Recent work of Alessandrini-Lee-Schaffhauser generalized the theory of higher Teichmüller spaces to the setting of orbifold surfaces. In particular, these authors proved that, as in the torsion-free surface case, there is a « Hitchin component » of representations into PGL(n,R) which is homeomorphic to a ball. They explicitly compute the dimension of Hitchin components for triangle groups, and find that this dimension is positive except for a finite number of low-dimensional examples where the representations are rigid. In contrast with these results and with the torsion-free surface group case, we show that the composition of the geometric representation of a hyperbolic triangle group with a diagonal embedding into PGL(2n,R) or PSp(2n,R) is always locally rigid.

Mini-Cours

We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is a plan for each session of the mini-course:

1) Motivation and ideas from mirror symmetry, main results.
2) Skeletal curves: a key notion in the theory.
3) Naive counts, tail conditions and deformation invariance.
4) Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras, applications to moduli spaces of Calabi-Yau pairs.

Registration is compulsory. Please click on the link below to receive the zoom link and password to join the mini-course online:

https://us02web.zoom.us/meeting/register/tZIvcOCorD8iG9ES5hqURXELfgJhQbXND8N1

 

Mini-Cours

We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is a plan for each session of the mini-course:

1) Motivation and ideas from mirror symmetry, main results.
2) Skeletal curves: a key notion in the theory.
3) Naive counts, tail conditions and deformation invariance.
4) Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras, applications to moduli spaces of Calabi-Yau pairs.

Registration is compulsory. Please click on the link below to receive the zoom link and password to join the mini-course online:

https://us02web.zoom.us/meeting/register/tZIvcOCorD8iG9ES5hqURXELfgJhQbXND8N1

 

 

——–  IMPORTANT INFORMATION  ——–

Due to the health situation related to the Coronavirus epidemic, the course has been cancelled and postponed at a later date to be confirmed.

Mini-Cours

We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized log Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is the plan for each session of the mini-course:

1. Motivation and ideas from mirror symmetry, main results.

2. Skeletal curves: a key notion in the theory.

3. Naive counts and deformation invariance.

4. Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras.

 

——–  IMPORTANT INFORMATION  ——–

Due to the health situation related to the Coronavirus epidemic, the course has been cancelled and postponed at a later date to be confirmed.

Mini-Cours

We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized log Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is the plan for each session of the mini-course:

1. Motivation and ideas from mirror symmetry, main results.

2. Skeletal curves: a key notion in the theory.

3. Naive counts and deformation invariance.

4. Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras.

——–  IMPORTANT INFORMATION  ——–

Due to the health situation related to the Coronavirus epidemic, the course has been cancelled and postponed at a later date to be confirmed.

Mini-Cours

We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized log Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is the plan for each session of the mini-course:

1. Motivation and ideas from mirror symmetry, main results.

2. Skeletal curves: a key notion in the theory.

3. Naive counts and deformation invariance.

4. Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras.

——–  IMPORTANT INFORMATION  ——–

Due to the health situation related to the Coronavirus epidemic, the course has been cancelled and postponed at a later date to be confirmed.

Mini-Cours

We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized log Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is the plan for each session of the mini-course:

1. Motivation and ideas from mirror symmetry, main results.

2. Skeletal curves: a key notion in the theory.

3. Naive counts and deformation invariance.

4. Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras.

In this talk, we will give a new interpretation of Shintani’s work concerning the generating function of nonpositive values of Hecke L-functions for totally real fields. In particular, we will construct a canonical class, which we call the Shintani generating class, in the cohomology of a certain quotient stack of an infinite direct sum of algebraic tori associated with a fixed totally real field. Using our observation that cohomology classes, not functions, play an important role in the higher dimensional case, we proceed to newly define the p-adic polylogarithm function in this case, and investigate its relation to the special value of p-adic Hecke L-functions. Some observations concerning the quotient stack will also be discussed. This is a joint work with Kei Hagihara, Kazuki Yamada, and Shuji Yamamoto.

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Bien cordialement,

Let p be a prime number and L a finite unramified extension of Q_p. We give a survey of past and new results on smooth admissible representations of GL_2(L) that appear in mod p cohomology. This is joint work with Florian Herzig, Yongquan Hu, Stefano Morra and Benjamin Schraen.

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In this talk, we will explain the final outcome on the Beilinson-Bloch-Kato conjecture for motives coming from certain automorphic representations of GL(n) x GL(n+1), of our recent project with Yichao Tian, Liang Xiao, Wei Zhang, and Xinwen Zhu. In particular, we show that the nonvanishing of the central L-value of the motive implies the vanishing of the corresponding Bloch-Kato Selmer group. We will also explain the main ideas and ingredients of the proof.

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We study the one-loop corrections to correlation functions in quantum field theories in the Anti de Sitter space-time. Our calculation shows the existence of non-local counterterms which however respect the AdS isometry. Our arguments are general and applicable to general non-conformal AdS field theories. We also explain why calculations in Euclidean and Lorentzian signatures should differ even at the leading order in non globaly hyperbolic manifolds.

For an arbitrary morphism of (super)manifolds, the pull-back is a linear map of the space of functions. In 2014 Th.Voronov have introduced thick morphisms of (super)manifolds which define a generally non-linear pull-back of functions. This construction was suggested as an adequate tool to describe $L_infty$ morphisms of algebras of functions provided with the structure of a homotopy Poisson algebra. It turns out that if you go down from `heaven to earth’, and consider usual (not super!) manifolds, then we come to constructions which have natural interpretation in classical and quantum mechanics. In particular in this case the geometrical object which defines the thick diffeomorphism becomes an action of classsical mechanics, and pull-back of the thick diffeomorphism with a quadratic action give a spinor representation. I will define a thick morphism and will tell shortly about their application in homotopy Poisson algebras. Then I will discuss the relation of thick morphisms with notions such as the action in classical mechanics and spinors.

The talk is based on the work: « Thick morphisms of supermanifolds, quantum mechanics and spinor representation’, J.Geom. and Phys., 2019, article Number: 103540,
https://doi.org/10.1016/j.geomphys.2019.103540,
arXiv:1909.00290
Authors: Hovhannes Khudaverdian, Theodore Voronov