Trimester I

From 1st September 2021 to 31st December 2021

Seminars in September 2021

Week 1

The talks on September 1 and 8 would be jointly organized by Bhaskaracharya Pratishthan and the Mathematics Department, IISER Pune.

1st September 2021, Wednesday 17:00 Hrs IST (GMT + 5:30)

Title- "Topological Underpinnings of the Modular Group". (video is here)

Speaker - Ravi S. Kulkarni, BP, Pune and CUNY, New York.

Chair for the Talk: Chandrasheel Bhagwat, IISER Pune.

Abstract: Abstract is here.

Week 2

7th September 2021

Talk-1: “Graphs as Riemann surfaces: Hurwitz type bound and Hyperelliptic Graphs". (Slides are here and video is here)

Speaker - Alexander Mednykh, Sobolev Institute of Mathematics, Novosibirsk.

Time: Tuesday 16:00 Hrs IST (GMT + 5:30) .

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: Abstract of this talk can be accessed here.

Talk-2: “Elliptic Functions and the Modular Group.“ (Slides are here and video is here)

Speaker - John Parker, Durham University.

Time: Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: In this talk I will introduce elliptic functions and the differential equation they solve, mainly using the Weirstrass p-function. I will also discuss the associated modular forms g_2 and g_3 as well as the modular function J and their relation with the modular group. I will use viewpoint of complex analysis, but I will also try to explain the geometric meaning behind this material. (Slides of this talk

8th September 2021

Title- "Monodromy and Triangle Groups". (Slides are here and video is here )

Speaker : T. N. Venkataramana , TIFR Mumbai.

Time: Wednesday 17:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Supriya Pisolkar, IISER Pune.

Abstract: We recall briefly the concept of monodromy associated to differential equations on open sets in the plane.

We then compute the monodromy of the Gauss hypergeometric function in terms of its parameters.

Week 3

14th September 2021

Talk-1: “Graphs as Riemann surfaces: Cyclic group action". (Slides are here and Video is here)

Speaker - Alexander Mednykh - Sobolev Institute of Mathematics, Novosibirsk.

Time: Tuesday 16:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: Abstract of this talk can be accessed here.

Talk-2: “The Hyperbolic Plane, Isometries and Triangle Groups". (Slides are here and Video is here)

Speaker - John Parker, Durham University.

Time: Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: In this talk I will give a basic introduction to several models of the hyperbolic plane and the corresponding matrix representations of its isometries. My main goal will be to show how triangle groups can be represented in the different models and how to pass between them.

Week 4

21st September 2021

Talk-1: "Enumeration of Maps and Coverings". (Video is here and Slides are here)

Speaker - Alexander Mednykh - Sobolev Institute of Mathematics, Novosibirsk.

Time: Tuesday 16:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Supriya Pisolkar, IISER Pune.

Abstract: Abstract of this talk can be accessed here.

Talk-2: "Divisors and Line bundles on Compact Riemann Surfaces". (Video is here, and Slides are here)

Speaker - Sudarshan Gurjar, IIT Bombay.

Time: Tuesday 18:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Supriya Pisolkar, IISER Pune.

Abstract: In this talk I will introduce divisors and line bundles on compact Riemann surfaces and discuss the correspondence between them. This will prepare ground for a discussion of the Riemann Roch theorem in the subsequent lecture.

Week 5

28th September 2021

Talk-1: "Riemann-Roch Theorem for Compact Riemann Surfaces". (Video is here and Slides are here)

Speaker - Sudarshan Gurjar, IIT Bombay.

Time: Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Supriya Pisolkar, IISER Pune.

Abstract: In this talk, I will discuss the Riemann Roch theorem for compact Riemann Surfaces, along with some applications.

Talk-2: "Hyperbolic Tessellation". (Video is here and Slides are here)

Speaker - Neelima Borade, Princeton University.

Time: Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Supriya Pisolkar, IISER Pune.

Abstract: In this expository talk I will briefly discuss the history of hyperbolic geometry and introduce the concept of tessellations.

I will define triangle groups focusing on the case of hyperbolic triangle groups and their subgroups called Fuchsian groups.

