Travel plans, Talks, Seminars and Conferences:

After 2015 I have included the slides of my talks in this page. For the slides of the previous talks see the link Talks and Seminars

18-29 July 2011:

1. I will presnt  a mini course with the title "Multiple integrals and modular differential equations" in 28o Colóquio Brasileiro de Matemática IMPA, Rio de Janeiro, 18 a 29 de julho de 2011

29 July-14 August 2011:

1. University of Alberta (Collaboration with Charles Doran)

14-27 August- 2011

1. Participating the conference Workshop on Arithmetic and Geometry of K3 surfaces and Calabi-Yau threefolds  to be held at the University of Toronto and the Fields Institute 

28 August-02 September 2011:  

1. Back to Rio

02-13 September- 2011:

1. Mainz University (Collaboration with Duco van Straten and Stefan Reiter).

2. 12 September, University of Kaiserslautern, presentation of the library foliation.lib. Here is the Title: Calculating Gauss-Manin connections, Picard-Fuchs equations and modular
   differential equations by Singular Abstract: In this talk we are going to introduce a library written in Singular which deals with the following topics:
   1. For a family of hypersurfaces X over T we calculate the Gauss-Manin connection on T.
   2. For one parameter family of varieties we calculate the Picard-Fuchs equation.
   3. In the multi parameter case T, we calculate certain vector fields whose ODE's is solved by modular-type functions.
   
   The case of Weierstrass family of elliptic curves will be explained in more detail.  

14-18 September- 2011:

1. Participating the conference Variation of Cohomology: D-Modules, Monodromy and Arithmetic,  University of Bayreuth   Title: Halphen-type equations attached to Calabi-Yau equations: After an introduction to Darboux-Halphen-Ramanujan equations which are attached to Picard-Fuchs equations coming from the variation of elliptic curves, I explain how such an attachment is generalized to an arbitrary Calabi-Yau equation. The resulting differential equation in this case is on ODE in seven variables. I explain modular properties of a particular solution of such an ODE with respect to the monodromy group of the Calabi-Yau equation. The case of mirror quintic Calabi-Yau equation will be explained in more details. 

18-25 September- 2011:

1. Max Planck Institute for Mathematics, (Collaboration with Prof. Don Zagier)

01-10 December-2011:   Canceled

1. Edmonton-Canada

2. Hodge theory and string duality, Banff, Canada

25 September-25 October 2011 : Canceled

1. During this period I will visit SUT, Sharif University of Technology and IPM, Institute for Physics and Mathematics. The following mini courses are planned:

• Title: Five lectures in Hodge theory
  Place: Sharif University of Technology
  Abstract: The objective of the course is to introduce Hodge theory and its central challenge, namely the Hodge conjecture which is one of the millennium conjectures of Clay Mathematical Institute. It plays an important role between many areas like Algebraic Geometry, topology and recently differential equations. The course will consist of:
  Lectute 1: Basics of singular homology and cohomology
  Lecture 2: Topology of algebraic varieties, Lefschetz theorems
  Lecture 3: Hodge decomposition, Hodge conjecture, Hodge filtration, 
  Lecture 4: Lefschetz (1,1) theorem,
  Lecture 5: Hodge conjecture for product of elliptic curves
  The course is mainly  for undergraduate students with a basic knowledege in complex analysis and topology. We will mainly focus on the historical aspects of the theory and will skip proofs.
  
  References: Lectures in Hodge Theory

•  Title: Modular forms and string theory
   Place: IPM
  Abstract: In the last decades, we  have seen many interactions between Number Theory, Arithmetic and Algebraic Geometry, and Theoretical Physics (in particular, String Theory). For instance, the classical modular forms and quasi-modular forms appear in the calculation of higher genus topological string partition functions in Quantum Field Theory. The objective of the course is to explore this interaction from  the mathematical point of view. We will mainly focus on three examples of target manifolds, namely, elliptic curves, K3 surfaces and  Calabi-Yau varieties. In the case of elliptic curves M. R. Douglas found explicit calculable formulas for topological partition functions in terms of generalized theta functions. This leads in a natural way to the polynomial structure of partition functions in terms of three Eisenstein series. Since we have the Ramanujan differential equation between Eisenstein series, we conclude that  derivatives of partition functions in the Ka"hler modulus have still such polynomial structures, and hence, they can be calculated in an effective way. In the case of K3 surfaces the calculations lead  to quasi-modular forms and this is explained by the fact that the mirror of the generic family of K3 surfaces can be constructed from elliptic curves. In 1991 there appeared the article of Candelas, de la Ossa, Green and Parkes, in which they  calculated in the framework of mirror symmetry the genus zero partition function, called also the Yukawa coupling,  which predicts the number of rational curves of a fixed  degree in a generic quintic three fold. Since then there was some effort to express the Yukawa coupling in terms of classical modular or quasi-modular forms, however, there was no success. The final objective of this course is to introduce a new theory of (quasi) modular forms attached to
  the mirror of quintic Calabi-Yau varieties.
  
  
  References: 

• H. Movasati, Eisenstein type series for Calabi-Yau varieties ,Nuclear Physics B, 847, (2011) 460–484.

• H. Movasati, Quasi-modular forms attached to elliptic curves, I, lecture notes at Besse, France 2010 and this  workshop held at IMPA 2011.  

17 June -23 June 2012: 

Canadian Number Theory Association XII Meeting
University of Lethbridge
June 17-22, 2012

Title: Modular forms for triangle groups
Abstract:
In this talk we first describe a solution of the Halphen equation which has modular properties with respect to a group which is not discrete in general. We show that in the particular case of triangular groups, the Halphen equation gives us a basis of the algebra of modular and quasi-modular forms. Some statements and conjectures about the q-expansion of such modular forms will be also presented. This is a work under preparation jointly with Ch. Doran, T. Gannon, Kh. M. Shokri.
  

 

28 June -26-July 2012: 

Together with Khosro Monsef Shokri I am going to give some lectures at IMCA-Peru:

Title: Rank of Elliptic curves
By:    Hossein Movasati, Khosro Monsef Shokri


Abstract:

The aim of these lectures is to introduce the arithmetic of elliptic curves and
in particular problems and theorems related to their ranks. The most important
property of elliptic curves which specialize them among algebraic curves is the
abelian group structure. The rational points of an elliptic curve form a subgroup
and it is finitely generated (Mordell Theorem). One of the main problems in number
theory is to find the rank of this group. By analytic  methods one can associate
to any elliptic curve over rational numbers, an L-function which is holomorphic on
the whole plane and modularity theorem shows that, this L function,  essentially
is the L function of some modular form. The Birch-Swinerton-Dyre conjecture predicts
that the rank of an elliptic curve is the zero order of this L function at the point
s=1. The only general result is due to Gross-Zagier-Kolyvagin for rank equal to zero or
one elliptic curves. In these lectures we explain all these ideas in more detail.

