department of mathematicsseminar/colloquium/colloquia-spring-2014.pdf · department of mathematics...

Department of MathematicsIndian Institute of Technology Bombay

Abel Prize 2014

[http://www.abelprize.no/]

The Department of Mathematics at IIT Bombay is organizing a lecture

by

Professor S. G. Daniintroducing the Abel Prize Laureate 2014.

Title : Introducing Ya. G. Sinai

Date : Thursday, April 17, 2014

Time : 16:00 - 17:00Venue : Ramanujan Hall, Dept. of Mathematics

About the Speaker:

Professor Shrikrishna G. Dani is currently DistinguishedVisiting Professor in the Department of Mathematics atIIT Bombay.


Department Colloquium

Speaker:

Prof. Roger WiegandWilla Cather Professor Emeritus of Mathematics

Department of Mathematics

University of Nebraska

Lincoln, USA

Title: Non-uniqueness of Direct-sum Decompositions

Abstract: Let R be a local domain of dimension one. For example, R might be the local ring of asingular point on an algebraic curve, or a localization of an order in an algebraic number field. We are particularlyinterested in the situation where the completion of the ring is not an integral domain. Examples include the nodalcubic curve C[x, y]/(y2− x3− x2) and rings such as Z[5

√−1] (suitably localized).

Let C(R) be the set of isomorphism classes of finitely generated torsion-free R-modules. We make C(R) anadditive semigroup, using the direct-sum relation: [M ]+[N ] = [M⊕N ], where “[ ]” denotes the isomorphismclass of a module. This semigroup encodes all direct-sum relations among finitely generated torsion-free modules.

The structure of this semigroup has been worked out in two antipodal cases: (1) when all branches of thecompletion have infinite representation type, and (2) when R has finite representation type. The intermediatecase, where R has infinite representation type but at least one branch has finite representation type seems to bemuch more difficult, but some progress has been made.

In this talk I will survey some of these results and give concrete examples to show spectacular failure ofuniqueness of direct-sum decompositions. For example, given any integer n ≥ 2 one can find a ring R (asabove) and indecomposable modules M,N, V such that M ⊕N is isomorphic to the direct sum of n copiesof V .

Date : Wednesday, April 16, 2014




Speaker:

Prof. S. RamananChennai Mathematical Institute

Chennai, INDIA

Title:

Quadric Geometry and Vector Bundles on Curves

Abstract:There is a close relationship between pencils of quadrics (particularly in odd-dimensionalprojective spaces) and hyper-elliptic curves. Classically, it is known that pencils in P 3 give allelliptic curves. Hyper-elliptic curve of genus g as well as their Jacobians can be geometricallydescribed in terms of a pencil in P 2g+1. The moduli of vector bundles of rank 2 on them canalso be described in a similar fashion.

Hitchin gave a map of the cotangent bundle of these moduli spaces (which is an open densesubset of the Higgs moduli) into an affine space and one can describe this also geometrically.I shall try and illustrate this in the particular case g = 2.





Speaker:

Prof. Satadal GangulyStatistics and Mathematics Unit

Indian Statistical Institute

Kolkata, INDIA

Title:

Oscillations in Arithmetic Sequences

Abstract:Measuring oscillations in arithmetic sequences often leads to deep and challenging problemswith interesting applications. For instance, having a good understanding of the sign-changes inthe values of the Mobius function µ(n) will lead to a resolution of the Riemann Hypothesis.I shall describe some of the basic ideas involved in studying oscillations along with examplesand applications with special emphasis on the problem of studying sign-changes in the valuesof the Ramanujan Tau function τ (n), and more generally to coefficients of modular forms.No prerequisites will be assumed.





Speaker:

Dr. Nikhil SrivastavaResearcher, Algorithms Group

Mcrosoft Research

Bangalore, INDIA

Title: The Solution of the Kadison-Singer Problem

Abstract: The Kadison-Singer problem is a question in operator theory which arose in1959 while trying to make Dirac’s axioms for quantum mechanics mathematically rigorous inthe context of von Neumann algebras. It asks whether every pure state on a discrete maximalabelian subalgebra of B(H) extends uniquely to a pure state on all of B(H), where H is aseparable complex Hilbert space. In the 70’s and 80’s, it was realized that the linear-algebraiccore of the problem lies in understanding when an arbitrary finite set of vectors in Cn can bepartitioned into two disjoint subsets each of which approximate it spectrally.

