ATMW Almora Mathematical Surveys (2012)

Almora Mathematical Surveys will bring together leading mathematicians and young researchers to learn about important developments in several areas of mathematics.
The symposium will also provide an opportunity to young researchers to showcase their latest research.

Abstracts

• Wednesday, 3 October 2012

(1) Speaker: M. S. Raghunathan, IIT Bombay
Title : The Congruence Subgroup Problem
Abstract : In 1962 Bass-Milnor-Serre and independently Mennicke showed that every subgroup H of finite index in SL(n, Z), for n > 2, contains the subgroup of all matrices in SL(n, Z), which are congruent to 1 modulo N, for a suitable positive integer N. Since then generalizations of this result has be obtained by various people for the so called arithmetic subgroups of algebraic groups. In this talk I will survey these developments over the last 50 years.

(2) Speaker: Vijay Kodiyalam, The Institute of Mathematical Sciences, Chennai
Title : Algebraic invariants of subfactors
Abstract : The theory of subfactors was initiated by Vaughan Jones in the early 80’s and led to his famous polynomial invariant for knots - the Jones polynomial. A factor is a von Neumann algebra (a ∗-algebra of operators on Hilbert space that is closed in the topology of pointwise convergence) which has trivial centre. We will concentrate only on II1 -factors which are factors on which there is a (necessarily unique)trace. A unital inclusion of factors is called a subfactor. Any II1 -factor has modules that are characterized by their dimension which can be any non-negative real number. The index of a II1 -subfactor N ⊆ M is the dimension of (roughly speaking) M as an N -module. A striking result of Jones is that the index is quantised - it is either larger than 4 or takes a discrete set of values between 1 and 4. Much of the algebraic aspects of subfactor theory lie in understanding, describing and characterising the ways N ⊆ M . We may even assume that both N and M are the hyperfinite II1 -factor - in some sense the analytically simplest II1 -factor.
Thus, the first invariant of a subfactor N ⊆ M is its index [M : N ] which is a multiplicative invariant. The next, is a sequence of finite-dimensional C ∗ -algebras arising as follows. Given a subfactor N = M0 ⊆ M = M1 , the Jones basic construction yields a
new subfactor M1 ⊆ M2 (of the same index) together with a distinguished projection e2 ∈ M2 . This construction can clearly be iterated to give a whole tower of factors M0 ⊆ M1 ⊆ M2 ⊆ · · · , together with the Jones projections e2 , e3 , e4 , · · · . The algebras
Pn = N ∩ Mn are finite dimensional C ∗ -algebras. These algebras along with their inclusions are clearly invariants of the original subfactor N ⊆ M . Comparing the subfactor N ⊆ M with a field extension K ⊆ L and the index [M : N ] with the field extension degree [L : K], one is naturally led to ask for the analogue of the Galois group and this led Ocneanu to propose an object that he called a paragroup as an invariant for the subfactor. This includes the data of the finite-dimensional algebras Pn but is a little more.
Meanwhile it was observed by Kauffman that the remarkable relationships satisfied by the Jones projections
ei ej= ej ei for |i − j| ≥ 2,
ei ei±1 ei = τ ei ,
(where τ is the reciprocal of the index) have a beautiful pictorial interpretation. From here, with a lot of hindsight, it is quite a natural question to ask whether all elements of the algebras Pn also have a pictorial interpretation.The (affirmative) answer to this leads to Jones’ planar algebras - an algebraic and plane topological object that encodes paragroups in a form amenable to easy computation.

• Thursday, 4 October, 2012

(1) Speaker: Mahan Mj. RKM Vivekananda University, Belur
Title : Recent Developments inKleinianGroups : a tribute to Bill Thurston
Abstract : The field ofKleiniangroups has seen a number of major developments during the last decade: both for infinite covolume groups as well as for lattices. For infinite covolume discrete groups the major open problem was to determine the possible
isometry types of complete hyperbolic 3-manifoldshomeomorphic to the product of a surface with an open interval. This was settled by Minsky and Brock-Canary-Minsky in their proof of Thurston’s Ending Lamination Conjecture. Using their work, the author
was able to settle another conjecture of Thurston: Connected limit sets of finitely generated Kleinian groups are locally connected. A couple of recent developments for lattices in SO(3, 1) have been the following:
(1) The surface subgroup conjecture of Thurston, asserting that every uniform lattice contains a surface subgroup. This was settled by Kahn and Markovic.
(2) The virtual Haken and virtual fibering conjecturesof Thurston, asserting that every closed hyperbolic 3-manifold has a finite-sheeted cover that fibers over the circle. This was settled by the work of Wise followed by work of Agol.

(2) Speaker : C. S. Rajan, TIFR, Mumbai
Title : Geometry and Arithmetic: More than analogies?
Abstract : Analogies between geometry and arithmetic have been a constant source of inspiration for theories and theorems. We will sketch some of these analogies. We speculate whether the analogy between spectrum, or more generally the geometry associated to the canonical metric can possibly lead to a deeper connection between geometry and arithmetic.

