CRM-Montreal-Quebec Analysis Seminar

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 Mathematical Analysis Lab playlist.

Upcoming Talks

Friday, April 5, 1:30pm, hybrid seminar at McGill, Burnside 1104 *Note earlier time*

Joint with the Geometric Analysis Seminar

Yi Wang (Johns Hopkins)

Yamabe flow of asymptotically flat metrics 

In this talk, we will discuss the behavior of the Yamabe flow on an asymptotically flat (AF) manifold. We will first show the long-time existence of the Yamabe flow starting from an AF manifold and discuss the uniform estimates on manifolds with positive Yamabe constant. This would allow us to prove global weighted convergence along the Yamabe flow on such manifolds. We will also talk about the case when the Yamabe constant is nonpositive. This is joint work with Eric Chen and Gilles Carron.

Past Talks

Winter 2024

Thursday, March 28, 2:30 pm, hybrid seminar at the CRM, room 5340 *Note unusual day*

Adam Black (Yale)

Dispersion for Coulomb wave functions

We study the Schödinger equation with a repulsive Coulombic potential on $\mathbb{R}^3$ . For radial data, we obtain an $L^1\rightarrow L^\infty$ dispersive estimate with the natural decay rate $t^{-\frac{3}{2}}. Our proof uses the spectral theory of strongly singular potentials to obtain an expression for the evolution kernel. A semiclassical turning point analysis of the kernel then allows the time decay to be extracted via oscillatory integral estimates. This is joint work with E. Toprak, B. Vergara, and J. Zou.

Friday, March 15, 2:30 pm, hybrid seminar at Concordia, room LB 921-4
Sébastien Darses (Université Aix-Marseille)
On generalizations of the Nyman-Beurling criterion for RH

The Nyman-Beurling criterion for the Riemann hypothesis is an approximation problem in L2 involving the fractional part function and some parameters.  Randomizing these parameters generates new criteria and new structures. Those ones are related to a variety of topics: the density of polynomials in weighted L2 spaces, the moment problem (in Analysis and Number Theory), Fourier Analysis and the analytic continuation of a « period » function.  We will review some results and new questions. Joint works with F. Alouges, E. Hillion, and J. Najnudel.

Friday, March 1, 2:00 pm, hybrid seminar at Laval, Alexandre-Vachon, VCH-3840
Bartosz Malman (Mälardalen University)
Clumps, and spectral clumps, for functions on the real line

In Fourier analysis, there is variety of statements which postulate that a function f and its Fourier transform \widehat{f} cannot simultaneously be too small, or that one of them has to be large if the other is small. What small or large means depends on context. The most famous such statement surely is the Fourier analytic version of Heisenberg's quantum mechanical uncertainty principle, but there is an abundance of other interpretations. In my talk, I want to review some other manifestations of the uncertainty principle, including theorems of Benedicks and Volberg, and I want to contribute a new interpretation which I recently stumbled upon. In my context, smallness will be interpreted in terms of a one-sided rapid decay condition on a function f living on the real line \mathbb{R}, and largeness will be interpreted in terms of the existence of a{spectral clump for f: an interval on which \widehat{f} is large enough to have an integrable logarithm. If time permits, I will discuss how this result finds applications in the theories of subnormal operators and de Branges-Rovnyak spaces.

Friday, February 16, 2:00 pm, hybrid seminar at the CRM, room 5340
José Manuel Palacios (University of Toronto)
Local Energy control in the presence of a zero-energy resonance

We consider the problem of stability and local energy decay for co-dimension one perturbations of the soliton of the cubic Klein-Gordon equation in 1+1 dimensions. Our main result gives a weighted time-averaged control of the local energy over a time interval which is exponentially long in the size of the initial (total) energy. A major difficulty is the presence of a zero-energy resonance in the linearized operator, which is a well-known obstruction to improved local decay properties. We address this issue by using virial estimates that are frequency-localized in a time-dependent way and introducing a "singular virial functional" with time-dependent weights to control the mass of the perturbation projected away from small frequencies.