We give a positive solution to the problem by proving essentially the strongest possiblepartitioning theorem of this type. The proof is based on two significant ingredients: a newexistence argument, which reduces the problem to bounding the roots of the expected char-acteristic polynomials of certain random matrices, and a general method for proving upperbounds on the roots of such polynomials. The techniques are elementary, mostly based on toolsfrom the theory of real stable polynomials, and the talk should be accessible to a broad audience.

Joint work with A. Marcus and D. Spielman.

Date : Wednesday, March 26, 2014




Speaker:

Prof. Graeme FairweatherMathematical Reviews

American Mathematical Society

Ann Arbor, USA

Title:

Alternating Direction Implicit Orthogonal Spline Collo-cation Methods for Time-Dependent Problems

Abstract:The formulation, analysis and implementation of efficient numerical techniques for the solutionof time-dependent problems in two space variables are described. The basic approach is todiscretize in space using orthogonal spline collocation (OSC) (also known as spline collocationat Gauss points), and to advance in time using an alternating direction implicit (ADI) method.OSC has several advantages over finite difference and finite element methods, while an ADImethod reduces a multidimensional problem to sets of independent one-dimensional problemsin the coordinate directions which can be solved efficiently using existing software.

After an overview of the development of ADI OSC methods for parabolic problems, extensions topartial integro–differential equations and a class of two-component nonlinear reaction-diffusionproblems are presented. Numerical results demonstrate the efficacy of the methods.





Speaker:

Prof. S. M. BhatwadekarProfessor of Mathematics

Bhaskaracharaya Pratishthana

Pune, INDIA

Title:

Projective Modules over the Kernel of a Locally Nilpo-tent Derivation on an Affine Space

Abstract:Let k be an algebraically closed field of characteristic zero, D be a locally nilpotent derivationon the polynomial algebra k[X1, · · · , Xn] andA be the kernel ofD. In this set up a questionof Miyanishi asks whether finitely generated projective modules over A are free. Note that forn = 3, by a result of Miyanishi, the kernel A is a polynomial algebra in two variables over kand hence an answer to the question is affirmative. In my talk I will address this question forn = 4.





Speaker:

Prof. A. RaghuramProfessor and Coordinator (Mathematics)

Indian Institute of Science Education and Research

Pune, INDIA

Title:

From Calculus to Number Theory

Abstract:In our study of integral calculus, we come across various convergent series. Some of themare very well-known, having been made famous by the works of Euler, Leibniz, MacLaurin andMadhava (from the 14th century Kerala School of Mathematics). Such series happen to beprototypes of a subject within modern number theory called Special Values of L-functions. AnL-function is a function of a complex variable that is attached to some interesting arithmeticor geometric data. The special values of these functions give structural information about suchdata. In this talk, I will weave a storyline starting with ancient formulas in Calculus leading upto some theorems and conjectures in modern Number Theory.





Speaker:

Prof. M. S. RaghunathanNASI Platinum Jubliee Chair


Indian Institute of Technology Bombay

Mumbai, INDIA

Title:

Imbedding Harish-Chandra Modules in Principal Series

Abstract:Let G be a connected semisimple Lie group with a finite centre and g its Lie algebra. Let Kbe a maximal compact subgroup. Let U be the enveloping algebra of g. A (g,K)-module isa module M over U with an action of K on it compatible with the U -module structure. A(g,K)-module M is a Harish-Chandra module if it decomposes as a K-module into a directsum of finite dimensional irreducible representations of K with each irreducible representationoccuring with finite multiplicity. Let P be a minimal parabolic subgroup of G and p its Liealgebra. Let ρ be a finite dimensional continuous irreducible representation of P on a vectorspace E. The representation I(ρ) of U induced by ρ is the U -module Homp(U,E) - U isregarded as a p-U -bimodule with p acting on the left and U acting on the right. A theoremof Casselman and Milicic asserts that any irreducible Harish-Chandra module imbeds in I(ρ)for a suitable ρ. In this talk we give a simple algebraic proof of this theorem in the special casewhen G has rank 1.

