Réunion d'hiver SMC 2025
Toronto, 5 - 8 decembre 2025
Développements récents en analyse complexe et géométrie
Org: Alexander Brudnyi (University of Calgary), Rasul Shafikov (Western University) et Mahishanka Withanachchi (University of Calgary)
[PDF]
Org: Alexander Brudnyi (University of Calgary), Rasul Shafikov (Western University) et Mahishanka Withanachchi (University of Calgary)
[PDF]
- HARSHITH ALAGANDALA, Western University
Local Polynomial Convexity at Hyperbolic CR-singularity in $M^n \subset \mathbb{C}^n$ [PDF]
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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]
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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]
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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]
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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.
- BLAKE BOUDREAUX, Arkansas
- DEBRAJ CHAKRABARTI, Central Michigan University
Restricted type estimates and the Bergman Projection [PDF]
- DEBRAJ CHAKRABARTI, Central Michigan University
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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
- DAN COMAN, Syracuse University
Tian’s theorem for Grassmannian embeddings and degeneracy sets of random sections [PDF]
- DAN COMAN, Syracuse University
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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.
- JESSE HULSE, University of Manitoba
A Formula for the Pluricomplex Green Function of the Bidisk [PDF]
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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
- PIERRE-OLIVIER, UQTR
- ANDY RAICH, University of Arkansas
Tower multitype and compactness of the dbar-Neumann operator in complex manifolds [PDF]
- PIERRE-OLIVIER, UQTR
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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
- LIS VIVAS, Ohio State
- YUNUS ZEYTUNCU, University of Michigan-Dearborn
Spectral Theory of the Kohn Laplacian on Quotient Manifoldsv [PDF]
- LIS VIVAS, Ohio State
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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.