The only prerequisite for the course is a basic knowledge in Algebra and Complex Analysis.  

30 July-02 August:

 Folheações Holomorfas e Geometria Algébrica em Minas,  60 anos do Prof. Márcio Gomes Soares

Title: Review of abelian and  iterated integrals in holomorphic foliations

Abstract: In this talk I am going to give an overview of abelian and iterated integrals
which appear in the deformation of holomorphic foliations with a first integral.
We present also some results jointly obtained by L. Gavrilov and I. Nakai.
 

20-21 August 2012:    

I am organizing the following small workshop:

Arithmetic and Geometry of Picard-Fuchs Differential Equations

01-29 September 2012:    

Visiting The Institute of Mathematical Sciences, Hong-Kong. I will give the following series of lectures

 
Lecture 1: Quasi-modular forms attached to Elliptic curves I:
Gauss-Manin connection and Picard-Fuchs equation of families of elliptic curves,
Lecture 2: Quasi-modular forms attached to Elliptic curves II:
Geometric definition of Quasi-Modular forms and  Ramanujan and Darboux-Halphen differential equations
Lecture 3: Modular-type functions  attached to mirror quintic Calabi-Yau threefolds
Lecture 4:  Modular-type functions  attached to Calabi-Yau linear differential equations:
Almkvist-Enckevort-Straten-Zudilin classification
Lecture 5: Partition functions and Bershadsky-Cecotti-Ooguri-Vafa anomaly equation
Lecture 6: A work of Yamaguchi-Yau and modular-type functions

References:
H. Movasati, Eisenstein type series for Calabi-Yau varieties , Nuclear Physics B, 847, (2011) 460–484.
H. Movasati, Modular-type functions attached to mirror quintic Calabi-Yau varieties I,
H. Movasati, Quasi-modular forms attached to Hodge structures I , To appear in Fields Communication Series
H. Movasati, Quasi-modular forms attached to elliptic curves, I, To appear in Annales Mathématique Blaise Pascal.
H. Movasati, Anomaly equation, partition functions and Modular-type functions, preprint.
Except the last one, all the articles are available at: http://w3.impa.br/~hossein/publication.html


HongKong Geometry Colloquium 22/09/2012:


Title: Differential equations for Humbert surfaces
Abstract:
In the moduli of principally polarized Abelian surfaces, we have Humbert surfaces which parametrize
the Abelian surfaces with an extra endomorphism. In this talk we first describe a Hodge theoretic
correspondence between principally polarized Abelian surfaces and N-polarized K3 surfaces of rank 17 and
then we introduce a collection of partial differential equations in the moduli of such N-polarized K3 surfaces
whose algebraic solutions are the Humbert surfaces. This is a joint work under preparation
with Ch. Doran and U. Whitcher.



 

15-19 October 2012: Canceled

Feuilletages et équations différentielles complexes
   

17 December 2012 -17 January 2013 :   

Particpating Frontiers in Mathematical Sciences, a conference in honor of Siavash Shahshahani's 70th birthday. During this period I will also visit
SUT, Sharif University of Technology and IPM, Institute for Physics and Mathematics.



The following mini courses are planned:

• Title: Five lectures in Hodge theory
  Place: IPM
  Abstract: The objective of the course is to introduce Hodge theory and its central challenge, namely the Hodge conjecture which is one of the millennium conjectures of Clay Mathematical Institute. It plays an important role between many areas like Algebraic Geometry, topology and recently differential equations. The course will consist of:
  Lectute 1: Basics of singular homology and cohomology
  Lecture 2: Topology of algebraic varieties, Lefschetz theorems
  Lecture 3: Hodge decomposition, Hodge conjecture, Hodge filtration, 
  Lecture 4: Lefschetz (1,1) theorem,
  Lecture 5: Hodge conjecture for product of elliptic curves
  The course is mainly  for undergraduate students with a basic knowledege in complex analysis and topology. We will mainly focus on the historical aspects of the theory and will skip proofs.
  
  References: Lectures in Hodge Theory

•  Title: From modular forms and elliptic curves  to Calabi-Yau threefolds 
   Place: Sharif University of Technology 
  Abstract: In the last decades, we  have seen many interactions between Number Theory, Arithmetic and Algebraic Geometry, and Theoretical Physics (in particular, String Theory). For instance, the classical modular forms and quasi-modular forms appear in the calculation of higher genus topological string partition functions in Quantum Field Theory. The objective of the course is to explore this interaction from  the mathematical point of view. We will mainly focus on three examples of target manifolds, namely, elliptic curves, K3 surfaces and  Calabi-Yau varieties. In the case of elliptic curves M. R. Douglas found explicit calculable formulas for topological partition functions in terms of generalized theta functions. This leads in a natural way to the polynomial structure of partition functions in terms of three Eisenstein series. Since we have the Ramanujan differential equation between Eisenstein series, we conclude that  derivatives of partition functions in the Ka"hler modulus have still such polynomial structures, and hence, they can be calculated in an effective way. In the case of K3 surfaces the calculations lead  to quasi-modular forms and this is explained by the fact that the mirror of the generic family of K3 surfaces can be constructed from elliptic curves. In 1991 there appeared the article of Candelas, de la Ossa, Green and Parkes, in which they  calculated in the framework of mirror symmetry the genus zero partition function, called also the Yukawa coupling,  which predicts the number of rational curves of a fixed  degree in a generic quintic three fold. Since then there was some effort to express the Yukawa coupling in terms of classical modular or quasi-modular forms, however, there was no success. The final objective of this course is to introduce a new theory of (quasi) modular forms attached to the mirror of quintic Calabi-Yau varieties.
  
  
  References: 

• H. Movasati, Eisenstein type series for Calabi-Yau varieties ,Nuclear Physics B, 847, (2011) 460–484.

• H. Movasati, Quasi-modular forms attached to elliptic curves, I,  To appear in Annales Mathématique Blaise Pascal .

I will give the talk:
     Title: Modular forms: From arithmetic to Physics applications.
     Abstract: In this talk I will review many applications of modular forms
     ranging from number theory, enumerative Algebraic Geometry, representation theory of groups
     and Lie algebras  to Topological String Theory.