I’ll end with some results connecting Fuchsian groups and hyperbolic tessellations.

Seminars in October 2021

Week 6

5th October 2021

Talk-1: "Modular Curves: Analytic and Algebraic aspects". (Video is here and Slides are here)

Speaker - Mihir Sheth, IISc Bangalore.

Time: Tuesday 16:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: I will begin by explaining how to compactify quotients of the upper half plane by congruence subgroups of the modular group. The resulting spaces are compact Riemann surfaces known as modular curves. I will then focus on the modular curve X_0(N) of level N and discuss its canonical model over rational numbers.

Talk-2: "Regular Tessellations". (Video is here and Slides are here)

Speaker - Gianluca Faraco, MPI, Bonn.

Time: Tuesday17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: I begin by recalling the notion of regular tessellations already introduced the last week. Then I’ll focus on planar and spherical tessellations making comparisons with their hyperbolic counterpart. We finally discuss Galois covering of \CP^1.

Week 7

12th October 2021

Talk-1: "Arithmetic Fuchsian Groups Through Quaternion Algebras". (Video is here and slides are here)

Speaker - Bram Petri, University of Paris 6 Sorbonne.

Time: Tuesday 16:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Shripad Garge, IIT Bombay.

Abstract: In this talk I will describe how to construct arithmetic Fuchsian groups out of quaternion algebras and

how to derive geometric properties of the corresponding surfaces from the underlying algebra.

Talk-2: "The Klein Quartic". (Video is here and slides are here)

Speaker - Bram Petri, University of Paris 6 Sorbonne.

Time: Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Shripad Garge, IIT Bombay.

Abstract: This talk will be about the Klein quartic, an example of many of the phenomena studied in this program. It's a Belyi surface, it corresponds to a finite index subgroup of the (2,3,7)-triangle group (and hence is arithmetic) and it can be explicitly written down as an algebraic curve. In the first of these lectures, I will discuss the various descriptions of this surface and link them up.

Week 8

19th October 2021

Talk-1: "Arithmeticity and Takeuchi's Theorem". (Video is here and Slides are here)

Speaker - John Parker - Durham University.

Time: Tuesday 16:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Anuradha Garge, Mumbai University.

Abstract: I will discuss the definition of an arithmetic group using quadratic forms (following Vinberg) and I will make this explicit in the case of triangle groups. As a consequence I will discuss Takeuchi's theorem. I will also try to explain the connection between the quadratic form method and the quaternion algebra method, as used by Takeuchi.

Talk-2: "Hypergeometric Functions as Modular forms on Shimura curves". (Video is here and Slides are here)

Speaker - Yifan Yang, National Taiwan University.

Time: Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Dipendra Prasad, IIT Bombay.

Abstract: Shimura curves are generalizations of classical modular curves. Because of the lack of cusps on Shimura curves, there have been very few explicit methods for them. In this talk, we will explain how we might use hypergeometric functions to express modular forms on a Shimura curve when the Shimura curve has genus 0 and three elliptic points.

Week 9

26th October 2021

Talk-1: Triangular Modular Curves-I. (Video is here)

Speaker - Pete Clark, University of Georgia.

Time: Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Dipendra Prasad, IIT Bombay.

Abstract: In this talk we will discuss Belyi-Wolfart curves, namely compact Riemann surfaces X of genus at least 2 such that X -> X/Aut(X) is a Belyi map. For each g \geq 2 this is yields a finite, nonempty set of isomorphism classes algebraic curves, and of course each is defined over Q-bar, so it is interesting to ask for the fields of moduli of such curves.

These curves include as special cases several classical families: the Fermat curves, the Hurwitz curves and the classical modular curves X(N) (for N \geq 7). The latter family motivates us to introduce a family of Belyi-Wolfart curves with Galois group PSL_2(F_q) or PGL_2(F_q) studied by the speaker and John Voight.

Talk-2: Triangular Modular Curves-II. (Video is here)

Speaker - Pete Clark, University of Georgia.

Time: Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Dipendra Prasad, IIT Bombay.