Friday, February 2, 2:00 pm, hybrid seminar at Laval, Alexandre-Vachon, VCH-3840
Ryan O'Loughlin (Université Laval)
Symmetric Tensor Products: An Operator Theory Approach

Although tensor products and their symmetrisation have appeared in mathematical literature since at least the mid-nineteenth century, they rarely appear in the function-theoretic operator theory literature. This talk will introduce the symmetric and antisymmetric tensor products from an operator theoretic point of view. Results concerning fundamental operator-theoretic questions, such as finding the norm and spectrum of the symmetric tensor products of operators will be presented. If time permits, the properties of the symmetric tensor products of familiar concrete operators, such as unilateral shifts, adjoints of shifts, and diagonal operators will be discussed.


Friday, January 19, 2:00 pm, hybrid seminar at the CRM, room 5340
Raphaël Ponge (Sichuan University)
Semiclassical Analysis and Noncommutative Geometry

Semiclassical analysis and noncommutative geometry are distinct fields within the wider area of quantum theory. Bridges between them have been emerging recently. This lays down on operator ideal techniques that are used in both fields. In this talk we shall present semiclassical Weyl’s laws for Schrödinger operators on noncommutative manifolds (i.e., spectral triples). This shows that well known semiclassical Weyl’s laws in the commutative setting ultimately holds in a purely noncommutative setting. This extends and simplifies previous work of McDonald-Sukochev-Zanin. There are numerous examples (including sub-Riemannian geometry, open manifolds with conformally cusp metrics, noncommutative tori). The approach relies on spectral asymptotics for some weak Schatten class operators. The talk should be accessible to a wide audience. 

Fall 2023

Friday, September 22, 2:00 pm, hybrid seminar at Concordia room LB 921-4
Maria Ntekoume (Concordia)
Critical well-posedness for the derivative nonlinear Schrödinger equation on the line
This talk focuses on the well-posedness of the derivative nonlinear Schrödinger equation on the line. This model is known to be completely integrable and L2-critical with respect to scaling. However, until recently not much was known regarding the well-posendess of the equation below H1/2. In this talk we prove that the problem is well-posed in the critical space on the line, highlighting several recent results that led to this resolution. This is joint work with Benjamin Harrop-Griffiths, Rowan Killip, and Monica Visan.

Friday, September 29, 2:00 pm, hybrid seminar at the CRM room 5340
Liz Vivas (The Ohio State University)

Wiegerinck conjecture on Bergman Spaces

Wiegerinck proved that the Bergman space over any domain in the complex plane is either trivial or infinite dimensional. In this talk I will discuss various generalizations and open questions related to this theorem. I will survey the case of the complex plane being replaced by C^n as well as a domain in a compact Riemann Surface.

Friday, October 6, 2:00 pm, hybrid seminar at the CRM room 5340
Nathaniel Sagman (University of Luxembourg)

Minimal surfaces in symmetric spaces.

For S a closed surface of genus at least 2, Labourie proved that every Hitchin representation of pi_1(S) into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3), and explained that if true, then the space of Hitchin representations admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmuller space.

After giving the relevant background, we will discuss the analysis and geometry of minimal surfaces in symmetric spaces, and explain how certain large area minimal surfaces give counterexamples to Labourie’s conjecture. 

Friday, October 13, No seminar

Friday, October 20, 1:30 pm, hybrid seminar at Laval, Alexandre-Vachon, VCH-3870
Gisèle Ruiz Goldstein (University of Memphis)
Chaos and deterministic PDEs in Mathematical Finance

Major developments in mathematical Finance have come from the study of two deterministic parabolic partial differential equations, the Nobel Prize winning Black-Scholes equation for stock options,

$$

\frac{\partial u}{\partial t}=\frac{\sigma^2}{2}x^2 \frac{\partial^2 u}{\partial x^2}+rx \frac{\partial u}{\partial x}-rx

$$

and the Cox-Ingersoll-Ross equation for zero coupon bonds,

$$

\frac{\partial u}{\partial t}=\frac{\sigma^2}{2}x \frac{\partial^2 u}{\partial x^2}+(\beta x+\gamma)\frac{\partial u}{\partial x}-xu,

$$

where $(x,t) \in (0,\infty) \times [0,\infty)$. Each has a particular initial condition $u(x,0) = u_0(x)$ of relevance in economics. In both models \sigma is the volatility, r is an interest rate, \beta and \gamma are also parameters given by the economic modeling.

