Réunion d'hiver SMC 2025

Toronto, 5 - 8 decembre 2025

Formulaire Conférences plénières, de lauréats et publique Par sessions Par conférenciers Affiches

Développements récents en analyse complexe et géométrie
Org: Alexander Brudnyi (University of Calgary), Rasul Shafikov (Western University) et Mahishanka Withanachchi (University of Calgary)
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HARSHITH ALAGANDALA, Western University Local Polynomial Convexity at Hyperbolic CR-singularity in $M^n \subset \mathbb{C}^n$ [PDF]

Let $M$ be a smooth manifold of dimension $n$ embedded in $\mathbb{C}^n$. If $T_pM \subset T_p\mathbb{C}^n$ is a totally real subspace for $p\in M$, then $M$ is locally polynomially convex at $p$. For a generic embedding $M$, we are interested in assessing polynomial convexity of $M$ at a CR-singularity, i.e., at a point $p\in M$ where $T_pM$ is not totally real. An order one CR-singularity in $M$ can be broadly classified as elliptic and hyperbolic. It is known that elliptic points give obstruction to polynomial convexity. In the case $n=2$, $M^2 \subset \mathbb{C}^2$ is locally polynomially convex at a hyperbolic complex point. We investigate local polynomial convexity of $M^n \subset \mathbb{C}^n$ at hyperbolic points in higher dimension.

ROBERTO ALBESIANO, University of Waterloo From division to extension [PDF]

Extending holomorphic data from subvarieties ($L^2$ extension) and lifting holomorphic sections of quotient bundles ($L^2$ division) are fundamental problems in complex geometry and several complex variables. They are also intimately related: in fact, Ohsawa showed that a version of the $L^2$ division theorem can be proved as a corollary of the $L^2$ extension theorem. We will see how, conversely, a version of the extension theorem can be obtained as an easy corollary of a division theorem with bounded generators.

TATYANA BARRON, University of Western Ontario Vanishing of Poincare series, revisited [PDF]

Let $H$ be the complex vector space of holomorphic sections of the k-th tensor power of the canonical bundle on a compact ball quotient. It is well known that if k is sufficiently large, then the Poincare series map $P:W \to H$ is surjective (here $W$ is the corresponding weighted Bergman space on the ball). I will discuss a criterion for $f\in W$ to be in $\ker(P)$.

ILIA BINDER, University of Toronto SLE as critical interface limits: power law rate of convergence [PDF]

This talk explores the rate of convergence of critical interfaces in various lattice models to Schramm–Loewner Evolution (SLE). We present a general framework for establishing power law convergence rates, offering a unified approach applicable across multiple models. As a central example, we examine the exploration process in critical percolation, demonstrating that for any “reasonable” critical percolation model, convergence to SLE is guaranteed and the polynomial rate follows automatically. This result holds unconditionally for critical site percolation on the hexagonal lattice and several of its generalizations, which will be discussed in detail. We further illustrate the applicability of the framework to other models, including the Harmonic Explorer and the Ising model.

This talk is based on joint projects with L. Chayes, D. Chelkak, H. Lei, and L. Richards.

DEBRAJ CHAKRABARTI, Central Michigan University Restricted type estimates and the Bergman Projection [PDF]

We obtain (weighted) restricted-type estimates for the Bergman projection operator on monomial polyhedra, a class of domains generalizing the Hartogs triangle. A restricted-type estimate is an estimate in the $L^p$-norm on an operator, which however holds only on characteristic functions. From these restricted-type estimates, we recapture $L^p$-boundedness results of the Bergman projection on these domains. On some monomial polyhedra, we show that the Bergman projection could fail to be of weak type $(q_*,q_*)$ , where $q_*$ denotes the right end-point of the interval of $L^p$-boundedness of the Bergman projection.

ISABELLE CHALENDAR, Université Gustave Eiffel RECENT PERSPECTIVES ON THE INVARIANT SUBSPACE PROBLEM [PDF]

We review recent work connected with the invariant subspace problem for operators and new developments in the last 15 years. In particular, we include discussions of almost-invariant subspaces, universal operators, specific classes of operators and new results in the framework of Banach space

Joint work with Jonathan R. Partington

DAN COMAN, Syracuse University Tian’s theorem for Grassmannian embeddings and degeneracy sets of random sections [PDF]

Let $(X,\omega)$ be a compact Kähler manifold, $(L,h^L)$ be a positive line bundle, and $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$. We show that the pullback by the Kodaira embedding associated to $L^p\otimes E$ of the $k$-th Chern class of the dual of the universal bundle over the Grassmannian converges as $p\to\infty$ to the $k$-th power of the Chern form $c_1(L,h^L)$, for $0\leq k\leq r$. As a consequence we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers $L^p\otimes E$ is $c_1(L,h^L)^r$. Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections. This is joint work with Turgay Bayraktar, Bingxiao Liu and George Marinescu.

