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
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
University of Alberta (Collaboration with Charles Doran)
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
Back to Rio
Mainz University (Collaboration with Duco van Straten and Stefan Reiter).
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.
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.
Max Planck Institute for Mathematics, (Collaboration with Prof. Don Zagier)
Edmonton-Canada
Hodge theory and string duality, Banff, Canada
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.
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.
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.
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.
I am organizing the following small workshop:
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.
Feuilletages
et équations différentielles complexes
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.
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.
Sabbatical year at Harvard
Talk at Brandeis University:
Title: A common
framework for automorphic forms and topological partition functions.
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.
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.
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.
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
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.
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
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.
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
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.
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
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
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.
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.
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.
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
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.
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,
Title: Modular and automorphic forms & beyond
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.
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,
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.
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.
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.
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.
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.
Mirror Symmetry and Related stuff, Sanya, China on January 7-11,
2019.
Title: CY modular
forms
Modular structures in Gromov-Witten theory and related
topics,
Title: CY
modular forms (on blackboard and this is different from China
talk)
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.
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.
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).
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.
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.
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
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.
. 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.
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.
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.