambient homology, must have an infinite number of simple periodic points. This is a higher-dimensional homological generalization of a celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in 1994. Time permitting, we may discuss further, very recent, related results.

2020 CRM/Montreal/Quebec Analysis Seminar

After a break due to COVID-19, the seminar is resuming on zoom, organized jointly with Laval university in Quebec city. Please, contact one of the organizers for the seminar zoom links.

Joint with Geometris Analysis Seminar
Wednesday, April 29, 13:30-14:30, Zoom seminar
Julian Scheuer (Freiburg)
Concavity of solutions to elliptic equations on the sphere
Abstract: An important question in PDE is when a solution to an elliptic equation is concave. This has been of interest with respect to the spectrum of linear equations as well as in nonlinear problems. An old technique going back to works of Korevaar, Kennington and Kawohl is to study a certain two-point function on a Euclidean domain to prove a so-called concavity maximum principle with the help of a first and second derivative test.
To our knowledge, so far this technique has never been transferred to other ambient spaces, as the nonlinearity of a general ambient space introduces geometric terms into the classical calculation, which in general do not carry a sign.
In this talk we have a look at this situation on the unit sphere. We prove a concavity maximum principle for a broad class of degenerate elliptic equations via a careful analysis of the spherical Jacobi fields and their derivatives. In turn we obtain concavity of solutions to this class of equations. This is joint work with Mat Langford, University of Tennessee Knoxville.

Friday, May 1, starts at 13:00 (Eastern time), on zoom
13:00. Alexandre Girouard (Universite Laval)
Homogenization of Steklov problems with applications to sharp isoperimetric bounds, part I
Abstract: The question to find the best upper bound for the first nonzero Steklov eigenvalue of a planar domain goes back to Weinstock, who proved in 1954 that the first nonzero perimeter-normalized Steklov eigenvalue of a simply-connected planar domain is 2*pi, with equality iff the domain is a disk. In a recent joint work with Mikhail Karpukhin and and Jean Lagacé, we were able to let go of the simple connectedness assumption. We constructed a family of domains for which the perimeter-normalized first eigenvalue tends to 8π. In combination with Kokarev's bound from 2014, this solves the isoperimetric problem completely for the first nonzero eigenvalue. The domains are obtained by removing small geodesic balls that are asymptotically densely periodically distributed as their radius tends to zero. The goal of this talk will be to survey recent work on homogenisation of the Steklov problem which lead to the above result. On the way we will see that many spectral problems can be approximated by Steklov eigenvalues of perforated domains. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than 2*pi. This talk is based on joint work with Antoine Henrot (U. de Lorraine), Mikhail Karpukhin (UCI) and Jean Lagacé(UCL).
13:50. Jean Lagacé (UCL)
Homogenization of Steklov problems with applications to sharp isoperimetric bounds, part II.
Abstract: Traditionally, deterministic homogenisation theory uses the periodic structure of Euclidean space to describe uniformly distributed perturbations of a PDE. It has been known for years that it has many applications to shape optimization. In this talk, I will describe how the lack of periodic structure can be overcome to saturate isoperimetric bounds for the Steklov problem on surfaces. The construction is intrinsic and does not depend on any auxiliary periodic objects or quantities. Using these methods, we obtain the existence of free boundary minimal surfaces in the unit ball with large area. I will also describe how the intuition we gain from the homogenization construction allows us to actually construct some of them, partially verifying a conjecture of Fraser and Li. This talk is based on joint work with Alexandre Girouard (U. Laval), Antoine Henrot (U. de Lorraine) and Mikhail Karpukhin (UCI).

Thursday, May 7, starts at 13:00 (Eastern time), Zoom seminar
Steve Zelditch (Northwestern)
Spectral asymptotics for stationary spacetimes
Abstract: We explain how to formulate and prove analogues of the standard theorems on spectral asymptotics on compact Riemannian manifolds -- Weyl's law and the Gutzwiller trace formula-- for stationary spacetimes. As a by-product we prove a semi-classical Weyl law for the Klein-Gordon equation where the mass is the inverse Planck constant.