13-31 July- 2013:

I will be visiting  IMCA-Peru. Four talks with title PROBLEMAS SUBSTANCIALES DE OLIMPIADAS DE MATEMÁTICA, TEÓRICA Y PRÁCTICA. It is based on     Picard-Lefschetz Theorie der Anordnungen, a text which I wrote in German when I was at MPIM in 2001.  

02 September 2013- 30 August 2014. 

Sabbatical year at Harvard   

 

29 October 2013: 

Talk at Brandeis University: 
Title: A common framework for automorphic forms and topological partition functions.

02 September 2013- 30:   

November 18-22, 2013
Workshop on Hodge Theory in String Theory
a joint workshop with PIMS CRG Program “Geometry and Physics”
Principal Organizers: Charles F. Doran, David Morrison, Radu Laza, Johannes Walcher.


  Title: A common framework for automorphic forms and topological partition functions.
Abstract: Classical modular forms and in general automorphic forms enjoy q-expansions with
fruitful applications in different branches of mathematics. From another side we have q-expansions coming
from the B-model computations of  mirror symmetry which, in general, are believed to be new functions.
In this talk I will present a common algebro-geometric framework for all these q-expansions.
This is based on the moduli of varieties with a fixed topological data and  enhanced  with a basis of
the algebraic de Rham cohomology, compatible with the Hodge filtration and with a constant intersection matrix.
In our way, we will  also enlarge the algebra of automorphic forms to a bigger algebra which is
closed under canonical derivations.  I will mainly discuss three examples:
1. Elliptic curves and classical modular forms,
2. Principally polarized abelian varieties, lattice polarized K3 surfaces and Siegel modular forms
3. Mirror quintic Calabi-Yau varieties, Yukawa coupling and topological partition functions.

 

06-09 December 2013:

2013 CMS Winter Meeting in Ottawa,  Session: Modular forms and Physics.

Title: Integrality properties of automorphic forms for  triangle groups
Abstract:
This talk is based on the joint work with Shokri      arXiv:1307.4372 and Doran, Gannon, Shokri.  arXiv:1306.5662.
We consider the integrality properties of the coefficients of the mirror map attached to
 the generalized hypergeometric function  with rational parameters and with a maximal unipotent monodromy.
We  present a conjecture on the $p$-integrality of the mirror map which can be verified experimentally. 
We prove  it for $n=2$ and prove its consequence on the $N$-integrality of the mirror map  for the particular cases $1\leq n\leq 4$.
 This was a conjecture in mirror symmetry which was first proved in particular cases by Lian-Yau.
The general format was formulated  by Zudilin and finally established by Krattenthaler-Rivoal.
 For $n=2$ we  obtain  the  Takeuchi's classification of arithmetic triangle
groups with a cusp, and for $n=4$ we prove  that  $14$ examples of hypergeometric Calabi-Yau equations are the full classification
of hypergeometric mirror maps with integral coefficients.
For our purpose we state and prove a refinement of a theorem of Dwork which largely simplifies many existing proofs in the literature.  

17-24 February 2014:

University of Texas at Austin (and Boston University 25 February 2014)

Title: A common framework for automorphic forms and topological string partition functions.
Abstract: Classical modular forms and in general automorphic forms enjoy q-expansions with
fruitful applications in different branches of mathematics. From another side we have q-expansions coming  from the B-model computations of  mirror symmetry which, in general, are believed to be new functions.
In this talk I will present a common algebro-geometric framework for all these q-expansions.
This is based on the moduli of varieties with a fixed topological data and  enhanced  with a basis of
the algebraic de Rham cohomology, compatible with the Hodge filtration and with a constant intersection matrix.  In our way, we will  also enlarge the algebra of automorphic forms to a bigger algebra which is
closed under canonical derivations.  I will mainly discuss three examples:
1. Elliptic curves and classical modular forms,
2. Principally polarized abelian varieties, lattice polarized K3 surfaces and Siegel modular forms
3. Mirror quintic Calabi-Yau varieties, Yukawa coupling and topological string partition functions.

01-30 July 2014:

Taida Institute for Mathematical Sciences, Taipei, Taiwan.

Title: Modular forms and Calabi-Yau varieties

Abstract: Classical modular forms and in general automorphic forms enjoy
q-expansions with fruitful applications in different branches of
mathematics. From another side we have q-expansions coming from the B-model
computations of mirror symmetry which, in general, are believed to be new
functions. In this series of talks I will present a common
algebro-geometric framework for all these q-expansions. This is based on
the moduli of varieties with a fixed topological data and enhanced with a
basis of the algebraic de Rham cohomology, compatible with the Hodge
filtration and with a constant intersection matrix. In our way, we will
also enlarge the algebra of automorphic forms to a bigger algebra which is
closed under canonical derivations. I will mainly discuss two examples: 1.
Elliptic curves and classical modular forms, 2. Mirror quintic Calabi-Yau
varieties, Yukawa coupling and topological partition functions. The talks
are based on the following articles available in arxiv:

         H. Movasati, Modular-type functions attached to mirror quintic
Calabi-Yau varieties,

         H. Movasati, Quasi-modular forms attached to elliptic curves I,
Annales Mathématique Blaise Pascal, v. 19, p. 307-377, 2012.

Date: July 07-09、July14-16 , 2014

Time: 13:30~15:00

Place: Room R440, Astronomy and Mathematics Building,  National Taiwan
University

Lecture 1: Modular forms and elliptic curves I, What is Gauss-Manin connection for Gauss? Deriving Darboux, Halphen and Ramanujan from Gauss-Manin.
Lecture 2: Modular forms and elliptic curves II, Ramanujan's contribution, Darboux and Halphen's contribution, Moduli space of enhanced elliptic curves,  Hodge filtrations, universal family of enhanced elliptic curves.
Lecture 3: An algebra of modular forms for Calabi-Yau varieties. Geometric Invariant theory and constructing moduli spaces T,
Lecture 4: An algebra of modular forms for mirror quintic Calabi-Yau varieties, I: What is mirror quintic Calabi-Yau threefold? Moduli space I, Gauss-Manin connection I, Intersection form, Hodge filtration, vector fields on moduli spaces, moduli space II
Lecture 5: An algebra of modular forms for mirror quintic Calabi-Yau varieties, II. Periods and modular-type functions, Integrality, functional equations, BCOV holomorphic anomaly
Lecture 6: An algebra of modular forms for mirror quintic Calabi-Yau varieties, III

01-30 August 2014:

Max-Planck Institute for Mathematics, Bonn, Germany.

