Seminars are usually held on Friday afternoon.
The seminar is currently held in hybrid mode, organized jointly with Laval University in Quebec City. In person seminars in Montreal are held at Concordia, McGill or the CRM/Université de Montréal; in person seminars in Quebec City are held at Laval.
To be added to the mailing list, attend a zoom session, and for suggestions, questions etc. please contact one of the organizers.
Some of the talks are recorded and posted on the CRM Youtube channel, on the Mathematical Analysis Lab playlist.
Friday, January 17, 2:00 pm, hybrid seminar at Concordia, room LB 921-4
Dmitry Pelinovsky (McMaster)
Stability of solitary waves and inverse scattering in the massive Thirring model
I will review the recent results on stability of exponential and algebraic solitons in the massive Thirring model in laboratory coordinates. The exponential solitons correspond to isolated eigenvalues in the Lax spectrum, whereas the algebraic solitons correspond to embedded eigenvalues on the imaginary axis, where the continuous spectrum resides. Besides single-solitons, we are also interested in multiple-soliton solutions corresponding to eigenvalues of higher multiplicity, which represent the stable dynamics of slow interactions between the solitons.
Friday, January 24, 11:00 am, hybrid seminar at Concordia, room LB 921-4 *Note earlier time*
Riikka Korte (Aalto University)
Conformal transformation of metric measure spaces
I will discuss the recent results on conformal deformations of metric measure spaces. Inspired by the stereographic projection and its inverse, the deformations that transform unbounded spaces into bounded ones are called sphericalizations and transformations that transform bounded spaces into unbounded ones are called flattenings. It is possible to construct sphericalizations and flattenings that preserve for example uniformity, doubling property, Poincaré inequality, p-energy and/or Besov energy. These transformations are useful for example in solving a Dirichlet problem on unbounded domains. I will discuss our recent results that are based on joint work with A. and J. Björn, R. Gibara, S. Rogovin, N. Shanmugalingam and T. Takala.
Friday, January 31, 2:10 pm, hybrid seminar at UdeM, room 5183
Vukasin Stojisavljevic (CRM)
Counting zeros coarsely vs classically
Classical Nevanlinna theory concerns counting zeros of a holomorphic map inside of a disc of increasing radius. A straightforward generalization of this idea to complex vector spaces of higher dimension fails, as shown by Cornalba and Shiffman in the 1970s. In a recent work, we showed that a certain higher-dimensional generalization is possible, if one counts zeros coarsely, i.e. by ignoring small oscillations. The same philosophy pertains to a number of other problems in analysis, such as counting zeros of harmonic maps. I will explain the general idea of coarse counting and demonstrate its utility with concrete examples. The talk is partially based on a joint work with L. Buhovsky, I. Polterovich, L. Polterovich and E. Shelukhin.
Friday, February 14, 2:15 pm, hybrid seminar at UdeM, room 5183
Lysianne Hari (Université Marie et Louis Pasteur)
Propagation of coherent states and problems of conical intersections
In this talk, we will study the propagation of coherent states (or wavepackets) for a system of coupled Schrödinger equations, in the semi-classical limit.
These equations appear in Quantum Chemistry to describe large molecules - namely with a very small electronic mass / nucleus mass ratio (semi-classical parameter).
The coupling between these equations can generate interesting phenomena: the Born-Oppenheimer Approximation is not valid anymore and transitions between energy levels can occur.
Usually, these couplings are induced by a matrix-valued potential with "eigenvalue crossings": they are equal on some region of the phase space.
We will focus on a precise question about the stability of the solution. Consider an initial coherent state (one can think of a gaussian), well-localized in phase space, "living" in a chosen eigenspace of the potential (ie at some given energy level). Does the associated solution keep the same localization property at leading order and does it stay in the same eigenspace (adiabaticity) ?
Without technical details, we will study various situations for several types of crossings leading to transition phenomena.