Abstract: In this talk we define and study the algebraic curves mentioned above. As Riemann surfaces they are uniformized by principal congruence subgroups of hyperbolic triangle groups. As such they provide an interesting challenge to the arithmetic/non-arithmetic dichotomy for Riemann surfaces: each of these Riemann surfaces is naturally associated to a quaternion algebra over a totally real number field, but nevertheless all but finitely many of them are non-arithmetic. After recalling what this means, we attempt to show that this family of curves is still arithmetically interesting: in particular, they all have field of moduli contained in a quadratic extension of an abelian number field.

Seminars in November 2021

Week 10: Diwali Vacation

Week 11

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9th November 2021

Talk-1: "Algebraic Curves and Belyi's theorem". (Video is here, Slides are here and some notes are here)

Speaker: Anand Deopurkar, Australian National University.

Time: Tuesday 16:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Nitin Nitsure, (retd.) TIFR Mumbai..

Abstract: Belyi's theorem states that a smooth projective curve $X$ over the complex numbers is defined over $\overline{\mathbb Q}$ if and only if it can be expressed as a finite cover of the projective line ramified over at most three points. As applications, we get that every such curve can be expressed as a compactification of the upper half plane by a finite index subgroup of the modular group. I will prove Belyi's theorem and discuss some of these applications.

Talks-2: "Belyi's Theorem and Grothendieck dessin d'enfant". (Video is here and Slides are here)

Speaker - Claire Burrin, ETH Zurich.

Time: Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Amita Malik, MPI Bonn.

Abstract: We will discuss the Grothendieck correspondence between dessins d'enfants and ramified covers of the sphere, its relation to Belyi’s theorem, and describe some features and invariants of the associated Galois action.

Week 12

16th November 2021

Talk -1: "Fundamental domains and Poincar\'e's polygon theorem". (Video is here)

Speaker: John Parker, Durham University.

Time: Tuesday 16:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Anima Nagar, IIT Delhi.

Abstract: A fundamental set S for a properly discontinuous group action is a set that containes exactly one point in each orbit of the group. A fundamental polyhedron D (polygon in dimension 2) is a closed polyhedron containing a fundamenal set S with the property that S contains the interior of D. The Poincare polygon theorem is a powerful tool for constructing fundamental polygons and is one of the principal techniques for showng that a group acts properly discontinuously. It takes some time to state the theorem properly, and do so involves some subtle points. In this talk I will explain this statement and show how to use the Poincare polygon theorem in practice.

Talk-2: "Belyi Map Computation". (Video is here)

Speaker - Hartmut Monien, Bethe Center for Theoretical Physics, Bonn University.

Time: Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Ajith Kumar, ICT Mumbai.

Abstract: In this talk I plan to give a short introductory survey of analytic methods, and will discuss about calculation of Belyi maps, modular forms and functions of non-congruence subgroups.

23rd November 2021

Talk-1: "Calculation of Three-point Branched Cover of Projective Line". (Video is here and Slides are here)

Speaker - Hartmut Monien, Bethe Center for Theoretical Physics, Bonn University.

Time: Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Akhilesh P., Kerala School of Mathematics, Kerala.

Abstract: In this lecture we discuss calculation of three point branched covers and present results on sporadic groups of genus zero.

Talk-2: Computing a database of Belyi maps. (Video is here and Slides are here)

Speaker - Samuel Schiavone, MIT, Massachusetts.

Time: Tuesday 19:00 Hrs IST (GMT +5:30).

Chair for the Talk: Akhilesh P., Kerala School of Mathematics, Kerala.

Abstract: We present a general numerical method for computing Belyi maps that employs their relationship with triangle groups, permutation triples, and dessins d'enfants. Using this method, we have computed an exhaustive database of Belyi maps in low degree, which is accessible both as a Magma package, and on the web as a part of the L-functions and modular forms database (LMFDB). We showcase some features of the database both in Magma and on the LMFDB website.

Week- 14

Talk-1: Some Problems and Questions. (Video is here and Slides are here)

Speaker - John Voight, Dartmouth College, Hanover.