Datum:  Die, 2014-08-12 15:00 - 16:00
Location:  MPIM Lecture Hall
Parent event:  Seminar on Algebra, Geometry and Physics
In this talk I will first remind how to derive the Ramanujan relations between Eisenstein series and the
Darboux-Halphen differential equation from the Gauss-Manin connection of families of elliptic curves.
Then I will explain a generalization of this fact in  the case of Calabi-Yau threefolds. In this way one gets an algebra
which generalizes the algebra of quasi-modular forms. Genus g topological string partition functions turn
out to be elements of this new algebra and the corresponding Bershadsky-Cecotti-Ooguri-Vafa anomaly equation
can be written in terms of certain vector fields derived from the Gauss-Manin connection. The talk is based on
the papers  arXiv:1111.0357 and  arXiv:1110.3664 and a joint work under preparation with
M. Alim, E. Scheidegger, S.T. Yau.

24 March 2015:

UFF, Niterio.

Mirror symmetry in higher genus

In this talk I will first recall the predictions of mirror symmetry for elliptic curves. This gives a recipe for  counting  the number of ramified curves over a torus with simple ramification points (after Douglas, Dijkgraaf, Kaneko, Zagier). Then,  I will discuss a similar topic in the case of Calabi-Yau threefolds, and in particular, I will introduce the algebraic anomaly equation for higher genus Gromov-Witten invariants. I will also discuss the ambiguity problem in this context. The talk is based on the monograph and the paper below:
http://w3.impa.br/~hossein/myarticles/GMCD-MQCY3.pdf
http://w3.impa.br/~hossein/myarticles/GMCD-CY3.pdf

06-10 May 2015:   

  Arithmetic and Algebraic Differentiation: Witt vectors, number theory and differential algebra In honor of Alexandru Buium, Berkeley,


Title: Gauss-Manin connection in disguise: a tale of a differentiation 

Abstract: In this talk I will first remind two historical, but not so well-known facts, namely, how to derive
the Ramanujan relations between Eisenstein series and the  Darboux-Halphen differential equation of  theta series, 
from the Gauss-Manin connection of families of elliptic curves.
The rest of the talk will be dedicated to generalizations of this beyond elliptic curves and classical modular/automorphic  forms.
This includes the case  of Calabi-Yau threefolds and the corresponding generating function of the virtual number of rational curves.

19-26 July 2015:

Lima-Peru

MINICURSO: ALGUNOS PROBLEMAS DE OLIMPIADAS DE MATEMÁTICAS ORIGINADOS DE LA TEORÍA DE HODGE
FECHA: Lunes 20 de Julio – 14:00 Horas
              Martes 21 de Julio – 10:00 Horas
              Miércoles 22 de Julio - 10:00 Horas
REQUISITOS : CONOCIMIENTOS BÁSICOS SOBRE MATRICES (TALES COMO RANGO DE UNA MATRIZ), POLINOMIOS.

First Class: Tow elementary problems, problem 1, problem 2
Second Class: Rudiments of Algebraic Geometry, Algebraic Topology and the desire of classifying subvarieties of a given variety.
Third Class: The solutions to the problems of the first class and more elementary problems

31-August-18 September 2015:

Harvard university.
08 September 2015. Differential Geometry Seminar.

Title: Why should one compute the periods of algebraic cycles?
Abstract: Let $X$ be a smooth projective  variety of dimension $2k$ over complex numbers
and let $Z$ be a subvariety of $X$ of dimension $k$. One says that the infinitesimal
Hodge conjecture (IHC) holds if the deformation space of the pair $(X,Z)$ inside the moduli
space of $X$ is the same as the deformation space of the Hodge cycle $[Z]$ induced by $Z$.
Hodge conjecture does not imply IHC, however, verifications of IHC in many explicit situations
imply the Hodge conjecture for deformed Hodge cycles. In this talk I am going to explain how
explicit computations of periods of differential forms over cycles $[Z]$ lead to the
verifications of IHC. I will also prove that IHC holds for linear projective spaces inside hypersurfaces.
This in two dimensional case ($k=1$), where the Hodge conjecture is well-known as Lefschetz $(1,1)$ theorem,  is a result of Green and Voisin in 1990.  Some partial results concerning complete intersections
inside sextic hypersurfaces fourfolds (which are Calabi-Yau) will be given. One of the  basic ingredients of
the proof is the so called infinitesimal  variation of Hodge structures.   The talk is partially based on  arXiv:1411.1766v1.

19-24 October 2015:

Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives (October 19-24, 2015)


Title: Two aspects of the project Gauss-Manin Connection in Disguise, Talk No. 1, Talk. No. 2,  No1+No2


In this talk I will first formulate an algebro-geometric framework in which
$q$-expansions of the B-model of topological string theory become natural generalizations
of elliptic modular forms. These $q$-expansions will be elements of the  algebra
of the so called Calabi-Yau modular forms. I will discuss many similarities and differences between
Calabi-Yau and elliptic modular forms. This includes Ramanujan-type differential equations,
functional equations, conifold cusp, some product formulas, gap condition, Hecke operators, growth
of the coefficients  of $q$-expansions etc. The talk is mainly based on the monograph Gauss-Manin connection in disguise: Calabi-Yau modular forms available at http://w3.impa.br/~hossein/myarticles/GMCD-MQCY3.pdf

04-08 January 2016:

Title:  Some elementary problems arising from Hodge theory
Prerequisite: Basic linear algebra


Description: Quite often in mathematics, one finds problems which
can be understood and solved with the knowledge of high school mathematics,
however, in order to explain their origin, one needs many advanced courses.
In this mini course I will explain few such problems arising from Hodge theory
and the topology of algebraic varieties. Some rudiments of Hodge theory will
be presented in order to give a flavor of the origin of such problems.

Source:
1) Multiple Integrals and Modular Differential Equations, 28 Colóquio Brasileiro de Matemática, p. 168, 2011.
2) http://w3.impa.br/~hossein/WikiHossein/OlympiadProblems.html

21 January-18 February 2016:

CMSA, Harvard University.
8 February 2016.
Title: Calabi-Yau modular forms: Are they as useful as elliptic modular forms?
Abstract: In this talk we give an overview of the theory of Calabi-Yau modular forms based on the
references [1,2,3,4]. The main ingredient of the theory is a family of non-rigid Calabi-Yau varieties.
We give many analogies of this theory with the classical theory of elliptic
modular forms. In the case of elliptically fibered Calabi-Yau four and three folds we explain
how both theories are related to each other.
1. H. Movasati, Modular-type functions attached to mirror quintic Calabi-Yau varieties ,
    Math. Zeit., Vol 281, Issue 3, 2015, pp. 907-929.
2  H. Movasati, Gauss-Manin connection in disguise: Calabi-Yau modular forms
3. M. Alim,  H. Movasati, E. Scheidegger, S-T Yau, Gauss-Manin connection in disguise: Calabi-Yau threefolds
4. B. Haghighat, H. Movasati, S-T Yau, Calabi-Yau modular forms in limit: Elliptic Fibrations.

