Number theory seminar:

Apart from regular seminars which are announced here, each semester we choose a book and we review its content.

  1. August-November 2010:   Fred Diamond, Jerry Shurman,  A First Course in Modular Forms (Graduate Texts in Mathematics)
  2. March-June: John Cremona, Algorithms for Modular Elliptic Curves. Some other useful links: Modular Forms: A Computational Approach by William Stein and Lectures on Serre's conjectures, by William Stein and  Ken Ribet 
  3. June-November,  2012, Let us explore the relation of modular forms and physics learning Feynman method from Mirror Symmetry,Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Ravi Vakil


Reading Seminar: Topological String Theory and modular forms
  1.  
    During the next year we are planning to read
    [A]: Chapters 9, 10, 11 of the book [1]
    [B]: Chapters 31, 32, 33, 34, 35 of the book [1] and the articles [2],[3],[4] and [5].
    The idea is to understand how and why (quasi-) modular forms and their generalizations appear in the context of
    mirror symmetry and topological string theory calculations. We also wish to understand the PHYSICAL proof of many
    enumerative statements in Algebraic Geometry, for instance "a generic quintic threefold contains 2875 lines". We will start
    with [A], however, each two weeks there will  be a talk regarding the MATHEMATICAL content of [B]. The idea is to follow [A]-part both from Mathematics and Physics point of view, and to follow [B] only from Mathematics point of view (this will mainly include mathematical computations in the B-model of Topological String Theory).  We aim to to see the contents of both [A]-talk and [B]-talks  converge to each other. After each eight talks we will plan the next eight talks. I will be mainly responsible for the first few [B]-talks. 
    The first 8 slots:  
  2. [A]-talks:
    From the book [1] Mirror Symmetry
    Chapter 9. Quantun Field Theory  in dimension zero
        Session 1: Machinery of Feynman diagrams pages 151-154.              (Khosro M. Shokri, 22/01/2013)
        Session 2: Fermions, Supersymmetry, localization pages 155-169       (Khosro M. Shokri, 25/01/2013)
        Session 3: Explicit evaluation of partition functions 160-162.       (Younes Nikdelan,  29/01/2013)
        Session 4: Zero dimensional Landau-Ginzburg Theory,pages 162-167.    (Younes Nikdelan,  01/02/2013
    Chapter 10. Quantum Field Theory in dimension one.
       Session 5: Quantum Mechanics. pages 169-174                                         (Jethro ven Ekeren 19/02/2013)
       Session 6: Harmonic Oscillator. page 174-177                                         (Jethro ven Ekeren 22/02/2013)    
    [B]-talks
    Session 3: What is a propagator for Calabi-Yau threefolds?,         [2],[5]      (Hossein Movasati, 05/02/2013)
    Session 4: What is Feynman rule for Calabi-Yau threefolds?,         [2],[5]      (Hossein Movasati  15/02/2013)
    The second 8 slots
    [A] talks
       Session 1: Sigma Model on a circle and Real line, I page 177-182.  12/03/2013 (F. Noseda)
       Session 2: Sigma Model on a circle and Real line, II page 177-182. 15/03/2013 (F. Noseda)
       Session 3: The structure of supersymmetric Quantum Mechanics, I page 182-187  19/03/2013 (Y. Nikdelan)
       Session 4: The structure of supersymmetric Quantum Mechanics, II page 182-187     22/03/2013  (Y. Nikdelan)
       Session 5: Supercharges, I,   page 187-194.                                       26/03/2013         (Kh. Shokri)
       Session 6: Supercharges, II,   page 187-194.                                      29/03/2013          (Kh. Shokri)
    [B]-talks
    Session 1: Mirror symmetry and elliptic curves, I, [3]Dijkgraaf 1995                 02/04/2013  (H. Movasati)
    Session 2: Mirror symmetry and  elliptic curves, II, [3]Dijkgraaf 1995               05/04/2013  (H. Movasati)
       ...........................................
    References:
    [1], Mirror Symmetry,Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Ravi Vakil
    Complementary article:
    [2], M.~Bershadsky, S.~Cecotti, H.~Ooguri, and C.~Vafa. Holomorphic anomalies in topological field theories.
    Nuclear Phys. B, 405(2-3):279--304, 1993.
    Kodaira-{S}pencer theory of gravity and exact results for quantum
    string amplitudes.  Comm. Math. Phys., 165(2):311--427, 1994.
    [3] Robbert Dijkgraaf. Mirror symmetry and elliptic curve. Volume 129 of  Progr. Math., pages 149--163. Birkh\"auser Boston, Boston, MA, 1995.
    [4] Philip Candelas, Xenia~C. de~la Ossa, Paul~S. Green, and Linda Parkes.A pair of {C}alabi-{Y}au manifolds as an exactly soluble superconformal theory.
    Nuclear Phys. B, 359(1):21--74, 1991.
    [5] Satoshi Yamaguchi and Shing-Tung Yau. Topological string partition functions as polynomials.
    J. High Energy Phys., (7):047, 20, 2004.   

