Seminars

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Homotopy quantization of real plane curve enumeration

Speacker: Sergey Galkin - HSE, Moscow
Tue 13 Nov 2018, 15:30 - Sala 236Differential Geometry

Abstract: Observe that an integral Heisenberg group has a topological incarnation as a second homotopy group of a particular space relative to a two-torus. Similar construction and computation assigns Heisenberg-like groups (generated by broken lines) to toric surfaces. We construct a natural integer-valued functional on the center of this group. By re-embedding a southern hemi-sphere via a second Frobenius map, we can associate an element in this center to every real rational curve on a projective plane (or a toric surface), that passes through Menelaus-generic collection of real points on a toric boundary.

Thus to every such real rational curve we associate an integer invariant, a quantum index. We show that it coincides with Mikhalkin's logarithmic area, and consequently it coincides with a tropical refinement introduced by Block and Goettsche. Time permits I will explain another motivation for our constructions: a non-commutative deformation of mirror symmetry for symplectic del Pezzo surfaces.


UPCOMING



On secant defectivity of homogeneous varieties

Speacker: Alex Massarenti - UFF
Wed 14 Nov 2018, 15:30 - Sala 228Algebra

Abstract: Most of the seminar will be a basic introduction to classical concepts in algebraic geometry such assecant defectivity, dimension of linear systems, and existence of special subvarieties of a given projective variety.I will then comment on Terracini's lemma and explain how it has been used to attack the problem of secant defectivity ofcertain homogeneous varieties.Finally, I will give an idea of a new method, based on degenerations techniques, I recently introduced with Rick Rischterto tackle secant defectivity problems. Such method allowed us to improve a result on non secant defectivity of Grassmanniansdue to Abo, Ottaviani and Peterson.


A question of Norton-Sullivan and the rigidity of pseudo-rotations on the two-torus

Speacker: Jian Wang - IMPA
Tue 20 Nov 2018, 15:30 - Sala 228Ergodic Theory

Abstract: In 1996,A. Norton and D. Sullivan asked the following question: If $f : \mathbb{T}^2 \rightarrow \mathbb{T}^2$ is a diffeomorphism,$h : \mathbb{T}^2\rightarrow\mathbb{T}^2$ is a continuous map homotopic to the identity,and $hf = T_{\rho}h$ where $\rho \in \mathbb{R}^2$ is a totally irrational vector and $T_{\rho} : \mathbb{T}^2 \rightarrow \mathbb{T}^2,z \mapsto z + \rho$ is a translation,are there natural geometric conditions(e.g. smoothness)on $f$ that force $h$ to be a homeomorphism? In this talk,we give a negative answer to this question with respect to the regularity.We also show that under certain boundedness condition,a $C^r$(resp. Hölder)conservative irrational pseudo-rotation on $\mathbb{T}^2$ with a generic rotation vector is $C^{r-1}$-rigid(resp. $C^0$-rigid). These provide a partial generalization of the main results in[Bramham,Invent. Math. 199 (2), 561-580, 2015;A. Avila,B. Fayad,P. Le Calvez,D. Xu,Z. Zhang,arXiv: 1509.06906v1]. These are joint works with Zhiyuan Zhang and Hui Yang.