Friday, February 21, 2:15 pm, hybrid seminar at UdeM, room 5183
Misha Karpukhin (UCL)
Eigenvalues and minimal surfaces
Given a Riemannian surface, the study of sharp upper bounds for Laplacian eigenvalues under the area constraint is a classical problem of spectral geometry going back to J. Hersch, P. Li, S.-T. Yau and N. Nadirashvili. The particular interest in this problem stems from the remarkable fact that the optimal metrics for such bounds arise as metrics on minimal surfaces in spheres. In the talk I will survey recent results on the subject with an emphasis on the fruitful interaction between the geometry and spectral bounds. In particular, I will describe a surprisingly effective method of constructing new minimal surfaces based on the eigenvalue optimisation with a prescribed symmetry group.
Friday, March 14, 2:00 pm, hybrid seminar at UdeM, room 5183
Yana Teplitskaya (Paris-Saclay)
Irregular examples of indecomposable Steiner trees with infinite number of branching points
A general metric Steiner problem is a problem of finding a set S with the minimal length, such that S∪A is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. By indecomposable Steiner tree we mean such a Steiner tree S that S \ A remains connected. This means that such a tree cannot be obtained as the union of several Steiner trees for subsets of A.
I will talk about examples of indecomposable Steiner trees with infinite number of branching points in Euclidean plane: I will provide ''small'' (connecting countable number of the terminal points) and ''large'' (connecting totally disconnected set of positive Hausdorff dimension) examples. These questions were motivated by search of irregular structures contained in Steiner trees. If we have enough time, we will even go beyond the plane. Everybody is welcome, no preliminary math knowledge needed.
Based on joint works with D. Cherkashin, E. Stepanov, E. Paolini
Friday, March 21, 2:15 pm, hybrid seminar at UdeM, room 5183
Bartłomiej Zawalski (Kent State University)
On star-converx bodies with rotationally invariant sections
We will outline the proof that an origin-symmetric star-convex body K with sufficiently smooth boundary and such that every hyperplane section of K passing through the origin is a body of affine revolution, is itself a body of affine revolution. This will give a positive answer to the question asked by G. Bor, L. Hernández-Lamoneda, V. Jiménez de Santiago, and L. Montejano-Peimbert in their recent paper on the isometric Banach’s conjecture, though with slightly different prerequisites. The theorem may be also seen as a high-dimensional variant of Bezdek’s conjecture. Our argument is built mainly upon the tools of differential geometry and linear algebra, but occasionally we will need to use some more involved facts from other fields like algebraic topology or commutative algebra.
Friday, March 28, 2:15 pm, hybrid seminar at UdeM, room 5183
Mikolaj Sierzega (Cornell - University of Warsaw)
Li-Yau-Type Bounds for the Fractional Heat Equation
Differential Harnack bounds are a key analytical device that bridge partial differential equations of the elliptic or parabolic type with Harnack bounds, which provide pointwise estimates on the local variability of solutions. A prime example is the famous Li-Yau inequality, which applies to positive solutions of the classical heat equation.
The growing interest in the theory and applications of nonlocal diffusion models naturally raises questions about analogues of Li-Yau-type inequalities in the nonlocal setting. However, despite many parallels between local and nonlocal diffusion models, even the model case of fractional heat flow presents both conceptual and technical challenges.
In my talk, I will discuss recent progress on optimal differential Harnack bounds for fractional heat flow. In particular, I will show how the structural properties of these estimates offer new insights into classical results for the standard heat equation.
Friday, April 4, 2:15 pm, hybrid seminar at Concordia, room LB 921-4
Thierry Laurens (University of Wisconsin-Madison)
Continuum Calogero--Moser models
The focusing CCM equation is a completely integrable PDE that describes a continuum limit of a particle gas interacting pairwise through an inverse square potential. This system is well-posed in the scaling-critical space L^2 below the mass of the soliton, but above this threshold there are solutions that blow up in finite time.
In this talk, we will discuss some new and existing results about solutions below the soliton mass threshold. This is based on joint work with Rowan Killip and Monica Visan.