Friday, May 15, 14:30-15:30 Eastern time, Zoom seminar
Malik Younsi (Hawaii)
Holomorphic motions, conformal welding and capacity
Abstract: The notion of a holomorphic motion was introduced by Mane, Sad and Sullivan in the 1980's, motivated by the observation that Julia sets of rational maps often move holomorphically with holomorphic variations of the parameters. Even though the original motivation for their study came from complex dynamics, holomorphic motions have found over the years to be of fundamental importance in other related areas of Complex Analysis, such as the theory of Kleinian groups and Teichmuller theory for instance. Holomorphic motions also played a central role in the seminal work of Astala on distortion of dimension and area under quasiconformal mappings. In this talk, I will first review the basic notions and results related to holomorphic motions, including quasiconformal mappings and the (extended) lambda lemma. I will then present some recent results on the behavior of logarithmic capacity and analytic capacity under holomorphic motions. As we will see, conformal welding (of quasicircle Julia sets) plays a fundamental role. This is joint work with Tom Ransford and Wen-Hui Ai.

Friday, May 22, 11:00-12:00 Eastern time, Zoom seminar
Jeff Galkowski (UCL)
Viscosity limits for 0th order operators
Abstract: In recent work, Colin de Verdiere--Saint-Raymond and Dyatlov--Zworski showed that a class of zeroth order pseudodifferential operators coming from experiments on forced waves in fluids satisfies a limiting absorption principle. Thus, these operators have absolutely continuous spectrum with possibly finitely many embedded eigenvalues. In this talk, we discuss the effect of small viscosity on the spectra of these operators, showing that the spectrum of the operator with small viscosity converges to the poles of a certain meromorphic continuation of the resolvent through the continuous spectrum. In order to do this, we introduce spaces based on an FBI transform which allows for the testing of microlocal analyticity properties. This talk is based on joint work with M. Zworski.

Thursday, May 28, 12:00-13:00 Eastern time, Zoom seminar
Blair Davey (CUNY)
A quantification of the Besicovitch projection theorem and its generalizations
Abstract: The Besicovitch projection theorem asserts that if a subset E of the plane has finite length in the sense of Hausdorff and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every linear projection of E to a line will have zero measure. As a consequence, the probability that a line dropped randomly onto the plane intersects such a set E is equal to zero. Thus, the Besicovitch projection theorem is connected to the classical Buffon needle problem. Motivated by the so-called Buffon circle problem, we explore what happens when lines are replaced by more general curves. We discuss generalized Besicovitch theorems and, as Tao did for the classical theorem (Proc. London Math. Soc., 2009), we use multi-scale analysis to quantify these results. This work is joint with Laura Cladek and Krystal Taylor.

Wednesday, June 3, 13:30-14:30, ON ZOOM
Joint seminar with geometric analysis
Sagun Chanillo (Rutgers)
Bourgain-Brezis inequalities, applications and Borderline Sobolev inequalities on Riemannian Symmetric spaces of non-compact type.
Abstract: Bourgain and Brezis discovered a remarkable inequality which is borderline for the Sobolev inequality in Eulcidean spaces. In this talk we obtain these inequalities on nilpotent Lie groups and on Riemannian symmetric spaces of non-compact type. We obtain applications to Navier Stokes eqn in 2D and to Strichartz inequalities for wave and Schrodinger equations and to the Maxwell equations for Electromagnetism. These results were obtained jointly with Jean Van Schaftingen and Po-lam Yung.

Thursday, June 11, Time TBA, zoom seminar
Spyros Alexakis (Toronto)
Title TBA

FALL 2019 MONTREAL ANALYSIS SEMINAR

Friday, September 6, 13:30-14:30, McGill, Burnside Hall, Room 1104
Reem Yassawi (Open University)
Measure non-rigidity for linear cellular automata
Abstract: pdf

