Trimester III

From 1st May 2022 to 31st August 2022

Seminars in May 2022

Week 1

Talk-1: ''Orbifold Curves: Geometry and Arithmetic". (Video is here and Slides are here)

Speaker: Frederic Campana, University of Lorraine France.

Time: 3rd May 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: A. J. Parameswaran, TIFR Mumbai.

Abstract: Let $f:X\to B$ a 'fibration' between two complex projective connected manifolds. A fundamental problem of algebraic geometry consists in deriving the `qualitative geometry' of $X$ (as expressed by suitable invariants such as the fundamental group, the cohomology, or the Kodaira dimension) from the ones of the base $B$ and of $X_b$, the generic smooth fibre. Our objective here is to stress the r\^ole played by the multiple fibres of $f$, and especially the fact that the `classical' notion of multiplicity is not the appropriate one, in general. This geometric observation leads, among many other things, to a conjectural `orbifold' version of the Mordell/Lang `conjecture'. We restrict here to the technically much simpler case when $B$ is a curve, the ideas in the general case being similar.

Talk-2: "Modularity of Calabi-Yau Varieties." (Video is here and Slides are here)

Speaker: Noriko Yui, Queen's University in Kingston Ontario.

Time: 5th May 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Pilar Bayer, University of Barcelona.

Abstract: Let X be a Calabi-Yau variety of dimension d. We will confine ourselves to Calabi-Yau varieties of small dimensions, e.g., d ≤ 3. Dimension one Calabi–Yaus are elliptic curves, those of dimension two are K3 surfaces, and dimension three ones are Calabi-Yau threefolds. Geometry and physics are both very much in evidence in Calabi-Yau varieties over the field of complex numbers. Today I will focus on Calabi-Yau varieties defined over the field Q of rational numbers (or number fields), and will discuss the modularity/automorphy of Calabi-Yau varieties in the framework of the Langlands Philosophy. In the last thirty years or so, we have witnessed tremendous advances on the modularity questions for Calabi-Yau varieties. Most of these results rest on the modularity of the two-dimensional Galois representations associated to them. In the first part of the lecture, I will present these fascinating modularity results for Calabi–Yau varieties over Q of dimension ≤ 3, as well as conjectures. Then I will discuss the current research in progess on the rank 4 weight 3 Calabi–Yau motives over Q and their Siegel modularity.

Week 2

Talk-1: Statement(s) of the Calabi conjecture(s) and examples-I.

Speaker: Vamsi Pritham Pingali, IISc Bangalore India. (Video is here and Slides are here)

Time: 10th May 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Diganta Borah, IISER Pune.

Abstract: After quickly reviewing the definition of complex manifolds and standard examples, I shall introduce Kahler metrics (and examples), holomorphic line bundles, the Chern connection, and curvature(s). I shall then state Calabi's conjectures. I shall try to present an example where Aubin-Yau's theorems are applicable. If time permits, I shall also reduce the conjectures to complex Monge-Ampere equations and mumble a few words about how can hope to solve such equations.

Talk-2: Hodge theory of Compact Kähler Manifolds. (Video is here and Slides are here)

Speaker: Rakesh Pawar, IISER Pune.

Time: 12th May 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Devendra Tiwari, BP Pune.

Abstract: I will discuss the Hodge decomposition theorem for compact Kähler manifolds. I will give an overview of the proof of the theorem and discuss some properties of the Hodge numbers as a consequence of the Hodge decomposition theorem.

Week 3

Talk-1: Statement(s) of the Calabi conjecture(s) and examples-II. (Video is here and Slides are here)

Speaker - Vamsi Pritham Pingali, IISc Bangalore India.

Time: 17th May 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Diganta Borah, IISER Pune.

Abstract: Same as the first talk by the speaker on 10th May.

Break- There won't be any talks on 19th, 24th, 26th May and 2nd June.

