Prochainement


Mercredi 10 juin 16:00-18:00 Yang Li (IAS (Princeton))
Weak SYZ conjecture for hypersurfaces in the Fermat family

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Lieu : Demander le lien zoom à jean-michel.bismut@u-psud.fr

Résumé : The SYZ conjecture predicts that for polarised Calabi-Yau
manifolds undergoing the large complex structure limit, there should be
a special Lagrangian torus fibration. A weak version asks if this
fibration can be found in the generic region. I will discuss my recent
work proving this weak SYZ conjecture for the degenerating hypersurfaces
in the Fermat family. Although these examples are quite special, this is
the first construction of generic SYZ fibrations that works uniformly in
all complex dimensions.

Weak SYZ conjecture for hypersurfaces in the Fermat family  Version PDF



Passés

Mercredi 27 mai 16:00-18:00 Aaron Charles Naber  (Northwestern)
Ricci Curvature and Differential Harnack Inequalities on Path Space

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Lieu : Demander le lien Zoom à jean-michel.bismut@u-psud.fr

Résumé : There has been an observation of late that many analytic estimates on manifolds M with lower Ricci curvature bounds have counterparts on the path space PM of the manifold when there are two sided bounds on Ricci curvature. We will begin reviewing some of these, in particular the estimates of [Nab],[Has-Nab] which generalize the Bakry-Emery-Ledoux estimates to path space. We will then discuss new results, which are joint with Haslhofer and Knofer, which generalize the Li-Yau differential harnack inequalities to the path space, under the assumption of two sided Ricci curvature bounds.
To accomplish this, we will introduce a family of Laplace operators on PM, built from finite dimensional traces of the Markovian hessian. The differential harnacks will take the form of differential inequalities for these operators, and will recover the classical Li-Yau when applied the simplest functions on path space, namely the cylinder functions of one variable.

Ricci Curvature and Differential Harnack Inequalities on Path Space  Version PDF

Mercredi 20 mai 16:15-18:15 Sebastian Goette  (Freiburg)
Extra twisted connected sums and their $\nu$-invariants

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Lieu : Demander le lien Zoom à jean-michel.bismut@u-psud.fr

Résumé : Joyce’s orbifold construction and the twisted connected sums by Kovalev and Corti-Haskins-Nordström-Pacini provide many examples of compact Riemannian 7-manifolds with holonomy $G_2$ (i.e., $G_2$-manifolds). We would like to use this wealth of examples to guess further properties of $G_2$-manifolds and to find obstructions against holonomy $G_2$, taking into account the underlying topological $G_2$-structures.
The Crowley-Nordstr\"om $\nu$-invariant distinguishes topological $G_2$-structures. It vanishes for all twisted connected sums. By adding an extra twist to this construction, we show that the $\nu$-invariant can assume all of its 48 possible values. This shows that $G_2$-bordism presents no obstruction against holonomy $G_2$. We also exhibit examples of 7-manifolds with disconnected $G_2$-moduli space. Our computation of the $\nu$-invariants involves integration of the Bismut-Cheeger $\eta$-forms for families of tori, which can be done either by elementary hyperbolic geometry, or using modular properties of the Dedekind $\eta$-function.

Extra twisted connected sums and their $\nu$-invariants  Version PDF

Mercredi 13 mai 16:00-18:00 Felix Schulze  (Warwick)
Mean curvature flow with generic initial data

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Lieu : Demander le lien Zoom à jean-michel.bismut@u-psud.fr

Résumé : We show that the mean curvature flow of generic closed
surfaces in ℝ³ avoids asymptotically conical and non-spherical compact
singularities. The main tool is a long-time existence and uniqueness
result for ancient mean curvature flows that lie on one side of
asymptotically conical or compact shrinking solitons. This is joint work
with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis.

Mean curvature flow with generic initial data  Version PDF

Mercredi 6 mai 16:00-18:00 Leonid Polterovitch  (Tel Aviv)
Geometric facets of quantization

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Lieu : Demander le lien Zoom à jean-michel.bismut@u-psud.fr

Résumé : I will touch upon several topics on the crossroads of quantization
and geometry. In particular, I will explain why compatible almost-complex
structures on symplectic manifolds correspond to optimal quantizations,
and discuss spectral geometry of the Berezin transform describing
quantization followed by dequantization. Joint works with Louis Ioos,
David Kazhdan, Viktoria Kaminker and Dor Shmoish.

Geometric facets of quantization  Version PDF

Mercredi 29 avril 16:00-18:00 Tristan Clifford Collins  (MIT)
The deformed Hermitian-Yang-Mills equation and geometric invariant theory

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Lieu : Zoom - Demander le lien à
jean-michel.bismut chez math.u-psud.fr

Résumé : I will discuss the deformed Hermitian-Yang-Mills (dHYM) equation, a nonlinear PDE which plays an important role on the complex side of mirror symmetry, analogous to the role of special Lagrangians in symplectic geometry. In particular, I will explain how ideas from Geometric Invariant Theory can be used to study the correspondence between algebro-geometric notions of stability and existence of solutions to the dHYM equation. This is joint work with S.-T. Yau.

The deformed Hermitian-Yang-Mills equation and geometric invariant theory  Version PDF

Mercredi 22 avril 13:55-16:00 Andras Szenes  (Université de Genève)
Thom polynomials and multipoint formulas

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jean-michel.bismut chez math.u-psud.fr

Résumé : Global singularity theory deals with topological obstructions to the existence of various types of singularities of maps.
The subject has its beginnings in the works of Thom in the 50s, then Damon in 70s, who described the general form of the single point formulas.
Multipoint formulas are a classical subject which were the systematically studied by Kleiman and Katz in the 80’s. Finally, in the last 20 years, Kazarian and Rimanyi came up with a stunning set of conjectures linking the two problems. I will describe all this, as well as recent joint work with G. Berczi on a promising approach to these conjectures.

Thom polynomials and multipoint formulas  Version PDF