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

Chair for the Talk: A. Raghuram, Fordham University, New York.

Abstract: We will present some questions and open problems related to the subjects of the trimester, with a focus on computational aspects.

Talk-2: Farey Symbols and Finite Index Subgroups of Modular Groups. (Video is here and Slides are here).

Speaker: Chaitanya Ambi, CMI, Chennai.

Time: 4th December, Saturday at 19:00 Hrs (IST).

Abstract: Farey symbols, originally introduced as a computational device, satisfy very interesting arithmetic properties.

We shall discuss the correspondence between Farey symbols and subgroups of finite index of the modular group. We shall also touch upon some arithmetic aspects of generalized Farey symbols.

Seminars in December 2021

Week - 15

Talk-1: Grothendieck's path from Topology to Arithmetic: Dessins d'enfants and the game of Lego-Teichmüller.

Speaker - Leila Schneps, CNRS France. (Video is here and Slides are here)

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

Chair for the Talk: Nitin Nitsure, TIFR Mumbai (retd.).

Abstract: I will discuss Grothendieck-Teichmüller theory and introduce Grothendieck's original text, the Esquisse d'un Programme.

I will show how he envisaged to deepen the theory of dessins d'enfants into Grothendieck-Teichmüller theory.

Grothendieck was interested in surgery on topological surfaces and how this action translates into maps between moduli spaces of Riemann surfaces. These topological operations become not only geometric but even arithmetic, and give information about the absolute Galois group. This interplay between pure topology and arithmetic, fascinated Grothendieck. It would be a partly historical talk about the origins of these ideas.

Talk-2: "Enumeration of Low Genus Triangular Modular Curves". (Video is here and Slides are here).

Speaker - Juanita Duque Rosero, Darthmouth College, Hanover.

Time: 7th December, Tuesday 20:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves are the domain of Belyi maps with monodromy $\text{PGL}_2(\mathds{F}_q)$ or $\text{PSL}_2(\mathds{F}_q)$. We will present an enumeration of all triangular modular curves of genus 0, 1, and 2. We will focus on the computational aspects of this project.

Week - 16

Talk-1: Hecke group- Arithmetic and Geometry-I. (Video is here and Slides are here)

Speaker: Ser Peow Tan, National University of Singapore.

Time: 14th December, Tuesday 16:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: Abstract is here.

Talk-2: Hecke group- Arithmetic and Geometry-II. (Video is here and Slides are here)

Speaker: Ser Peow Tan, National University of Singapore.

Time: 14th December, Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: Abstract is here.

Week - 17

Talk-1: "ABC Implies Mordell". (Video is here and Slides are here)

Speaker - Anirudh Gurjale, University of Rochester.

Time: 21st December, Tuesday 17:30 Hrs IST (GMT + 5:30).

Chair for the Talk: Jitendra Bajpai, MPI, Bonn.

Abstract: Mordell conjecture (Falting's theorem) states that a curve of genus $\geq 2$ over $\Q$, or a number field, has finitely many rational points. On the other hand, the ABC conjecture implies that for any $\epsilon > 0$, and relatively prime nonzero $A, B, C \in \Z$ such that A + B = C,

$$rad(ABC) >> max(|A|, |B|, |C|)^{1 - \epsilon).$$

Noam Elkies uses Belyi's theorem and theory of heights to conclude that ABC implies Mordell conjecture. This is interesting because effective versions of ABC would then imply effective height bounds for the rational points. Don Zagier states an equivalent and amusing formulation 'Mordell is as easy as ABC!'.

Talk-2: "Finite Index Subgroups of the Modular Group and Modular Forms". (Video is here & Slides are here)

Speaker - Ling Long, Louisiana State University, USA.

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

Chair for the Talk: Jitendra Bajpai, MPI, Bonn.

Abstract: We will give an overview of finite index subgroups of the modular group and some properties of their corresponding modular forms based on joint projects with A.O.L. Atkin, Wen-Ching Winnie Li, Tong Liu, and Zifeng Yang.

Here concludes the first trimester of the Year-long series of virtual seminars.