23 February 2016:

IMPA:
Title: Early history of singular homology and de Rham cohomology
Abstract:
In this talk I will give an overview of the origin  of singular homology
and de Rham cohomology during 19th century, when both concepts were
not yet defined. This is namely the study of elliptic and Abelian integrals.
This is also the early history of topology under the old name "Analysis Situs".
Then I will discuss many contributions of the following books to these concepts:
E. Picard, G. Simart,  Théorie des fonctions algébriques de deux variables indépendantes.
Vol. I, II. 1897, 1906.

26 February 2016:

Workshop de Topologia & Dinâmica (TopDin) de 22 (segunda) a 26 (sexta) de fevereiro de 2016

Title: As origens da conjetura de Hodge
Nesta palestra vou falar sobre os livros,
Théorie des fonctions algébriques de deux variables indépendantes. Vol. I, II.
publicado por Picard e Simart em 1897, 1906. Nestes livros se trata de estudo de integrais multiplas
e deles sairam os conceitos de numero e grupo de Picard na geometria algébrica, porém, poucas
pessoas sabem a relação destes conceitos com integrais. Vou explicar esta relação e 
no final vou falar sobre as  origens da conjetura de Hodge  que vem destes livros.

24 June 2016

Colga:

09-24 July 2016 Iran:

13th International Conference on Dynamical systems, Differential Equations and Applications, Isfahan,  13-15 July 2016.
Title: Differential Equations and Arithmetic

Differential equations are mainly studied from the point of view of dynamics.
In this talk I am going to discuss many differential equations which are mainly
used in Arithmetic Algebraic Geometry and Number Theory. These includes, various
types of Gauss-Manin connections and Picard-Fuchs linear differential equations,
differential equations solved by modular and quasi-modular forms (Darboux, Ramanujan and Halphen),
algebraic solutions of planar differential equations etc.
Frontiers in Mathematical Sciences 19-21 july, Tehran  2016. Three lectures plus a cconference talk

Title: An invitation to Hodge theory
         (On blackboard)
Abstract: The origin of Hodge theory goes back to many works on elliptic, abelian
and multiple integrals. In this talk I am going to explain how Lefschetz
was puzzled with the computation of Picard rank and this led him to consider
the homology classes of curves inside surfaces. This was ultimately formulated
in Lefschetz (1,1) theorem and then the Hodge conjecture which is one of the
millennium problems of Clay Mathematical Institutue. The talk is based on my
book under preparation: http://w3.impa.br/~hossein/myarticles/hodgetheory.pdf

25-30 September 2016:  (Canceled)

Modular Forms in String Theory Banff, Canada

08-13 January 2017:  Mini course in Freibrug, Germany:

Title: Modular and automorphic forms & beyond

Is it worth to elaborate a (new) mathematical theory which is a huge generalization of
the theory of (holomorphic) modular/automorphic forms, without knowing if at some point
you will have fruitful applications similar to those of modular forms? If your answer is yes, this
talk might be useful for you. This new theory starts with a moduli space of  projective varieties
enhanced with elements in their algebraic de Rham cohomology and with some compatibility with the Hodge filtration and
the cup product. These moduli spaces are conjectured to be affine varieties, and their ring of functions are candidates for
the generalization of automorphic forms. Another main ingredient of this theory is a set of certain vector
fields on such moduli spaces which are named "Gauss-Manin connection in disguise".
I will explain this picture in three examples.
1. The case of elliptic curves and the derivation of the algebra of quasi-modular forms (due to Kaneko and Zagier). 
2. The case of Calabi-Yau varieties and the derivation of generating function for Gromov-Witten invariants.
3. The case of principally polarized abelian surfaces and the derivation of Igusa's generators for the algebra of genus $2$ Siegel
modular forms.

In the follow-up lectures I will try to explain the three cases above in more details.

Lecture 1: Ramanujan's relations between Eisenstein series is derived from the Gauss-Manin connection of a family of elliptic
curves. A similar discussion will be done for Darboux and Halphen equations. I will also give some applications regarding
modular curves.
Lecture 2: I will explain a purely algebraic version of the Bershadsky-Cecotti-Ooguri-Vafa anomaly equation using
a Lie algebra on the moduli of enhanced Calabi-Yau varieties.
Lecture 3: In this lecture, I will explain how automorphic forms, and in particular Siegel modular forms, fit well
to the geometric theory explained in the previous lectures.

References:
The manuscript of my lecture notes
Gauss-Manin Connection in Disguise: Calabi-Yau modular forms,
Surveys in Modern Mathematics, International Press, Boston, 2017.
Gauss-Manin connection in disguise: Calabi-Yau threefolds 
(with Murad Alim, Emanuel Scheidegger, Shing-Tung Yau), CMP, 2016.
Quasi-Modular forms attached to elliptic curves: Hecke operators, JNT, 2015.
A course in Hodge Theory: With Emphasis on Multiple Integrals, Book under preparation.


 

9 January 20: Differential Geometry Seminar, Freiburg.

Title: Periods of algebraic cycles

The origin of Hodge theory goes back to many works on elliptic, abelian
and multiple integrals (periods). In this talk, I am going to explain how Lefschetz
was puzzled with the computation of Picard rank (defined using periods)
and this led him to consider the homology classes of curves inside surfaces.
This was ultimately formulated in Lefschetz (1,1) theorem and then the Hodge conjecture. In the second half of the talk
I will discuss periods of algebraic cycles and will give some applications in identifying
some components of the Noether-Lefschetz and Hodge locus.  The talk is based on my book
under preparation: A course in Hodge Theory: With Emphasis on Multiple Integrals, 

 

17 January 2017:   Seminar on algebra, Physics and Geometry, MPIM.

Title: Modular and automorphic forms & beyond

26 January 2017: Oberseminar MPIM

21 February 2017:  Seminario de Folheações holomorfas

Título: Computing a Taylor series

Is it worth to spend more than a year and compute a Taylor series of a theoretically-easy-to-describe holomorphic function in many variables? In this talk I am going to report on such a computation for periods of  hypersurfaces. Similar well-known series in the literature range from Gauss hypergeometric function to Gelfand-Kapranov-Zelvinsky (GKZ) hypergeometric functions.  Then I will give some applications of this computation in studying  reducedness and smoothness of  components of   Noether-Lefschetz and Hodge loci. 

