Conference Home

Algebraic Geometry and Number Theory conference

Dates: 14th December to 20th December 2017


Venue: Indian Statistical Institute, Bangalore Center

Schedule of talks

Registration will begin at 9:30 am on 14th December.

Thu 14 Fri 15 Sat 16 Sun 17 Mon 18 Tue 19 Wed 20
Welcome
9:45-10.45 Dalawat Park Parameswaran H Fakhruddin Krishna Biswas
Coffee O
11:05-12:05 Ramadoss Hanumanthu Kedlaya L Choudhury Bucur Majumder
12:10-1:10 Kummini Khuri-Makdisi Heim I Iyengar Madapusi-Pera Vaish
Lunch D
2:30-3:30 Hogadi Singh Free A Purnaprajna Das Free
Coffee Y
3:45-4:45 RECEPTION TBD Free TBD From 4:30 at PJA Free
Post Session Banquet Special talk by Purnaprajna

Talks will be held in the second floor auditorium of the main building.



Abstracts

Name: Indranil Biswas
Title: Isomonodromic deformations and very stable vector bundles of rank two.
Abstract: For the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two, consider the family of holomorphic vector bundles over curves underlying this universal deformation. We prove that the vector bundle corresponding to a general parameter is in fact very stable. (Joint work with Viktoria Heu and Jacques Hurtubise.)

Top

Name: Alina Bucur
Title: Statistics for points on curves over finite fields
Abstract: A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families. Our main focus will be the family of cyclic prime degree covers, which can be approached both via the original combinatorial/analytic approach and via maps from the idele class group. This is joint work with Chantal David, Brooke Feigon, Nathan Kaplan, Matilde Lalin, Ekin Ozman and Melanie Matchett Wood.

Top

Name: Utsav Choudhury
Title: Motivic Galois group and periods of Kontsevich-Zagier.
Abstract: The relation between Kontsevich-Zagier period algebra and motivic Galois group of Nori will be described. We will see the relation between Nori motives and Voevodsky motives and then deduce the relation between Nori's Galois group and Galois group constructed by Ayoub.

Top

Name: Chandan Singh Dalawat
Title: Primitive p-extensions of p-fields.
Abstract: We give a canonical parametrisation of the set of primitive extensions (those which have no intermediate extensions) of a local field with finite residue field. The parameter can be used to compute various invariants of the extension, such as its Galois group with the ramification filtration.

Top

Name: Mrinal Kanti Das
Title: Stably free modules over smooth real affine algebras.
Abstract: Let $R$ be a smooth affine algebra of dimension $d$ over the field real numbers. We shall talk about the structure of isomorphism classes of stably free $R$-modules of rank $d$.

Top

Name: Najmuddin Fakhruddin
Title: Auxiliary primes and deformations of Galois representations
Abstract: I will discuss some results towards Mazur's conjecture on the dimension of deformation rings associated to mod $p$ Galois representations of a number field (proved by Wiles for the representations associated to classical modular forms). Our approach is based on the construction of optimal level lowering congruences between certain Galois representations and some generalisations, partly still conjectural, of commutative algebra results of Wiles . The talk will be based on work in progress with Chandrashekhar Khare and Ravi Ramakrishna.

Top

Name: Krishna Hanumanthu
Title: Seshadri constants and SHGH conjecture.
Abstract: Let $X$ be a projective variety and let $L$ be a line bundle on $X$. The Seshadri criterion for ampleness says that $L$ is ample if and only if there is a positive real number $e$ such that for every pair $(C,x)$ of a curve $C$ on $X$ and point $x$ on $C$, the ratio $(L.C)/m$ is at least $e$, where $(L.C)$ denotes the intersection product of $L$ and $C$ and $m$ is the multiplicity of $C$ at $x$. Inspired by this criterion, Demailly introduced the notion of Seshadri constants in the 1990s. They have interesting connections to the geometry of projective varieties. In the case of the complex projective plane ${\mathbb P}^2$, Seshadri constants are related to the famous Nagata conjecture and the more general Segre-Harbourne-Gimigliano-Hirschowitz (SHGH) conjecture. We will discuss these connections and talk about a recent result which exhibits irrational Seshadri constants on blow ups of ${\mathbb P}^2$ assuming a part of the SHGH Conjecture. This talk is based on joint work with Brian Harbourne.