Week 5

Talk: Counting Points on Calabi-Yau Varieties over Finite Fields. (Video is here and Slides are here)

Speaker - Ursula Whitcher, Mathematical Reviews, American Mathematical Society.

Time: 31st May 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Manami Roy, Fordham University New York.

Abstract: Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. We describe methods to extract point-counting information and make arithmetic predictions using classical and modern mirror constructions.

Seminars in June 2022

Week 6

Talk-1: Hypergeometric Motives - I. (Video is here and Slides are here)

Speaker - Fernando Rodriguez Villegas, ICTP Trieste, Italy.

Time: 7th June 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: One can associate to the classical generalized hypergeometric differential equation with rational parameters a family of motives (the hypergeometric motives of the title). These are highly accessible for both analysis and computations and provide a rich and very diverse environment in which to study general features of motives. In these talks we will describe the construction of hypergeometric motives and go over some of its most prominent characteristics. If time permits we will also discuss the extension to hypergeometric families in more than one variable.

Talk-2: Hypergeometric Motives - II. (Video is here and Slides are here)

Speaker - Fernando Rodriguez Villegas, ICTP Trieste, Italy.

Time: 9th June 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Sudarshan Gurjar, IIT Bombay.

Abstract: Same as the first talk by the speaker on 7th June.

Week 7

Talk-1: Hypergeometric Functions in One Variable. (Video is here and Slides are here)

Speaker - Frits Beukers, Utrecht University.

Time: 14th June 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: T. N. Venkataramana, TIFR Mumbai.

Abstract: We give an introduction to the theory of both Gauss hypergeometric functions and hypergeometric functions of higher order. In this introduction we shall concentrate on the determination of the monodromy group of the hypergeometric equations, and pay particular attention to the cases where the monodromy group is finite. Of course the theory of second order (Gauss) hypergeometric functions is closely linked to triangle groups. We shall only indicate this and not repeat the theory that has been discussed in Trimester I.

Talk-2: Hypergeometric Functions in Several Variables. (Video is here and Slides are here)

Speaker - Frits Beukers, Utrecht University.

Time: 16th June 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: John Parker, Durham University.

Abstract: Hypergeometric functions in several variables, such as Appell, Lauricella, Horn, Kampé-Fériet functions, form a veritable zoo of functions. By the end of the 1980's Gel'fand, Kapranov and Zelevinsky propoposed a scheme with subsumes the above examples and more. It turns out that many of the properties of the hypergeometric systems of equations are governed by the combinatorics of a certain polytope, which is called 'A'. The associated functions are called A-hypergeometric functions. In this lecture we give a gentle introduction to the basics of these functions.

Week 8

Talk-1: An Overview of Dualities in String Theory. (Video is here and Slides are here)

Speaker - Shiroman Prakash, Dayalbagh Educational Institute, Agra.

Time: 21st June 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Utkarsh Mishra, UESTC Chengdu.

Abstract: Calabi Yau manifolds play an important role in string theory and M-theory. In this talk, I will attempt to provide an introductory overview of M-theory as the UV completion of 11-dimensional supergravity, and the five 10-dimensional string theories (mainly focusing on string theories known as type IIA and type IIB). These string theories are often studied in geometries in which one or more of the dimensions are taken to be small and compact. A remarkable feature that has numerous implications for mathematics is a phenomenon known as duality -- different superstring theories compactified on different spaces can give rise to the same low-energy theory. We will give an overview of superstring dualities and describe the simplest example of duality, which is T-duality, in detail. We will hopefully conclude with a very brief overview of how dualities in string theory lead to mirror symmetry in Calabi Yau manifolds. The talk will be aimed at non-specialists.

Talk-2: Supercongruences for Rigid Hypergeometric Calabi–Yau Threefolds.

Speaker - Fang Ting Tu, Louisiana State University USA. (Video is here and Slides are here)

Time: 23rd June 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Supriya Pisolkar, IISER Pune.