01, June-31 August 2017:   Paris VII

06 June 2017 Paris VII, 24 June 2017 Toulouse

Title: From Picard and Simart's books  to periods of algebraic cycles.

The origin of Hodge theory goes back to many works on elliptic, abelian
and multiple integrals (periods). In particular, Picard and Simart's book 
"Théorie des fonctions algébriques de deux variables indépendantes. Vol. I, II."
published in 1897, 1906, paved the road for modern Hodge theory.
The first half of the talk is mainly about these books, for  instance,   I am going to explain
how Lefschetz was puzzled with the computation of Picard rank (by Picard  and using periods)
and this led him to consider the homology classes of curves inside surfaces.
This was ultimately formulated in Lefschetz (1,1) theorem and then the Hodge conjecture.
In the second half of the talk I will discuss periods of algebraic cycles and will give some applications in identifying
some components of the Noether-Lefschetz and Hodge locus.  The talk is based on my book:
A course in Hodge Theory: With Emphasis on Multiple Integrals,

29 June 2017:   Paris VI

Title: Modular and automorphic forms & beyond

Is it worth to elaborate a (new) mathematical theory which is a huge generalization of
the theory of (holomorphic) modular/automorphic forms, without knowing if at some point
you will have fruitful applications similar to those of modular forms? If your answer is yes, this
talk might be useful for you. This new theory starts with a moduli space of  projective varieties
enhanced with elements in their algebraic de Rham cohomology and with some compatibility with the Hodge filtration and
the cup product. These moduli spaces are conjectured to be affine varieties, and their ring of functions are candidates for
the generalization of automorphic forms. Another main ingredient of this theory is a set of certain vector
fields on such moduli spaces which are named "Gauss-Manin connection in disguise" and are natural generalizations of Ramanujan
relations between Eisenstein series and Darboux-Halphen differential equation of theta series.
I will explain this picture in three examples.
1. The case of elliptic curves and the derivation of the algebra of quasi-modular forms (due to Kaneko and Zagier). 
2. The case of Calabi-Yau varieties and the derivation of generating function for Gromov-Witten invariants.
3. The case of principally polarized abelian surfaces and the derivation of Igusa's generators for the algebra of genus $2$ Siegel
modular forms.

10-14 July 2017:   Hamburg

Title: Noether-Lefschetz and Hodge loci

Abstract: In this talk I will talk about identifying components of the Hodge loci which  live in the
parameter spaces of hypersurfaces. For surfaces this is known as Noether-Lefschetz loci.  The main
tools are the infinitesimal variation  of Hodge structures and the notion of modular foliations.

04-June 25 July 2018:   YMSC, Tsinhua university, China

Title: Modular and automorphic forms & beyond

Abstract:  The guiding principal in this lecture series is to develop a new theory of modular forms which
encompasses most of the available theory of modular forms in the literature, including Calabi-Yau modular forms with
its examples such as   Yukawa couplings and topological string partition functions,  and even go beyond all these cases.
We will first  use the available tools in Algebraic Geometry, such as Geometric  Invariant Theory, and construct the
moduli space T of projective varieties enhanced with elements in their algebraic de Rham cohomology ring.
The new theory of modular forms lives on the moduli space T. It turns out that such moduli spaces are of high dimension
and enjoy certain foliations, called modular foliations,  which are of  high codimension, and are constructed from the
underlying Gauss-Manin connection.  The mincourse will be mainly focused on three independent topics:

1. Hilbert schemes and actions of reductive groups and the construction of the moduli space T.
2. The theory of foliations of arbitrary codimensions on schemes and its relation with Noether-Lefschetz  and Hodge loci
in the case of modular foliations.
3. To rewrite available theories of automorphic forms, such as Siegel modular forms, Hilbert modular forms, modular forms
for congruence groups, and in general automorphic forms on Hermitian symmetric domains,  using the moduli space T. This
will produce a geometric theory of differential equations of automorphic forms.

The seminar is based on a book that I am writing and its preliminary draft will be distributed
between participants. It involves many reading activities on related topics, and contributions are most welcome.


References:
H. Movasati, Modular and automorphic forms & beyond, manuscript under preparation.

H. Movasati, Gauss-Manin connection in disguise: Calabi-Yau modular forms (Book),
with appendices by Khosro Shokri and Carlos Matheus, Surveys in Modern Mathematics, Vol 13, International Press, Boston.

B. Haghighat H. Movasati, S.-T. Yau. Calabi-Yau modular forms in limit: Elliptic fibrations, Communications in Number
Theory and Physics, Vol. 11, Number 4, 879-912, 2017.

M. Alim, H. Movasati, E. Scheidegger, S.-T. Yau. Gauss-Manin connection in disguise: Calabi-Yau threefolds,
Comm. Math. Phys.  344, (2016), no. 3, 889-914.

H. Movasati. Quasi-modular forms attached to elliptic curves, I, Annales Mathematique Blaise Pascal,  v. 19, p. 307-377, 2012.

01-November 2018 28 February 2019:   CMSA, Harvard university

Title: Hodge and Noether-Lefschetz loci

Abstract: Hodge cycles are topological cycles which are conjecturally (the millennium Hodge conjecture) supported in algebraic
cycles of a given smooth projective complex manifold. Their study in families leads to the notion of Hodge locus, which is also known
as Noether-Lefschetz locus in the case of surfaces. The main aim of this mini course is to introduce a computational approach to the study
of Hodge loci for hypersurfaces and near the Fermat hypersurface. This will ultimately lead to the verification of the variational
Hodge conjecture for explicit examples of algebraic cycles inside hypersurfaces and also the verification of integral Hodge conjecture for
examples of Fermat hypersurfaces. Both applications highly depend on computer calculations of rank of huge matrices. We also aim to review
some classical results on this topic, such as Cattani-Deligne-Kaplan theorem on the algebraicity of the components of the hodge loci,
Deligne's absolute Hodge cycle theorem for abelian varieties etc.
In the theoretical side another aim is to use the available tools in algebraic geometry and construct the moduli space
of projective varieties enhanced with elements in their algebraic de Rham cohomology ring. These kind of moduli spaces have
been useful in mathematical physics in order to describe the generating function of higher genus Gromov-Witten invariants, and it turns out
that the Hodge loci in such moduli spaces are well-behaved, for instance, they are algebraic leaves of certain holomorphic foliations. Such
foliations are constructed from the underlying Gauss-Manin connection. This lectures series involves many reading activities on related
topics, and contributions by participants are most welcome.