Top

Name: Bernhard Heim
Title: Properties of powers of the Dedekind $\eta$-function.
Abstract:

Top

Name: Amit Hogadi
Title: Gabber's presentation lemma for finite fields.
Abstract: Gabber's presentation lemma, originally proved by Gabber for infinite fields, can be thought of as an algebra-geometric analogue of tubular neighbourhood theorem. In this talk I will outline a proof of this theorem for finite fields. This is joint work with my student Girish Kulkarni.

Top

Name: Srikanth Iyengar
Title: Morphisms of perfect complexes over commutative rings.
Abstract: This talk will be based on some recent work with Avramov and Neeman, contained in the preprint: https://arxiv.org/abs/1711.04052. The central result is a criterion for the tensor nilpotence of morphisms of perfect complexes over commutative noetherian rings, in terms of a numerical invariant of the complexes known as their level. Applications to local rings include a strengthening of the Improved New Intersection Theorem, short direct proofs of several results equivalent to it.

Top

Name: Kiran Kedlaya
Title: Fargues-Fontaine curves are geometrically simply connected
Abstract: Given an algebraically closed field $F$ of characteristic $p$ complete for a nonarchimedean absolute value, Fargues and Fontaine construct a certain "curve" over ${\mathbb Q}_p$ which, in the language of perfectoid spaces, parametrizes the untilts of $F$ to characteristic zero. We show that this space is geometrically simply connected; that is, if one extends the coefficient field from ${\mathbb Q}_p$ to any algebraically closed nonarchimedean field, then the resulting space admits no nontrivial finite etale coverings. This builds upon work of Fargues-Fontaine and Weinstein; it also leads to a version of Drinfeld's lemma (on products of etale fundamental groups in positive characteristic) in analytic geometry, generalizing an observation of Scholze.

Top

Name: Kamal Khuri-Makdisi
Title: Periods of modular forms and identities between Eisenstein series.
Abstract: Borisov and Gunnells observed in 2001 that certain linear relations between products of two holomorphic weight 1 Eisenstein series had the same structure as the relations between periods of modular forms; a similar phenomenon exists in higher weights. We give a conceptual reason for this observation in arbitrary weight. This involves an unconventional way of expanding the Rankin-Selberg convolution of a cusp form with an Eisenstein series. This is joint work with Wissam Raji.

Top

Name: Amalendu Krishna
Title: Bloch-Quillen formula for 0-cycles on singular schemes.
Abstract: The Bloch-Quillen formula provides a description of the Chow group of 0-cycles on smooth schemes in terms of the cohomology of K-theory sheaves. But this formula is still conjectural for singular schemes. We shall prove this formula for many singular schemes and provide its applications to Chow groups with modulus.

Top

Name: Manoj Kummini
Title: F-rationality of Rees algebras
Abstract: We will discuss some necessary and some sufficient conditions for Rees rings to be F-rational. This is joint work with M. Koley.

Top

Name: Keerthi Madapusi-Pera
Title: Stratifications of Shimura varieties
Abstract: We will explain some new advances in the understanding of stratifications of Shimura varieties using Scholze's powerful theory of perfectoid spaces.

Top

Name: Souradeep Majumder
Title: Parabolic bundles in positive characteristic.
Abstract: In this talk algebraic parabolic bundles on smooth projective curves over algebraically closed field of positive characteristic is defined. We will show that the category of algebraic parabolic bundles is equivalent to the category of orbifold bundles defined by Kumar and Parameswaran. Tensor, dual, pullback and pushforward operations are also defined for parabolic bundles.

Top

Name: Jinhyun Park
Title: On motivic cohomology of truncated polynomials.
Abstract: We propose an approach for studies of motivic cohomology of truncated polynomials. To test this approach, we compute some basic cases. This is a joint work with Sinan Unver.

Top

Name: A.J. Parameswaran
Title: Autoduality of compactified Jacobian and Chern classes of pull back Picard Bundles.
Abstract: Let $Y$ be an integral nodal curve. We show that the component of the moduli of torsion free sheaves on the compactified Jacobian $\bar{J}(Y)$ of $Y$ which contains Pic$^0 \bar{J}(Y)$ is isomorphic to $\bar{J}(Y)$ under the map induced by the Abel-Jacobi map $Y \to \bar{J}(Y)$. We determine the Chern classes (in Chow group) of the Picard bundles on the desingularisation of the compactified Jacobian over a nodal curve $Y$. We give an explicit description of the compactified Jacobian of a nodal curve and compute its singular cohomology.