Abstract: This is a a joint work with Ling Long, Noriko Yui, and Wadim Zudilin. In this work, we give a full solution of Rodriguez-Villegas' rigid Calabi-Yau threefolds supercongruence conjecture. Two different approaches are implemented, and they both successfully apply to all the fourteen cases. Essential ingredients in executing the both approaches are the modularity of the underlying Calabi-Yau threefolds and a $p$-adic perturbation method applied to hypergeometric functions. In this talk, I will briefly introduce the finite hypergeometric functions, the existence of these supercongruences conjectured by Rodriguez-Villegas, and give an example to illustrate how we proved these supercongruences.

Week 9

Talk-1: `The First K in K3'. (Video is here and Slides are here)

Speaker - Ramesh Sreekantan, ISI Bangalore.

Time: 28th June 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Rakesh Pawar, IISER Pune.

Abstract: K3 surfaces are examples of 2 dimensional Calabi-Yau manifolds. Andre Weil named them after Kummer, Kahler, Kodaira and the mountain K2. In this talk we will discuss the construction of the Kummer Surface of a principally polarised Abelian Surface and some classical geometry related to this.

Talk-2: A Whipple Formula Revisited. (Video is here and Slides are here)

Speaker - Fang Ting Tu, Louisiana State University USA.

Time: 30th June 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Rupam Barman, IIT Guwahati.

Abstract: In this talk, I will introduce hypergeometric functions over finite fields and the joint work with Wen-Ching Winnie Li and Ling Long regarding a finite field Whipple evaluation. In this work, we consider the hypergeometric data corresponding to a formula due to Whipple which relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. When the hypergeometric data are primitive and defined over $\mathbb Q$, we explain a special structure of the corresponding Galois representations behind Whipple's formula leading to a decomposition that can be described by the Fourier coefficients of Hecke eigenforms. In this talk, I will use an example to demonstrate our approach and relate the hypergeometric values to certain periods of modular forms.

Week 10

Talk-1: Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds.

Speaker - Yifan Yang, National Taiwan University. (Video is here and Slides are here)

Time: 5th July 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Fang Ting Tu, Louisiana State University USA.

Abstract: In this talk we are concerned with the monodromy of Picard-Fuchs differential equations associated with one-parameter families of Calabi-Yau threefolds. We show that in the hypergeometric cases the matrix representations of monodromy relative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds. Furthermore, we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of Sp(4,Z) and whose levels are related to the geometric invariants of the Calabi-Yau threefolds. This is a joint work with Yao-Han Chen and Noriko Yui.

Talk-2: `K_1 and K3'. (Video is here and Slides are here)

Speaker - Ramesh Sreekantan, ISI Bangalore.

Time: 7th July 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Rakesh Pawar, IISER Pune.

Abstract: In this talk I will discuss the construction of elements of K_1 of a K3 surface and the relation of this with a conjecture of Gross, Kohnen and Zagier on algebraicity of values of Green's functions.

Week 11

Talk-1: Hecke Traces via Hypergeometric Character Sums. (Video is here and Slides are here)

Speaker - Ling Long, Louisiana State University USA.

Time: 12th July 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Siddhi Pathak, CMI Chennai.

Abstract: Hypergeometric character sums are useful for counting points on hypergeometric type Calabi-Yau varieties. In this process, modularity of rigid Calabi-Yau manifolds gives rise to ways expressing certain weight-4 Hecke traces in terms of explicit hypergeometric character sums.

In this talk we give formulas for traces of Hecke operators on spaces of modular forms in terms of hypergeometric character sums, extending earlier results of Ahlgren, Frechette-Ono-Papanikolas and Lennon. Especially we consider modular forms for subgroups related to the quaternion algebra $B$ over $\math Q$ with discriminant 6.

This is a joint work with Wen-Ching Winnie Li, Jerome Willian Hoffman and Fang-Ting Tu.

Talk-2: An Overview of Dualities in String Theory. (Video is here and Slides are here)

Speaker - Shiroman Prakash, Dayalbagh Educational Institute, Agra.