References:
M. Alim, H. Movasati, E. Scheidegger, S.-T. Yau. Gauss-Manin connection in disguise: Calabi-Yau threefolds, Comm. Math. Phys. 344, (2016), no. 3, 889-914.
E. H. Cattani, P. Deligne, and A. G. Kaplan. On the locus of Hodge classes. Amer. Math. Soc., 8(2):483--506, 1995.
B. Haghighat H. Movasati, S.-T. Yau. Calabi-Yau modular forms in limit: Elliptic fibrations, Communications in Number Theory and Physics, Vol. 11, Number 4, 879-912, 2017.
H. Movasati, Modular and automorphic forms & beyond, Book under preparation,2019.
H. Movasati. A Course in Hodge Theory: with Emphasis on Multiple Integrals.Book submitted,2018.
H. Movasati, On elliptic modular foliation, II, 2018
H. Movasati, R. Villaflor Loyola, Periods of linear algebraic cycles,, 2018.
H. Movasati, Gauss-Manin connection in disguise: Calabi-Yau modular forms, Surveys in Modern Mathematics, Vol 13, International Press, Boston.
H. Movasati, Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci, Asian Journal of Mathematics, Vol.21, No. 3, pp. 463-482, 2017.
C. Voisin. Hodge loci and absolute Hodge classes. Compos. Math., 143(4):945--958, 2007.
C. Voisin. Hodge loci. Handbook of moduli. Vol. III, volume 26 of Adv. Lect. Math. (ALM)}, pages 507--546. Int. Press, Somerville, MA, 2013.

16-19 December 2018:   Tokyo University

Hypergeometric functions and mirror symmetry, December 17-19, 2018


Title: B-Model of mirror symmetry for compact non-rigid Calabi-Yau manifolds
Abstract:
In B-model of mirror symmetry, period  manipulations play an important role for computing
the Gromov-Witten invariants of the A-model. This requires computing power series of periods,
finding a maximal unipotent monodromy, mirror map etc.   In this talk I will present a purely 
algebraic version of such  computations for Calabi-Yau  varieties  of arbitrary dimension.
It involves a construction of the moduli space of enhanced Calabi-Yau varieties and modular vector
fields on it. This will give us an algebraic BCOV anomaly equation and will eventually lead us to the
the theory of Calabi-Yau modular forms.

07-11 January 2019:   Sanya, China

Mirror Symmetry and Related stuff, Sanya, China on January 7-11, 2019.
Title: CY modular forms

25-27 January 2019:   Univeresity Michigan,  Ann Arbor

Modular structures in Gromov-Witten theory and related topics,
Title: CY modular forms (on blackboard and this is different from China talk)

08-12 April 2018:   Beijing International Center for Mathematical Research,   Beijing

Conference: Higher Genus Gromov-Witten invariants of Calabi-Yau threefold
Title: Polynomial structure of generating functions of higher genus GW invariants Abstract: Yamaguchi-Yau (2004) and Alim-Lange (2007) have computed the polynomial structure of the generating functions of higher genus GW invariants using the B-model of mirror symmetry. In this talk I will present a purely algebraic version of such computations for Calabi-Yau varieties of arbitrary dimension. It involves a construction of the moduli space of enhanced Calabi-Yau varieties and modular vector fields on it. This will give us an algebraic BCOV anomaly equation and will eventually lead us to the the theory of Calabi-Yau modular forms. The talk is partially based on my book "Gauss-Manin Connection in Disguise: Calabi-Yau Modular Forms".


Beida Talk:
Title: Hunting new Hodge cycles for cubic hypersurfaces
In this talk I will describe a computer assisted project in order to find new Hodge cycles for hypersurfaces. The talk is based on my book "A Course in Hodge Theory: with Emphasis on Multiple Integrals" and the article arXiv:1902.00831.

05 July 2019:   UFF, Niteroi

01-10 October CMSA Harvard, 10-12 October University of Missouri–St. Louis   

25 October UERJ   

06-21 December 2019, 1-5 January 2020, TIFR Mumbay, IMSc Chennai, India.

Title: Ramanujan's relations between Eisenstein series
Abstract: In 1916 S. Ramanujan discovered three identities involving the Eisenstein series $E_2,E_4,E_6$ and their derivatives. This can be seen as a vector field in the moduli space of an elliptic curve $E$ enhanced with a certain frame of the de Rham cohomology of $E$. For this one needs algebraic de Rham cohomology, cup product and Hodge filtration developed by Grothendieck and Deligne among many others. Viewed in this way, Ramanujan's differential equation can be generalized to an arbitrary projective variety. If time permits I will explain two generalizations of this picture in the case of Abelian varieties and Calabi-Yau threefolds.

06 January 2020 -07 Februry 2020, Isaac Newton Institute (INI), Cambridge UK.

INI 23 January 2020
Title: Variational Hodge conjecture and Hodge loci
Abstract: Grothendieck’s variational Hodge conjecture (VHC) claims that if we have a continuous family of Hodge cycles (flat section of the Gauss-Manin connection) and the Hodge conjecture is true at least for one Hodge cycle of the family then it must be true for all such Hodge cycles. A stronger version of this (Alternative Hodge conjecture, AHC), asserts that the deformation of an algebraic cycle Z togther with the projective variety X, where it lives, is the same as the deformation of the cohomology class of Z in X. There are many simple counterexamples to AHC, however, in explict situations, like algebraic cycles inside hypersurfaces, it becomes a challenging problem. In this talk I will review few cases in which AHC is true (including Bloch's semi-regular and local complete intersection algebraic cycles) and other cases in which it is not true. The talk is mainly based on the article arXiv:1902.00831.





27 January 2020, Plymouth, UK.
Title: What happens when a period vanishes?
A period is a number obtained by integration of an algebraic differential form over a topological cycle. In this talk I will review few phenomena which follow from the vanishing of periods. This includes the arise of limit cycles in planar differential equations, contraction of curves and the millennium Hodge conjecture.





29 January 2020, Loughborough, UK.
Title: Differential equations of modular forms,
Abstract: Examples of differential equations of modular forms go back to Darboux, Halphen, Chazy and Ramanujan among many others. It turns out that one can describe such differential equations without knowing about modular forms. This new point of view starts with a moduli space of projective varieties enhanced with elements in their algebraic de Rham cohomology and with some compatibility with the Hodge filtration and the cup product, and the computation of Gauss-Manin connection on such moduli spaces. In the first lecture I will explain this picture for Ramanujan and Darboux-Halphan differential equations. In the second lecture I will describe differential equation of Siegel modular forms using this geometric machinery (joint work with J. Cao and S.-T. Yau, arXiv:1910.07624).