Friday, April 11, 2:15 pm, hybrid seminar at UdeM, room 5183
Gautam Aishwarya (Michigan State University)
An entropy analogue of the Kneser-Poulsen conjecture
Does the volume of a finite union of balls decrease if their centres are moved pairwise closer together? This long-standing open question in discrete geometry is widely believed to have an affirmative answer, known as the Kneser-Poulsen conjecture. In this talk, we will discuss and prove a natural entropic version of this conjecture. Specifically, we show that if T is a 1-Lipschitz map, X is any random vector, and Z is an independent standard Gaussian random vector, then the entropy of X+ \sqrt{s} Z is always greater than the entropy of T(X) + \sqrt{s} Z, for all s>0. In analytic terms, this says that if two initial conditions f_{0} and g_{0} differ by a 1-Lipschitz change of variables, then the corresponding solutions to the heat equation f_{s} and g_{s} maintain their initial entropy comparison at all times s>0. We will also see a common unification of Costa's "concavity of entropy power" theorem and our result. This talk is based on joint work with Dongbin Li (UAlberta).
Thursday, April 17, 2:00 pm, hybrid seminar at UdeM, room 5448 *Note date and room change*
Shay Sadovsky (NYU)
Non-traditional optimal transport
In the classical theory of optimal mass transport, cafes in the measure space of cafes must, given a cost function c, buy croissants from the measure space of bakeries, with lowest total cost to the cafes. This is the traditional problem of optimal transport. If the cost is allowed to attain the value infinity, this problem, known as the non-traditional problem of optimal transport, becomes much harder to solve (and sometimes impossible). In this talk we will go into detail describing optimal transport, the difference between the traditional and non-traditional setting, the associated non-traditional cost, and a necessary condition for a solution to exist. We will present the general scheme of the proof, and our new methods.
Based on joint work with Shiri Artstein-Avidan and Katarzyna Wyczesany
Friday, April 25, 2:15 pm, hybrid seminar at UdeM, room 5183
Laurent Moonens (Paris-Saclay)
On the Lp behavior of rectangular differentiation processes
Differentiation processes are sequences of averaging operators Tk f := |Bk|-1 1Bk* f where Bk are (usually convex) sets shrinking to the origin (here |B| stands for the Lebesgue measure of B and * for the convolution).
As follows famously from the Hardy-Littlewood maximal inequality, the case where subsets Bk in question are balls whose radii tend to zero yields the almost convergence of Tkf for all f in L1.
When sets Bk belong to other classes of subsets of the Euclidean spaces, the Lp behavior of those operators depends heavily on their geometry, the way they are « oriented » and how both factors allow them to interact.
In this talk, we’ll discuss the general picture, relations between Lp a.e. convergence and maximal results, famous open questions in the field as well as results obtained jointly with E. D’Aniello, A. Gauvan and J. Rosenblatt related to the apparently simple case when sets Bk are rectangles in the Euclidean plane.
Friday, May 23, 2:15 pm, hybrid seminar at at Concordia, room LB 921-4
Jean-François Jabir (HSE)
Dynamics of McKean-Vlasov type with weak regularity conditions and related particle approximations
This talk is dedicated to present quantitative propagation of chaos results for smoothened interacting particle systems related to a class of singular kinetic/Langevin-type McKean-Vlasov SDEs driven by a non-degenerate symmetric stable Levy noise. Singularities in the mean-field limit emerges with a pairwise interaction kernel belonging to a class of mixed Besov spaces with negative smoothness. Specificall, we establish different weak and trajectorial propagation of chaos results in the Brownian and pure-jump regimes.
(This talk is based on a joint work with Zimo Hao, Stéphane Menozzi, Michael Röckner and Xicheng Zhang.)
Friday, September 13, 2:00 pm, hybrid seminar at the CRM, room 5183
Amir Moradifam (UC Riverside)
The Sphere Covering Inequality and Its Applications
We show that the total area of two distinct Gaussian curvature 1 surfaces with the same conformal factor on the boundary, which are also conformal to the Euclidean unit disk, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We refer to this lower bound of total areas as the Sphere Covering Inequality. This inequality and its generalizations are applied to a number of open problems related to Moser-Trudinger type inequalities, mean field equations and Onsager vortices, etc, and yield optimal results. In particular we confirm the best constant of a Moser-Truidinger type inequality conjectured by A. Chang and P. Yang in 1987. This is a joint work Changfeng Gui.