EUGENIO DELLEPIANE, Université Laval Boundedness, Compactness and Schatten class for Rhaly matrices [PDF]

We discuss the Rhaly operator $R_\alpha$, that acts on $\ell^2$ as the infinite matrix \begin{equation*} R_\alpha=\begin{pmatrix} \alpha_0 & 0 & 0 & \cdots\\alpha_1 & \alpha_1 & 0 & \cdots\\alpha_2 & \alpha_2 & \alpha_2 & \cdots \\vdots & \vdots & \vdots & \ddots \end{pmatrix} . \end{equation*} Rhaly matrices arise as a natural generalization of the Cesàro operator. We provide new characterizations of the boundedness and compactness of $R_\alpha$ on $\ell^2$, and we completely characterize its membership in the $p$-Schatten class $\mathcal{S}^p(\ell^2)$, for $1< p<\infty$. We also answer to an open question, posed by Mashreghi--Ransford in 2019. This talk is based on a joint work with Carlo Bellavita and Giorgos Stylogiannis.

JESSE HULSE, University of Manitoba A Formula for the Pluricomplex Green Function of the Bidisk [PDF]

A formula for the pluricomplex Green function of the bidisk with two poles of equal weights will be derived. We divide the bidisk into two open regions, where the formula is found explicitly on the first region, and the other region is the union of a family of hypersurfaces. Time permitting, the newly derived formula will be used to find a formula for the Caratheodory metric on the symmetrized bidisk (up to a fourth degree polynomial) that matches Agler and Young's formula but without the supremum.

LUKA MERNIK, Florida Polytechnic University Plurisubharmonic defining functions [PDF]

Necessary conditions for a domain $\Omega\subset \mathbb C^n$ admitting a local plurisubharmonic defining function on the boundary are given. In tandem, we give an algorithm to construct a local plurisubharmonic defining function on the boundary when one exists. The process involves solving partial differential equations. In some cases we show that the necessary conditions are also sufficient.

PIERRE-OLIVIER, Universite du Quebec a Trois-Rivieres Characterizing Rational Lemniscates: Is That Even Possible? [PDF]

A polynomial lemniscate is a level curve of the modulus of a complex polynomial. As an application of the Riemann–Hurwitz formula, such a lemniscate is a Jordan curve if and only if all critical values of the polynomial lie inside the unit disk. A natural question is whether this characterization extends to rational lemniscates, the level curves of the modulus of rational functions. Unfortunately, the critical-value condition is only necessary.

In this talk, I will show that by shifting the perspective, from rational lemniscates themselves to asking which compact sets of the Riemann sphere arise as rational lemniscates, we obtain a complete characterization through three conditions that are topological, analytic, and algebraic.

This is joint work with Kirill Lazebnik and Malik Younsi.

ANDY RAICH, University of Arkansas Tower multitype and compactness of the dbar-Neumann operator in complex manifolds [PDF]

In this talk, I will discuss a new approach to establishing compactness of the $\bar\partial$-Neumann operator on $(p,q)$-forms on not necessarily pseudoconvex domains in complex manifolds. I will introduce a construction called the tower multitype and use it to build a stratification of the boundary. The stratification implies Property $(P_q)$ which in turn implies compactness of the $\bar\partial$-Neumann operator. I will also provide examples.

The result is joint work with Professor Dmitri Zaitsev of Trinity College Dublin.

DROR VAROLIN, Stony Brook

LIZ VIVAS, Ohio State University Orthogonal Polynomials and Parabolic Implosion [PDF]

Orthogonal polynomials appear naturally on the study of composition of Moebius transformations. In this talk, we consider different classes of orthogonal polynomials related to the perturbation of one single parabolic Moebius map. The results can be seen as particular cases of non-autonomous parabolic implosions. We consider also a random situation and see how our results hold on this setting too.

YUNUS ZEYTUNCU, University of Michigan-Dearborn Spectral Theory of the Kohn Laplacian on Quotient Manifoldsv [PDF]

In this talk, we study the spectrum of the Kohn Laplacian on quotient manifolds. In particular, we relate the asymptotic properties of the Kohn Laplacian's eigenvalues on sphere quotients to the group action.

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