Time: 14th July 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Devendra Tiwari, BP Pune.

Abstract: Calabi Yau manifolds play an important role in string theory and M-theory. In this talk, I will attempt to provide an introductory overview of M-theory as the UV completion of 11-dimensional supergravity, and the five 10-dimensional string theories (mainly focusing on string theories known as type IIA and type IIB). These string theories are often studied in geometries in which one or more of the dimensions are taken to be small and compact. A remarkable feature that has numerous implications for mathematics is a phenomenon known as duality -- different superstring theories compactified on different spaces can give rise to the same low-energy theory. We will give an overview of superstring dualities and describe the simplest example of duality, which is T-duality, in detail. We will hopefully conclude with a very brief overview of how dualities in string theory lead to mirror symmetry in Calabi Yau manifolds. The talk will be aimed at non-specialists.

Week 12

Talk-1: Hypergeometric Groups and Their Arithmeticity. (Video is here and Slides are here)

Speaker - Sandip Singh, IIT Bombay.

Time: 19th July 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Supriya Pisolkar, IISER Pune.

Abstract: The hypergeometric groups, we will discuss in this talk, are the subgroups of $\mathrm{GL}_n(\mathbb{C})$ generated by the companion matrices of two monic self-reciprocal coprime polynomials (having integer coefficients) of degree $n$ so that their Zariski closures inside $\mathrm{GL}_n(\mathbb{C})$ are either symplectic or orthogonal groups. These groups arise as monodromy groups of hypergeometric differential equations, and in this talk we will discuss their arithmeticity and thinness. We will also discuss the dichotomy between arithmeticity and thinness of the fourteen hypergeometric groups associated to Calabi-Yau threefolds.

Talk-2: On Some Hypergeometric Supercongruence Conjectures of Long.

Speaker - Michael Allen, Oregon State University. (Video is here and Slides are here)

Time: 21st July 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Mihir Sheth, IISc Bangalore.

Abstract: In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin. In 2020, Long conjectured a number of further supercongruences for hypergeometric functions of a similar shape. In this talk, we extend the approach of Long, Tu, Yui, and Zudilin towards establishing six of Long's conjectures, and also discuss possibilities for further generalizations.

Week 13

Talk-1: Cohomology and Congruences. (Video is here and Slides are here)

Speaker - Masha Vlasenko, Institute of Mathematics of the Polish Academy of Sciences, Warsaw.

Time: 26th July 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Soumya Sankar, Ohio State USA.

Abstract: In his work on rationality of zeta functions of algebraic varieties Bernard Dwork discovered a number of remarkable p-adic congruences. In this lecture I will demonstrate these congruences and overview our recent work with Frits Beukers which explains their underlying mechanism. We will also discuss the phenomena of supercongruences and excellent Frobenius lifts.

Talk-2: Calabi-Yau Differential Operators. (Video is here and Slides are here)

Speaker - Masha Vlasenko, Warsaw.

Time: 28th July 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Ursula Whitcher, Mathematical Reviews, AMS.

Abstract: We will exploit the p-adic Cartier operation and other tools introduced in the first lecture in the context of families of Calabi-Yau hypersurfaces. Using them we can prove prove integrality of the mirror map and integrality of instanton numbers in some key examples of mirror symmetry. Many of our examples are hypergeometric. This is joint work with Frits Beukers.

Week 14

Talk-1: Calabi-Yau Manifolds, Modularity and Arithmetic Geometry. (Video is here and Slides are here)

Speaker - Albrecht Klemm, University of Bonn.

Time: 2nd August 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Fang Ting Tu, Louisiana State University USA.

Abstract: Using Mirror symmetry and Dworks p-adic deformation of the Gauss Manin connections we construct the Hasse Weil Zeta function of $\zeta(X/\mathbb{Q},s)$ for families of Calabi-Yau threefolds $X$ and explore the consequences of its modularity in special fibres at algebraic extension of $\mathbb{Q}$ on the physics of string compactifications on $X$ as well as on enumerative geometry on $X$.