(3) Speaker : Kaushal Verma, Indian Institute of Science, Bangalore
Title : Fatou-Bieberbach domains
Abstract : Fatou-Bieberbach domains are proper subdomains in Cn that are biholomorphic to Cn . This talk will outline the construction of such domains and their basic properties. Several examples will also be given.

• Friday, 5 October, 2012
(1) Speaker: S. Kesavan, The Institute of Mathematical Sciences, Chennai
Title : On the spectrum of the Laplacian
Abstract: Let Ω ⊂ Rn be a bounded domain. Consider the eigenvalue problem
                −∆u = λu in Ω
                    u = 0 on ∂Ω.
Using the theory of compact self-adjoint operators, we can show that there exists a sequence of positive reals {λn } which are the eigenvalues and a corresponding sequence of eigenfunctions {un } which forms an orthonormal basis for L2 (Ω).
Various properties of the eigenvalues ond eigenfunctions will be discussed. Open problems related to these will also be presented.

(2) Speaker: S. G. Dani, IIT Bombay
Title : Interface between hyperbolic geometry and Diophantine approximation
Abstract :The geodesic and horocycle flows associated to surfaces of constant negative curvature, and their analogues acting on homogeneous spaces of Lie groups, have played an important role in the study of various questions in Diophantine approximation, including qualitative aspects of approximation of irrational numbers by rationals, values of linear forms and quadratic forms at integral points, uniform distribution of various sequences in the ambient spaces, etc. Conversely some questions in hyperbolic geometry can be treated fruitfully by application of notions in Diophantine approximation such as continued fractions. In this survey we begin by introducing the basic framework of hyperbolic geometry, the geodesic and horocycle groups, the dynamical questions associated with them, and describe their relations with and applications to problems in number theory, highlighting the bridges between the two areas. A brief treatment of how the theory generalizes to homogeneous spaces of Lie groups, well-known applications in the theme, and open problems will also be included.

(3) Speaker: Parameswaran Sankaran, Institute of Mathematical Sciences, Chennai
Title : The Vector Field Problem : A Survey
Abstract : Let M be a (paracompact Hausdorff) smooth connected manifold of dimension n ≥ 1. A vector field on M is an association    p → v(p) of a tangent vector v(p) ∈ Tp M for each p ∈ M which varies continuously with p. In more technical language it is a (continuous) cross-section of the tangent bundle τ (M ) of M . The vector field problem asks: Given M , what is the largest possible number r
such that there exists vector fields v1 , . . . , vr which are everywhere linearly independent, that is, v1 (x), . . . , vr (x) ∈ Tx M are everywhere linearly independent for every x ∈ M . The number r is called the span of M , written Span(M ). It is clear that 0 ≤ Span(M ) ≤ dim(M ). The vector field problem is an important and central problem in differential topology. My talk will focus on the vector field problem and related questions of stable span, parallelizability, etc. In particular we will discuss the following: (i) a brief survey of the main results of general nature, (ii) state the result for the span of spheres and projective spaces, and, (iii) consider the class of homogeneous
spaces.

(4) Speaker: Anupam Saikia, Indian Institute of Technology Guwahati
Title: Application of Euler systems in recent advances on some well-known conjectures.
Abstract: The notion of Euler systems originated in the work of Thaine and Kolvagin in late 1980’s and has been successfully developed by Rubin, Perrin-Riou, Kaand others to obtain significant results relating arithmetic objects to their analytcounterparts. An Euler system of a p-adic representation T of the absolute Galogroup of a number field K is a collection of cohomology classes             cF ∈ H 1 (Gal(F /F ), Tfor a family of abelian extensions F of K with a relation between cF and cF whenevF ⊂ F . Euler systems provide a powerful tool in bounding class groups number fieldand Selmer groups associated with Galois representations. Some of the mportanapplications are as follows. cF resemble the Euler factors of an Euler product.

(i) Euler systems constructed from cyclotomic and elliptic units have been used bRubin to prove several cases of Iwasawa’s Main Conjecture, which relates the locanalytic behavior of a Galois representation to an associated Galois module.
(ii) Euler systems have been used by Kolyvagin, Rubin, Kato and others to provcertain cases of the Birch and Swinnerton-Dyer conjecture, which relates the MordeWeil and Shafarevich-Tate groups of an elliptic curve on one side and the criticL-values of the elliptic curve on the other side.
(iii) Thaine’s original work leading to Euler systems provides a crucial ingredient the recent proof of the Catalan’s conjecture by Mihailescu (2002).
(iv) In the proof of Bloch-Kato’s Tamagawa Number Conjecture for Dirichlet characteby Huber and Kings (2003), Euler systems are used to reduce the conjecture to thanalytic class number formula.