Friday, September 20, 10:00 am, hybrid seminar at the CRM, room 4186
Philippe Charron (Université de Genève)
Lower bounds on the inner radius of nodal domains of Laplace eigenfunctions
Consider nodal domains (connected components of the set where a function is non-zero) of Laplace eigenfunctions on a closed compact manifold. By Faber-Krahn's inequality, a nodal domain cannot contain a ball of size C λ^{-1/2}. However, it is harder to find lower bounds on the size of the biggest ball that can be contained inside a nodal domain.
In this talk, I will discuss a recent result (with Dan Mangoubi) that shows that it is always possible to find a ball of size C λ^{-1/2} (log λ)^{-(d-2)/2}. The proof uses a mixture of old and new techniques, some of which come from Brownion motion / heat kernel estimates, while others come from recent advances on solutions to elliptic equations.
Friday, September 27, 9:30 am, hybrid seminar at Concordia, room LB 921-4 *Note earlier time*
Shahab Shaabani (Concordia)
The operator norm of paraproductson bi-parameter Hardy spaces
In this talk, we discuss recent work on the operator norm of paraproducts on bi-parameter Hardy spaces.
A paraproduct is a bilinear form arising from the product of two functions, both expanded in either a wavelet basis, such as the Haar, or in Littlewood-Paley pieces. In the one-parameter theory, the frequency interactions in the product of two functions are divided into either low-low, low-high, or high-low interactions, and each gives rise to a bilinear form called a one-parameter paraproduct. Some of these forms behave much better than the product itself, and for them, Hölder’s inequality holds for the full range of exponents, provided that the Lebesgue spaces are replaced by Hardy spaces and the space of bounded functions is replaced by functions of bounded mean oscillation. Similar results hold for any number of parameters as well.
In our recent work, it is shown that for all positive values of p, q, and r with 1/q=1/p+1/r, the operator norm of the dyadic paraproduct π_g from the bi-parameter dyadic Hardy space H^p_d to H^q_d is comparable to ∥g∥_{H^r_d}. In addition, similar results are obtained for bi-parameter Fourier paraproducts of the same form.
Friday, October 4, 2:00 pm, hybrid seminar at the CRM, room 5183
Denis Grebenkov (PMC, CNRS - Ecole Polytechnique/CRM, Université de Montréal)
The exterior Steklov problem: asymptotic results
In this talk, I will present recent asymptotic results for the exterior Steklov problem in the complement of a compact Euclidean set. The Steklov problem is first rigorously posed for the modified Helmholtz equation (p-Delta)u = 0 with a strictly positive parameter p. The asymptotic behavior of the eigenvalues is then analyzed as p goes to 0. For unbounded domains, the asymptotic formulas depend on the space dimension and exhibit non-analytic terms, i.e., the eigenvalues are not analytic functions of the parameter p. Some open questions related to spectral, probabilistic and numerical aspects of this spectral problem will be outlined.
Friday, October 18, 10:00 am, hybrid seminar at the CRM, room 5340
Vitaliy Kurlin (University of Liverpool)
Can we geometrically sense the shape of a molecule?
Can we hear the shape of a drum? This question was negatively answered decades ago by many authors including Gordon, Webb, Wolpert, who constructed non-isometric planar shapes that have the identical eigenvalues of the Laplace operator (Bull. AMS, v.27 (1992), p.134-138). The more general question: can we sense the shape of a rigid object such as a cloud of atomic centers representing a molecule?
The SSS theorem from school geometry says that any triangles (clouds of 3 unordered points) are congruent (isometric) if and only if they have the same three sides (ordered by length). An extension of this theorem to more points in higher dimensions was practical only for clouds of m ordered points, which are uniquely determined up to isometry by a matrix of m x m distances. If points are unordered, comparing m! matrices under all permutations of m points is impractical.
We will define a complete (under rigid motion) and Lipschitz continuous invariant for all clouds of m unordered points, which is computable in polynomial time of m in any fixed Euclidean space, published in CVPR 2023. For the QM9 database of 130K+ molecules with 3D coordinates, the more recent invariants distinguished all clouds of atomic centers without chemical elements, which confirmed that the shape of a molecule including its chemistry is determined from sufficiently precise atomic geometry. The relevant papers are at https://kurlin.org/research-papers.php#Geometric-Data-Science.