Talk-2: Calabi-Yau threefolds coming from a hypergeometric family of K3 surfaces.

Speaker - Andrew Harder, Lehigh University. (Video is here and Slides are here)

Time: 4th August 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Devendra Tiwari BP, Pune.

Abstract: Classification of Calabi-Yau threefolds is a deep and interesting problem, both from the perspective of mathematics, and from the perspective of string theory. However very little is actually known about this problem. Even the question of whether the collection of all deformation families of Calabi-Yau threefolds is finite or infinite has not yet been settled. However, if we restrict our attention to specific subclasses of Calabi-Yau threefolds, the picture sometimes becomes much clearer. In this talk, I will discuss one such situation; the class of Calabi-Yau threefolds which are fibred over the projective line by "mirror quartic" K3 surfaces. Along the way, we make explicit use of a particular hypergeometric Picard-Fuchs equation. We find that there are only finitely many such families of Calabi-Yau threefolds, and we see that their classification is connected to the classification of Picard rank 1 Fano threefolds by mirror symmetry. This is based on joint work with C. Doran, A. Novoseltsev, and A. Thompson.

Week 15

Talk-1: Mirror symmetry and Calabi-Yau families over the thrice-punctured sphere. (Video is here and Slides are here)

Speaker - Charles Doran, University of Alberta, Canada.

Time: 9th August 2022, Tuesday at 19:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Noriko Yui, Queen's University in Kingston Ontario.

Abstract: Motivated by mirror symmetry for quintic Calabi-Yau threefolds, I will begin by describing my classification with John Morgan of “mirror compatible” integral variations of Hodge structure over the thrice-punctured sphere. Toric hypersurface and complete intersection realizations (and near-realizations!) suggest a regularity in these families — they are all fibered by high Picard rank K3 surfaces — and led Andrew Harder, Andrey Novoseltsev, Alan Thompson, and I to a complete classification and construction of such Calabi-Yau threefolds. More generally, codimension-one Calabi-Yau submanifolds induce fibrations, with the periods of the total space relating to those of the fibers and the structure of the fibration. I will describe a method of iteratively constructing Calabi-Yau manifolds in tandem with their Picard-Fuchs equations developed with Andreas Malmendier, with special attention to the case of families over the thrice-punctured sphere.

Talk-2: Mirroring towers: Fibration and degeneration in Calabi-Yau geometry. (Video is here and Slides are here)

Speaker -Charles Doran, University of Alberta, Canada.

Time: 11th August 2022, Thursday at 19:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Mainak Poddar, IISER Pune.

Abstract: What in general is the mirror symmetric meaning of a fibration on a Calabi-Yau manifold? With Andrew Harder and Alan Thompson, I propose a mirror interpretation involving degenerations of the mirror Calabi-Yau family. This both unifies the CY/CY and Fano/LG mirror constructions and answers an old question of Andrei Nicolaevich Tyurin. I will discuss our motivations and describe a variety of sources of evidence in favor of such an interpretation, involving topology, geometry, periods (with Jordan Kostiuk and Fenglong You), tropical geometry (with Lawrence Barrott), and even a mirror to the Clemens-Schmid sequence in Hodge theory that refines the usual mirror symmetry of Hodge diamonds.

Week 16

Talk-1: The arithmetic of families of Calabi-Yau manifolds: black holes and modularity- I

Speaker - Xenia de la Ossa, University of Oxford, Oxford. (Video is here and Slides are here)

Time: 16th August 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Ling Long, Louisiana State University USA.