Friday, September 20, 13:30-14:30, McGill, Burnside Hall, Room 1104
Damir Kinzebulatov (Laval)
Heat kernel bounds and desingularizing weights for non-local operators
Abstract: In 1998, Milman and Semenov introduced the method of desingularizing weights in order to obtain sharp two-sided bounds on the heat kernel of the Schroedinger operator with a potential having critical-order singularity at the origin. In this talk, I will discuss the method of desingularizing weights in a non-symmetric, non-local situation. In particular, I will talk about sharp two-sided bounds on the heat kernel of the fractional Laplacian perturbed by a Hardy drift. The crucial ingredient of the desingularization method is a weighted L^1->L^1 estimate on the semigroup, leading to the weighted Nash initial estimate. Milman and Semenov established this estimate appealing to the Stampacchia criterion in L^2. These arguments becomes quite problematic in the non-local non-symmetric situation (e.g. for a strong enough singularity of the drift, there is only L^p theory of the operator for p>2). The core of the talk will be the discussion of a new approach to the proof of this estimate. Joint with Yu.A.Semenov and K.Szczypkowsi (arxiv:1904.07363)

Monday, November 4, 13:30-14:30, McGill, Burnside Hall, Room 1104
Stephane Sabourau (U. Paris-Est)
Systolically extremal metrics on nonpositively curved surfaces
Abstract: The regularity of systolically extremal surfaces (i.e., surfaces of minimal area with fixed systole) is a delicate problem already discussed by M. Gromov in the 80's. We propose to study the problem of systolically extremal metrics in the context of generalized metrics of nonpositive curvature. A natural approach would be to work in the class of Alexandrov surfaces of finite total curvature, where one can exploit the tools of the completion provided in the context of Radon measures as studied by Reshetnyak and others. However the generalized metrics in this sense still don't have enough regularity. Instead, we develop a more hands-on approach and show that, for each genus, every systolically extremal nonpositively curved surface is piecewise flat with finitely many conical singularities. Joint work with M. Katz.

Friday, November 15, 14:30-15:30, Universite de Montreal, Pavillon Andre-Aisenstadt, Room 5183.
Almaz Butaev (U. Calgary)
Extension problem on subspaces of BMO on domains
Abstract: In joint work with Galia Dafni, we discuss the extension problem for some subspaces of functions of bounded mean oscillation (BMO). Based on the extension operator of Jones we construct a universal extension in the sense that it simultaneously extends certain natural subspaces of BMO. The presented results will show an interplay between approximation, extension and geometric properties of the domain.

Spectral Geometry Seminar
Tuesday, November 26, 14:00-15:00, Universite de Montreal, Pavillon Andre-Aisenstadt, Room 5448.
Olivier Lafitte (CRM)
Precise descriptions of bands of the Airy-Schrodinger operator on the real line
Abstract: Joint work with Hakim Boumaza, LAGA, Université Paris 13 In this talk, we present recent results on band spectrum generated by a Schrodinger operator with a non C^1 potential for which one has eigenfunctions described by special functions. This generalizes a result Harrell (1979) and in particular we are able to have a precise estimate on the validity regime of the semi-classical behavior as well as the exact width of each band. The ongoing work on a multiple-wells potential will be as well presented.

Friday, November 29, 14:00-15:00, Concordia, Library Building, Room LB921-4.
Ritva Hurri-Syrjanen (U. of Helsinki)
On the John-Nirenberg inequalities
Abstract: The goal of my talk is to address some inequalities which Fritz John and Louis Nirenberg proved to be valid for certain functions defined in a cube. I will discuss the validity of similar inequalities for functions dened in an arbitrary bounded domain. My talk is based on joint work with Niko Marola and Antti Vahakangas.

Friday, December 13, 13:30-14:30, McGill, Burnside Hall, Room 1104
Jean-Philippe Burelle (U. Sherbrooke)
Higher Teichmuller and higher rank Schottky groups
Abstract: Schottky groups are the simplest and most classical examples of Kleinian groups, that is, of discrete subgroups of Mobius transformations. I will explain several generalisations of this notion to subgroups of higher rank Lie groups. One of these generalisations leads to an explicit description of positive representations of surfaces with non-empty boundary, a type of higher Teichmuller representation introduced by Fock and Goncharov in 2003. I will show how this description allows the construction of fundamental domains for an open domain of discontinuity in the projective space or the sphere, depending on the dimension. This talk will feature joint work with N. Treib, F. Kassel and V. Charette.