Friday, October 25, 2:00 pm, hybrid seminar at Concordia, room LB 921-04
*Joint with the Applied Math Seminar
Dan Hill (Saarland University)
An Existence Proof for Fully Localised Planar Patterns
Spatially localised patterns are known to emerge in a variety of physical settings, ranging from dryland vegetation to vibrating fluids to the buckling of cylinders. While there are a number of mathematical tools for studying localised patterns in one spatial dimension, developing equivalent approaches in higher spatial dimensions remains a major challenge in the area of pattern formation. In this talk, we will focus on 2D patterns that are localised in the radial direction and present a novel formal approach to derive radial amplitude equations [D.J. Hill & D.J.B. Lloyd, SIAM J. Appl. Math. (2024)], which provides new insight into the emergence of fully localised planar patterns. As an example, we formally derive radial amplitude equations for fully localised hexagons and quasipatterns in the Swift--Hohenberg equation (SHE), for which we can obtain explicit localised solutions. We then present current work on an existence proof for fully localised stripes in the SHE, using modulation theory from water waves (e.g. see [B. Buffoni, M.D. Groves & E. Wahlen, J. Math. Fluid Mech. (2022)]) extended to polar coordinates. To the authors' knowledge, this would provide the first existence proof of radially-localised non-axisymmetric planar patterns. This work is in collaboration with Mark Groves (Universität des Saarlandes).
CANCELED Friday, November 15, 11:00 am, hybrid seminar at UQAM, room PK-5115
*Joint with the Geometric Analysis Seminar
Yannick Sire (Johns Hopkins)
Geometric measure of nodal, critical and singular sets for solutions of degenerate equations
I will describe recent results on the “size” of various sets associated to solutions of some elliptic PDEs whose coefficients are degenerate or singular. In the case of eigenfunctions of the associated second order operators, these estimates are related to some famous conjectures by Yau (formulated in a more classical setting). Degenerate PDEs appear in a lot of different contexts like conical spaces, realizations of Dirichlet-to-Neumann maps, Poincare-Einstein manifolds in conformal geometry, etc… I will describe several strategies to get these estimates, leading for some of them to sharp bounds. Along the way, I will also describe some eigenfunction and cluster estimates, which are very much related to this topic.
Friday, November 29, 2:00 pm, hybrid seminar at the CRM, room 5183
Lukas Bundrock (University of Alabama)
Spectral Optimization for the Robin Laplacian in Exterior Domains
The spectrum of the Laplace operator with Robin boundary conditions on bounded domains is a well-studied area with important physical applications. For example, in analyzing the long-term behavior of Brownian motion with particle creation at the boundary, the principal eigenvalue provides information about the expected number of particles within the domain.
When the domain is unbounded—specifically, the complement of a compact set—new questions and challenges emerge, holding potential relevance in quantum theory.
In this talk, I will discuss the existence of discrete eigenvalues s well as the problem of geometrically optimizing the lowest point of the spectrum. We will see that, while the ball serves as a local maximizer among domains of fixed measure, it does not necessarily achieve the global optimum.
Friday, December 6, 2:00 pm, hybrid seminar at the CRM, room 5183
David Sher (DePaul University)
On Pleijel's nodal domain theorem
A classical problem in spectral geometry is to study the number of nodal domains of eigenfunctions of the Laplacian. Courant's nodal domain theorem tells us that the kth eigenfunction of the Dirichlet Laplacian has at most k nodal domains. Pleijel's nodal domain theorem is instead an asymptotic statement, telling us that the ratio of the number of nodal domains to the index of the eigenfunction has limsup bounded above by a fixed constant less than 1. In this talk, we give a survey of recent extensions of and variations on Pleijel's theorem. As an example, we prove that Pleijel's nodal domain theorem holds for the Robin Laplacian on any Lipschitz domain. This is joint work with Katie Gittins (Durham), Asma Hassannezhad (Bristol), and Corentin Lena (Padova).
Dmitry Faifman (Université de Montréal)
Dmitry Jakobson (McGill)
Damir Kinzebulatov (Laval)
Maria Ntekoume (Concordia)