 The thirs 8 slots: 
Here is the plan for the third series of lectures on "Modular Forms and Physics". We are reading the Book
"Mirror Symmetry,Kentaro Hori,...Chapter 8,9,10" which deals with Quantum Field theory in dimensions d=0,1,2.
So far, we have learned some techniques with path integrals which in some interesting cases result in modular forms or
related functions.
From this point on we plan for talks once per week. We plan to meet 10:30 Fridays. If somebody is interested
to participate regularly, but this schedule is not convenient for him, let me know. The plan for the next 6 slots
are as below. Let me know if you want to take the responsibility of some.

19/04/2013:  The Review and discussion of what we have learned so far, 151-196,  Hossein Movasati
26/04/2013:  Perturbative Analysis 197-202,  Younes Nikdelan
03/05/2013:  Landau-Ginzburg Theory, 202-206, Kh. Monsef Shokri
10/05/2013:  Supersymmetric quantum mechanics on a Riemannian manifold. 206-212
17/05/2013:  Supersymmetric quantum mechanics on a Kahler manifold, 212-220.
24/05/2013:  Instantons, 220-...


Reading Seminar on Toric Calabi-Yau Varieties
Reading seminar on toric Calabi-Yau varieties
Reference:  Cox, David A. ; Katz, Sheldon,  Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs, 68. AMS, 1999 .
and the references therein.  We start from  Chapter 3: Toric Geometry Chapter 4: Mirror symmetry construction.
The seminar will take place each Friday from  11:00-12:30.

16/08/2013: Hossein Movasati, Cones and Fans, polytopes and homogeneous coordinates
23/08/2013: Khosro Monsef Shokri, Kahler cones and symplectic geometry.
30/08/2013:


Calabi-Yau Varieties Seminar, 23 August-10 December 2016
Automorphic forms and beyond
Tuesdays: 10:30-12:00

The main objective of this series of lectures is to construct a theory of (holomorphic) automorphic forms in an algebraic geometric context containing all the previously known automorphic form theories. This new theory is the main product of the project "Gauss-Manin connection in disguise". Some ingredients of this theory are algebraic de Rham cohomology, Hodge filtrations, Gauss-Manin  connection,  Monodromy  groups etc. Our correspondence between this new automorphic forms and algebraic varieties is purely of algebraic and complex nature, however, the  long term hope is to put it in a more arithmetic context, pushing it toward arithmetic modularities and Langlands correspondence. The first lectures will be dedicated to  general definitions, basic set up etc. Then we aim to recover many classical automorphic forms,  such as Siegel modular forms, Hilbert modular forms, modular forms for congruence  groups, automorphic forms for moduli of K3 surfaces etc.  The next step would be to develop a theory of auotmorphic forms for Calabi-Yau varieties. Depending on the level of participants some preliminaries in Hodge theory will be offered. Here are the main contributors  to the seminar.
Hossein Movasati: Main lecturer, the general theory
Roberto Villaflor: Background in Hodge theory
Younes Nikdelan: Examples of modular forms in the case of Calabi-Yau varieties
Yadollah Zare: Topology of algebraic varieties, Picard-Lefschetz theory.

06/09/2016: Hossein Movasati,
13/09/2016:
20/09/2016:
27/09/2016:

 



Calabi-Yau Varieties Seminar, 2018
Minicourse: Modular and automorphic forms & beyond
The main aim of this research/reading minicourse is to 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. 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 involve many reading activities on related topics. Some of the seminars, depending on the topic will be announced either in the seminar series "Folhações Holomorfas" or "Variedades de Calabi-Yau" or there will be no official announcement. It will be on Tuesdays 15:30-17:00 (and sometimes Thursdays), in case there is no other seminar of "Folhações Holomorfas".
First Lecture: Title: On a foliation of dimension 3 in 6 dimensions
This is the first talk of lecture series entitled "Modular and automorphic forms & beyond". I will try to explain the ingredients of the whole project by taking product $T$ of two copies of the moduli of enhanced elliptic curves. Almost all concepts of the minicourse will appear in this baby example. The moduli space is of dimension 6 and I will introduce counterparts $T_d$, $d$ a natural number, of modular curves $Y_0(d)$, in $T$. These are 3 dimensional affine subvarieties of $T$ and they are the only algebraic leaves of a foliation constructed from the Ramanujan vector field.