We study these problems in weighted sup norm Banach spaces whose functions are unbounded near infinity (and possibly also near 0). The Black-Scholes equation is governed by a semigroup that is strongly continuous, quasicontractive, and chaotic. Extentions to time dependent coe¢cients will be given for this model. Recent results embedding the chaotic behavior of the Black-Scholes equation into a class of equations which are of interest in mathematical finance will be discussed.

The Cox-Ingersoll-Ross equation is governed by a strongly continuous quasi-

contractive semigroup, and the solution is given by a new type of Feynman-Kac formula. Extensions to more general potential terms will be explained as well as extensions to time dependent coefficients.

Friday, October 20, 2:30 pm, hybrid seminar at Laval, Alexandre-Vachon, VCH-3870
Jerome Goldstein (University of Memphis)
Equipartition of Energy, old and new results

Equipartition of energy results for the wave equation on R^n (even with n = 1) could have been proved by Euler, but they were not discovered until the mid 1960s. For the usual wave equation on all of Euclidean space, the energy becomes half kinetic and half potential as t -> 1. This fails on bounded domains. The subject quickly evolved into a chapter in the theory of selfadjoint operators. We shall survey the history of the theory, some applications, and recent results. The newest results involve systems of wave equations with “cross friction” terms. These systems are ill posed and temporally inhomogeneous, yet equipartition of energy in a general context still makes sense and holds. The newest results comprise new joint work with Gisèle Goldstein, Sandra Lucente, and Silvia Romanelli.


Friday, October 27, 2:00 pm, hybrid seminar at Laval, Alexandre-Vachon, VCH-3870
Alexey Shevyakov (University of Saskatchewan)
Conservation laws of differential equations: computation, connections, and applications

Local conservation laws of a system of differential equations (DE) are given by one or several divergence expressions that hold on solutions of that system. For ordinary differential equations, conservation laws lead to first integrals and the reduction of order. For partial differential equations (PDE), they provide globally conserved quantities, such as energy, momentum, as well as more exotic ones. Conservation laws used for analysis of global solution behaviour, are related to multiple other analytical properties of PDEs, and play an important role in the numerical treatment of PDEs.

In this talk, we will review the general theory, including trivial and equivalent conservation laws, their characteristic form, relationships with integrability, symmetries of DEs, Hamiltonians, variational systems, Lagrangians, and the first and second Noether's theorems. A systematic procedure to seek conservation laws will be discussed, applicable to virtually any PDE system; it will be compared to the Noether's theorem approach to seek conservation laws of variational models. A symbolic implementation of the direct method of conservation law computation in Maple will be presented. Examples of conservation laws and conserved quantities for classical PDEs and some nonlinear models arising in contemporary work will be discussed.

Time permitting, we will consider a common framework for different types of conservation laws of PDE systems in three space dimensions, including their global and local formulations in static and moving domains given by volumes, surfaces, and curves. 

Friday, November 3, 2:00 pm, hybrid seminar at Laval, Alexandre-Vachon, VCH-3870
Malik Younsi (University of Hawaii)
Miracles of Holomorphic Motions

Holomorphic motions were introduced by Mane, Sad and Sullivan in the 1980's motivated by applications in holomorphic dynamics. They have since then been applied in various other areas such as Kleinian groups, Teichmüller theory and most notably quasiconformal mappings.

In this talk, I will discuss recent results on the variation of several quantities under holomorphic motions, such as logarithmic capacity, (continuous) analytic capacity, Minkowski/Packing/Hausdorff dimension, and Hausdorff measure. This is based partly on joint work with Wen-Hui Ai, Aidan Furher and Thomas Ransford.

*Monday*, November 6, 2:30 pm, hybrid seminar at McGill, Burnside 1104
Victor Ivrii (University of Toronto)

Pointwise Spectral Asymptotics near Singularity

In this talk we establish semiclassical asymptotics and estimates for the $e_h(x,x,\tau)$ where

 $e_h(x,y,\tau)$ is the Schwartz kernel of the spectral projector for a second order elliptic operator inside domain with the power singularity in the origin. While such asymptotics   for its trace $\N_h(\tau)= \int e_h(x,x,\tau)\,dx$ are well-known, the poinwise  asymptotics are much less explored.

Our main tools: microlocal methods, improved successive approximations and geometric optics methods.