(5) Speaker : Sukumar Das Adhikari, HRI, Allahabad
Title : Algebraic methods in Additive Combinatorics
Abstract : Here we try to have a glimpse of some classical problems and results in Additive Combinatorics and some tools from rudimentary abstract algebra which come in handy in dealing with these problems. A prototype of zero-sum theorems, the theorem of Erd ̋s, Ginzburg and Ziv (known as the EGZ theorem in the literature) has several proofs; some among the most interesting
proofs involve elementary algebraic techniques.The direct problem for addition in groups is to find a lower bound for |A + B| in terms
of |A| and |B|, where for a finite set A, we denote its cardinality by |A|. The classical Cauchy-Davenport theorem, which can be said to be the first theorem in additive group theory, gives a lower bound for |A + B| in terms of |A| and |B|, where A and B are nonempty subsets of Z/pZ. We shall see an elementary algebraic proof of this theorem. The Cauchy-Davenport can be also used to prove the EGZ theorem mentioned above.We shall also discuss an algebraic method due to Olson which gave the exact value of the Davenport constant for certain classes of finite groups and which also yields the best known general upper bound.The Erd ̋s-Heilbronn Conjecture which was proved in 1994 by Dias da Silva and Hamidoune was later proved by Alon, Nathanson and Ruzsa via polynomials. In a brief discussion of the polynomial method, we shall state a general result of Alon, which implies many of the elementary results we use during our discussion.

• Saturday, 6 October, 2012

(1) Speaker : Gadadhar Misra, Indian Institute of Science, Bangalore
Title : The Bergman kernel function
Abstract : The important role of the Bergman kernel function in several areas of analysis and geometry will be discussed. Several methods for computing the Bergman kernel function will be described explicitly.
(2) Speaker : M. Vanninathan, TIFR-CAM, Bangalore
Title : Methods for Elliptic Homogenization : A Brief Survey
Abstract : In this talk, we introduce the homogenization problem in the setting of elliptic partial differential equations (PDE)and state some of the objectives of the theory.In the literature, various aspects of such equations are discussed. Some of them are
existence and uniqueness of solutions,their stability with respect to data, dependence of solutions on parameters, apriori estimates on solutions, influence of regularity of data on the solution, influence of singularity of data on the solution etc. Homogeniza-
tion deals with an asymptotic property of PDE which is subtly different from classical aspects of PDE mentioned above. More precisely,the theory examines the effects of oscillations in the data on the solution. We also mention how the need for such a studyarises in applications.Throughout our discussion, we deal with a single model problem. Outlined are various methods of homogenization found in the literature.Without going through the details, we highlight the main ideas and steps involved in them.More precisely, we plan to discuss the following methods depending on the availability of time: Two scale Asymptotic Expansion, Oscillating test functions,Two scale convergence,Bloch Wave Method, Gamma Convergence, Compensated Compactness,H-Measures, Probabilistic Method, Periodic Unfolding Method etc. In any case, appropriate references from the literature are given for the convenience of the interested audience.Some open problems of homogenization theory are also mentioned.

(3) Speaker : V. Kumar Murty, University of Toronto
Title : The Prime Number Theorem and generalizations
Abstract : Denote by π(x) the number of primes ≤ x. The classical prime number theorem gives the asymptotic growth of π(x). It is the simplest case of a vast array of results and conjectures that ask for the distribution of primes with specific properties.
For example, the Chebotarev density theorem describes the distribution of prime ideals of a number field F that factor in a particular way in an extension field K of finite degree over F . The recently proved Sato-Tate conjecture is also a prime number theorem
which can be interpreted as the splitting of primes in a certain infinite extension of the rational numbers.
In this talk, we will discuss several prime number theorems, their relationship to L-functions and their applications to problems in number theory and arithmetic geometry.We shall also describe some open problems.

Time Table

Tea Break

 Lunch Break

 Tea Break

Patron

Mrs. Hemlata Dhoundiyal

     Acting Vice-Chanceller Kumaun University

 Scientific Committee

Convener	R. Balasubramaian	IMSc, Chennai
Secretary	J. K. Verma	IIT Bombay, Mumbai
Members	S. Kesavan	IMSc, Chennai
	Mahan Maharaj	RKMV University, Belur
	G. Misra	IISc, Bangalore
	M.S. Raghunathan	IIT Bombay, Mumbai

 Local Organising Committee

Convener	H. S. Dhami	Kumaun University
Secretary	Sanjay Pant	Delhi University
Members	V.P. Pandey	Kumaun University
	Jaya Upreti	Kumaun University

Participants  requested to confirm their participation by sending an email to Prof. Sanjay Pant on  Sanjpant at gmail.com by July 27, 2012.

Please mention following details in your email:

1. Your current position: research scholar, PDF,lecturer, assistant professor, associate professor, professor
2. Affiliation
3. Whether you will like to contribute a 3 page summary of an unpublished paper for the exhibition ?
    We will contact you later about it.
4. Arrival-departure date, time in Kathgodam and from Kathgodam

 From Uttarakhand

Almora is a hill station. There are trains from Delhi up to Haldwani/Kathgodam.

The names of trains are Ranikhet Express, Uttaranchal Sampark Kranti Express and AC express from Ananad Vihar in Delhi.

Air service is also available for Pantnagar from Delhi but one will have to check whether the facility is available

on the days of journey.

Taxi shall be available from Kathgodam or Haldwani which can be arranged for the experts after knowing their itinerary.