Top

Name: B.P. Purnaprajna
Title: Extremal varieties of general type in all dimensions. (with Jungkai Chen and Francisco Gallego)
Abstract: What is the analogue of genus 2 curves in higher dimensions? This talk answers this basic question among many other things. We consider the relations among fundamental invariants that play an important role in algebraic geometry. In particular, the relations between canonical volume and geometric genus for varieties of general type are established. We prove an inequality for a $n$-dimensional minimal Gorenstein variety of general type and investigate the compelling extremal case, when the inequality is an equality. These extremal varieties are natural higher dimensional analogue of Horikawa's surfaces whose invariants satisfy the equality in Noether's inequality. For extremal varieties of general type of arbitrary dimension, it is shown that their canonical linear system are base point free. We give a characterization of these varieties and study its deformation type. It is also proved that these extremal varieties of general type are simply connected in the case of terminal singularities, and otherwise have finite fundamental group. These results and those not mentioned here but will be dealt in the talk, give a complete generalization of Horikawa's results in Annals of Math (1976) for all dimensions!

Top

Name: Ajay Ramadoss
Title: Representation homology of spaces
Abstract: We discuss representation homology of topological spaces, which is a higher homological extension of the representation varieties of fundamental groups. We give a natural interpretation of representation homology as functor homology and relate it to other homology theories associated with spaces (such as Pontryagin algebras and $S^1$-equivariant homology of free loop spaces). One of our main results, which we call the Comparison Theorem, computes the representation homology of any simply connected space of finite rational homotopy type in terms of its Quillen and Sullivan models. We also compute representation homology for some interesting examples, such as spheres, Riemann surfaces, complex projective spaces and link complements in $ \mathbb{R}^3$. While the representation homology of spheres and complex projective spaces is related to the strong Macdonald conjecture of Feigin and Hanlon, the representation homology of link complements is a new homological link invariant similar to knot contact homology. This is joint work with Yuri Berest and Wai-Kit Yeung.

Top

Name: Anurag Singh
Title: Hankel determinantal rings.
Abstract: We will discuss various aspects of rings defined by minors of Hankel matrices of indeterminates, and sketch a proof that these have rational singularities. This is joint work with Aldo Conca, Maral Mostafazadehfard, and Matteo Varbaro.

Top

Name: Vaibhav Vaish
Title: Morel's weight truncations and the motivic intersection complex.
Abstract: We formalize a notion of ``punctual gluing'' of t-structures. This allows us to construct analogue of certain S.Morel's weight truncations in the motivic setting. As an application we can construct the analogue of a (suitably functorial) intersection complex for an arbitrary threefold in Voevodsky's triangulated category of mixed motives. We will also discuss an on going project on refining the construction to relative Chow motives, at least in the context of Shimura varieties.

Top



Name: Purnaprajna P. Bangere
Title: A Meta-Mathematical Framework for Integrating Indian Classical and Western Music.
(joint with David Balakrishnan, Director of Turtle Island String Quartet)
Abstract:
Math part: One can obtain deep insights about algebraic varieties defined over a field K by working in a more general setting where varieties are defined over rings containing K. In particular, the behavior of a variety as it moves in a family is of deep interest, and plays a vital role in algebraic geometry. This viewpoint has yielded many interesting results. Inspired by these viewpoints and by Grothendieck's writing on nilpotents and deformation theory in algebraic geometry, we develop a meta geometric framework for a music that integrates elements of Indian and western classical music, jazz, and the blues.
Music Part: `The study of the overtone series is well known as a foundational point of origin among western musicians who are drawn to explore the cross- civilizational integration of Indian classical music. Based on the work of Herman Helmholtz in the 1850s and further elucidated for the modern era with specific application for musicians by W.A. Mathieu's book, Harmonic Experience, this fundamental theoretical basis led to the creation of the Turtle Island String Quartet in 1985. When Prof. Purnaprajna approached me with the idea of working together on his Meta Raga system, I immediately realized its potential. I further perceived clear congruities in his theoretical framework to not only the approach taken in W.A. Mathieu's book, but also the work of western music giants such as Arnold Schoenberg and John Coltrane. The collaboration has yielded a fininished composition, as well as two others nearing completion'---By David Balakrishnan, multi Grammy winning director of Turtle Island String Quartet

Top

Statmath Unit | ISI Bangalore Centre | ISI Delhi Centre | ISI Kolkata Centre |