Abstract: The main goal of this talk is to explore questions of common interest for physicists, number theorists and geometers, in the context of the arithmetic of Calabi Yau 3-folds. The main quantities of interest in the arithmetic context are the numbers of points of the manifold considered as a variety over a finite field. We are interested in the computation of these numbers and their dependence on the moduli of the variety. The surprise for a physicist is that the numbers of points over a finite field are also given by expressions that involve the periods of a manifold. The number of points are encoded in the local zeta function, about which much is known in virtue of the Weil conjectures. In these talks we discuss a number of interesting topics related to the zeta function, the corresponding L-function, and the appearance of modularity for one parameter families of Calabi-Yau manifolds. We will discuss on an example for which the quartic numerator of the zeta function of a one parameter family factorises into two quadrics at special values of the parameter which satisfy an algebraic equation with coefficients in Q (so independent of any particular prime), and for which the underlying manifold is smooth. We note that these factorisations are due to a splitting of the Hodge structure and that these special values of the parameter are rank two black hole attractor points in the sense of type IIB supergravity. Modular groups and modular forms arise in relation to these attractor points. To our knowledge, the rank two attractor points that were found by the application of these number theoretic techniques, provide the first explicit examples of such attractor points for Calabi-Yau manifolds. Time permitting, we will describe this scenario also for the mirror manifold in type IIA supergravity.

Talk-2: The arithmetic of families of Calabi-Yau manifolds: black holes and modularity- II.

Speaker - Philip Candelas, University of Oxford, Oxford. (Video is here and Slides are here)

Time: 18th August 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Minhyong Kim, International Centre for Mathematical Sciences, Edinburgh.

Abstract: Same as the previous talk on 16th August.

Week 17

Talk-1: QM Abelian Varieties, Hypergeometric Character Sums and Modular Forms. (Video is here and Slides are here)

Speaker - Jerome Hoffman, Louisiana State University USA.

Time: 23rd August 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Masha Vlasenko, Institute of Mathematics of the Polish Academy of Sciences, Warsaw.

Abstract: This is a report of work in progress with Winnie Li, Ling Long and Fang-Ting Tu. The theme is to relate hypergeometric character sums to traces of Hecke operators on modular forms in interesting cases. These arise from certain arithmetic triangle groups. Especially we consider the quaternion algebra B over Q with discriminant D = 6. In that case, quotients of the upper half plane by the units in these algebras give rise to Shimura curves, which are moduli spaces for 2-dimensional abelian varieties with quaternion multiplication (QM).

In the talk, I will explain the geometric background of this problem, in particular the Eichler-Shimura theory relating modular forms to parabolic cohomology, both in the complex-analytic and in the `l-adic' etale setting. The key result, due to Kuga-Shimura, computing the zeta functions of the fiber spaces of abelian varieties in terms of Hecke polynomials, allows one to relate these hypergeometric character sums to traces of Hecke operators on spaces of modular forms.

Talk-2: Hodge Cycles for Cubic Hypersurfaces. (Video is here and Slides are here)

Speaker - Hossein Movasati, IMPA Rio de Janeiro.

Time: 25th August 2022, Thursday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Ramesh Sreekantan, ISI Bangalore.

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 describe a computer assisted project in order detect instances in which the deformation space of an algebraic Hodge cycle inside a hypersurface is larger than the deformation space of the expected algebraic cycle. One easy example is a Veronese algebraic cycle inside a cubic six fold. A more difficult and conjectural example is an algebraic Hodge cycle which is the sum of two projective spaces of dimension n/2 (lines for n=2 and planes for n=4) inside a Fermat cubic n-fold. 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.

Week 18 (Final Week of the Year-Long Program).

Talk: Modular Forms and Calabi-Yau Varieties. (Video is here and Slides are here)

Speaker - Kapil Paranjape, IISER Mohali.

Time: 30th August 2022, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: A. J. Parmeswaran, TIFR Mumbai.

Abstract: This is a report on a work done with Dinakar Ramakrishnan which was to explore the possibility of linking modular forms with certain special Calabi-Yau varieties. This led us to formulate a problem that we could solve in some cases and solve in a weaker form in other cases. In this talk we will look at the progress made on these questions.

Here concludes the third trimester as well as the Year-long series of virtual seminars.