31 January 2020, Nottingham, UK.
Title: Modular forms for triangle groups
Abstract: In this talk we first describe a solution of the Halphen equation which has modular properties with respect to a group which is not discrete in general. We show that in the particular case of triangular groups, the Halphen equation gives us a basis of the algebra of modular and quasi-modular forms. We then consider $p$- and $N$-integrality of Fourier coefficients of such modular forms using their relation with the Gauss hypergeometric function. As a corrolary we get Takeuchi's classification of arithmetic triangle groups with a cusp. For our purpose we state a refinement of a theorem of Dwork which largely simplifies many existing proofs in the literature. This talk is based on the joint work with Kh. Shokri (JNT 2014) and Ch. Doran, T. Gannon, Kh. Shokri (CNTP 2013).

Thanks to Covid19 I stopped traveling for a while and started the online seminar GADEPs.

10 March 2021. Algebraic Geometry Seminar at WUSTL

Title: Hodge cycles for cubic hypersurfaces

Abstract: Despite the abundant examples of Hodge cycles in the literature, finding them for smooth hypersurfaces of even dimension n is extremely difficult (of course if you do not pick up an algebraic cycle). In this talk I will consider the Hodge/algebraic cycle which is the sum of two projective space of dimension n/2 (lines for n=2 and planes for n=4) and describe a computer assisted project in order detect instances in which the deformation space of such a Hodge cycle inside a hypersurface is larger than the deformation space of the expected algebraic cycle. The talk is based on Chapter 19 of my book "A Course in Hodge Theory: with Emphasis on Multiple Integrals" which is also available in arXiv:1902.00831.

10 March 2021. Algebraic Geometry at Bogota

Title: Modular and automorphic forms & beyond;

Abstract: I will talk on a project which aims to develop a unified theory of modular and automorphic forms. It encompasses most of the available theory of modular forms in the literature, such as those for congruence groups, Siegel and Hilbert modular forms, many types of automorphic forms on Hermitian symmetric domains, Calabi-Yau modular forms, with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. Its main ingredient is the so- called ‘Gauss-Manin connection in disguise’. The talk is bases on the author's book with the same title. It is available in his webpage.

19 March 2021, Seminário de folhações holomorfas, IMPA

Title: Algebraic curves and foliations

Abstract: Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane ${\mathbb A}^2_k$ having $f=0$ as a fixed invariant curve is generated as $k[x,y]$-module by at most four elements, three of them are the trivial foliations $fdx,fdy$ and $df$. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen-Suslin theorem, we show that for a suitable field extension $K$ of $k$ such a module over $K[x,y]$ is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples $K=k$. This is a joint work with C. Camacho and with an appendix by C. Hertling, https://arxiv.org/abs/2101.08627

06 May 2021, GeMAT Seminar - Geometric Methods in Algebra and Topology

Title: Hodge cycles for cubic hypersurfaces

Abstract: Despite the abundant examples of Hodge cycles in the literature, finding them for smooth hypersurfaces of even dimension n is extremely difficult (of course if you do not pick up an algebraic cycle). In this talk I will consider the Hodge/algebraic cycle which is the sum of two projective space of dimension n/2 (lines for n=2 and planes for n=4) and describe a computer assisted project in order detect instances in which the deformation space of such a Hodge cycle inside a hypersurface is larger than the deformation space of the expected algebraic cycle. The talk is based on Chapter 19 of my book "A Course in Hodge Theory: with Emphasis on Multiple Integrals" which is also available in arXiv:1902.00831.

26 August 2021, Colóquio de Matemática, Recife

. Abstract: Examples of differential equations of modular forms go back to Darboux, Halphen, Chazy and Ramanujan among many others. It turns out that one can describe such differential equations as vector fields on certain moduli spaces. This new point of view starts with a moduli space of projective varieties enhanced with elements in their algebraic de Rham cohomology and with some compatibility with the Hodge filtration and the cup product, and the computation of Gauss-Manin connection on such moduli spaces. I will explain this picture for Ramanujan and Darboux-Halphan differential equations. The talk is based on my book "Modular and Automorphic Forms & Beyond, Monographs in Number Theory, World Scientific (2021)" in which the Tupi name ibiporanga (pretty land) for such a moduli space and atauúba (fire arrow) for such vector fields is suggested. The talk will be in Portuguese.

08 September 2021, BRAG: Brazilian Algebraic Geometry Seminar

Title: Ibiporanga: A moduli space for differential equations of automorphic forms

Abstract: In this talk I will consider a moduli space of projective varieties enhanced with a certain frame of its cohomology bundle. In many examples such as elliptic curves, abelian varieties and Calabi-Yau varieties, and conjecturally in general, this moduli space is a quasi-affine variety. There are certain vector fields on this moduli which are algebraic incarnation of differential equations of automorphic forms. Using these vector fields one can construct foliations with algebraic leaves related to Hodge loci. The talk is based on my book "Modular and Automorphic Forms & Beyond, Monographs in Number Theory, World Scientific (2021)" in which the Tupi name ibiporanga (pretty land) for such a moduli space is suggested.

13-17 September 2021, Algebraic Geometry session of CLAM 2020

Title: On reconstructing subvarieties from their periods

Abstract: We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. This is a joint work with Emre Sertoz.

27-29 October 2021, Two lectures on differential equations and algebraic geometry
Lecture 1 and Lecture 2 Both Lectures without handwritten text

02 November 2021, Algebraic Geometry in String Theory, CMSA Harvard.

17 October-30 December 2021, Paul Sabatier university, Toulouse, France

20 January 2022, Lisbon

16 March 2022, Pure Maths Colloquium at the University of Sheffield

05 April 2022, IMPA

02 May-29 July 2022, Paul Sabatier university, Toulouse, France

07 June 2022, IPM Algebraic Geometry Biweekly Webinar, Tehran Iran

June-July 2022, CIMI, Toulouse

12-28 July 2022 Germany,

25 August 2022,

26-30 September 2022,

18 December 2023-12 June 2023. Chinese University of Hong Kong-BIMSA, Beijing.

30 May 2023,

01 November 2023-29 December 2023,

24 December 2023,

02 February 2024,

01 January 2055 (or even earlier, maybe tomorrow) till forever: A travel with no return to nowhere with no baggage and no ticket and no body!