Friday, November 10, 2:30 pm, hybrid seminar at Laval, Alexandre-Vachon, VCH-3870
Linan Chen (McGill University)
Exceptional sets of Gaussian free fields: an example of multifractal analysis in random geometry  

Abstract: Gaussian free fields (GFFs) are multivariate versions of Brownian motion and important models in the study of random geometry. For the log-correlated GFF, the concept of "thick point" was introduced by Hu-Miller-Peres ('08) as the analog of "extremum" to characterize certain exceptional behavior of the GFF, and the study of thick point yielded interesting results such as multifractality on the geometry of GFF. We extend this study to GFFs in arbitrary dimensions, including the polynomial-correlated fields, and propose the notion of "steep point" to capture a more general class of exceptional sets. The results not only confirm multifractality for GFFs in higher dimensions, but also lead to geometric properties that are subtler than multifractality.

In this talk, we will introduce the mathematical formulation of GFFs, explain the regularization techniques, and discuss multifractality. If time permits, we will also mention the connections between GFFs and the KPZ relation.

Friday, November 17, *10:00 am*, hybrid seminar at McGill, Burnside 1214 (joint with the Probability Seminar)
Kodjo Raphaël Madou (McGill University)

Form-boundedness and singular SDEs: weak vs strong solutions

Abstract: This talk  focuses on  well-posedness of singular SDEs. We prove existence and uniqueness of weak solution to SDEs under minimal assumptions on singular drift (form-boundedness).  Additionally, we use the Rockner-Zhao approach  based on a compactness criterion for random fields in Wiener-Sobolev spaces, to establish strong well-posedness result for SDEs featuring singular drift term.
The talk is based on joint papers with Damir Kinzebulatov.

Friday, November 17, 2:30 pm, hybrid seminar at Laval, Alexandre-Vachon, VCH-3870
Pierre-Olivier Parisé (University of Hawaii)

Exploring Application of Summability Theory in Holomorphic Function Spaces

Abstract: In this talk, I will introduce the basic concepts in summability theory and present some interesting applications of this theory to the summability of Taylor series of functions in different spaces of holomorphic functions.

Friday, November 24, 2:00 pm, hybrid seminar at the CRM, room 5340
Dmitry Faifman (Tel Aviv University)

Nash and Whitney problems in convex valuation theory

Abstract: A (convex, smooth) valuation is a finitely additive measure on convex bodies, satisfying a smoothness condition; many interesting objects in convex and differential geometry are in fact valuations. 

Assume that a collection of valuations is given on a family S of subspaces of R^n . Are they the restrictions of a single valuation? Clearly, compatibility of the given data on intersections is a necessary condition. Is it sufficient? 

We will discuss several geometrically distinct instances of this problem, whence it acquires distinct flavors. 

When S is the whole k-grassmannian, and the valuations j-homogeneous, we will see that the condition is sufficient, provided k-j>1. This can be seen as a dimensional localization of the transition from densities to valuations. 

In another setting where S consists of pairwise non-intersecting subspaces, we again establish a positive answer. As a corollary, we will deduce a Nash embedding theorem for smooth valuations on manifolds. 

Finally, we will consider the setting of finite generic families of subspaces, giving rise to a surprising extension phenomenon.

Based on a joint work with Georg Hofstaetter.  



Thursday, December 21, 2 pm, hybrid seminar at Laval, Alexandre-Vachon, VCH-2870

Jade Brisson (Université de Neuchâtel)

Steklov eigenvalues in negatively curved manifolds

Abstract: In the setting of negatively curved manifolds of dimension $n\ge3$, we consider the Steklov eigenvalue problem on compact pinched negatively curved manifolds with totally geodesic boundaries.  We show that the first nonzero Steklov eigenvalue is bounded below in terms of the total volume and boundary area when the dimension is at least three. In particular, it shows that Steklov eigenvalues can only tend to zero when the total volume and/or boundary area go to infinity.  It can be seen as a counterpart of the lower bound for the first nonzero Laplace eigenvalues on closed pinched negatively curved manifolds of dimension at least three as proved by Schoen in 1982. We provide examples showing that the dependency on both volume and boundary area is necessary. This is joint work with Ara Basmajian, Asma Hassannezhad and Antoine Métras.

Organizers

Galia Dafni (Concordia & CRM)
Dmitry Jakobson (McGill)
Damir Kinzebulatov (Laval)
Maria Ntekoume (Concordia)