Date : Wednesday, February 26, 2014




Speaker:

Prof. E. K. NarayananDepartment of Mathematics

Indian Institute of Science

Bangalore, INDIA

Title:

Bounded Hypergeometric Functions Associated toRoot Systems

Abstract:A natural extension of Harish-Chandra’s theory of spherical functions on Riemannian symmetricspaces of non-compact type was introduced by Heckman and Opdam in the late eighties. In thistheory, the symmetric space G/K is replaced with a triple (a,Σ,m) where a is a Euclideanvector space with an inner product, Σ a root system in a∗ and m a multiplicity functionon Σ. Associated to this triple, there is a family of commuting differential operators (whichcoincide with left G-invariant differential operators on G/K when the triple is geometric)which admit joint eigenfunctions called hypergeometric functions (these functions coincidewith Harish-Chandra’s spherical functions in the geometric case). We study these functionsand characterize the bounded hypergeometric functions, thus establishing an analogue of thecelebrated theorem of Helgason and Johnson. This is joint work with Angela Pasquale andSanjoy Pusti.





Speaker:

Prof. Ravi RamakrishnaDepartment of Mathematics

Cornell University

Ithaca, USA

Title:

Galois Representations

Abstract:In the last 30 years representations of (infinite) Galois groups have played an increasinglyimportant role in number theory. Indeed, arithmetic objects such as the Diophantine equationy2 = x3 − x2 + 1 or xn + yn = zn often attached Galois representations that ‘know’the solutions. This talk will survey of a small slice of this theory and will be accessible tomathematicians in all disciplines.

Date : Thursday, February 13, 2014




Speaker:

Prof. Krishna B. AthreyaDistinguished Professor

Department of Statistics

Iowa State University

Ames, USA

Title:

Statistical Estimation of Integrals w.r.t. Infinite Mea-sures

Abstract:Given an L1 function f on an infinite measure space the problem of generating an infinitesequence of random variables and statistics of these random variables to consistently estimatethe integral of f and obtaining an approximate confidence interval will be addressed in thistalk. This is joint work with Vivek Roy of Iowa State Stat Dept.





Speaker:

Prof. Anish GhoshSchool of Mathematics

Tata Institute of Fundamental Research

Mumbai, INDIA

Title:

Effective Density of Lattice Orbits

Abstract:I will discuss the distribution of dense lattice orbits on homogeneous spaces of algebraic groups.The emphasis will be on examples, illustrating interesting connections with number theory andharmonic analysis. This is joint work with A. Gorodnik and A. Nevo.

Date : Wednesday, January 29, 2014




Speaker:

Prof. N. SaradhaSchool of Mathematics

Tata Institute of Fundamental Research

Mumbai, INDIA

Title:

Number of Representations of Integers by Binary Forms

Abstract:Let F (x, y) be an irreducible binary form of degree r ≥ 3 with integer coefficients and h anon-zero integer. In a seminal work in 1909, Thue proved that the equation F (x, y) = h hasonly fnitely many solutions in integers x and y. For this purpose, he employed a method basedon approximation of algebraic numbers by rationals. His method was developed by severalmathematicians to give better estimates for the number of solutions of Thue equations. In1987, Bombieri and Schmidt estimated the number of primitive solutions as cr1+w(h), wherec is an absolute constant andw(h) denotes the number of distinct prime divisors of h. Furtherthey showed that c = 215 if r ≥ r0 where r0 is unspecified. In a joint work with DivyumSharma, we showed that the above result of Bombieri and Schmidt is true with r0 = 23.Further, c may be taken as small as 10 provided the discriminant of F is large compared to r.In this talk, I shall indicate some salient features of Thue’s method and how the improvementin our work has been obtained.




Institute Colloquium

Speaker:

Prof. M. Ram MurtyProfessor and Queen’s Research Chair


Queen’s University

Kingston, CANADA

Title:

Ramanujan and the Zeta Function

Abstract:In the 18th century, Euler proved that the Riemann zeta function evaluated at even argumentsis always a rational multiple of a power of π. Hence, these values are all transcendental.The values of the zeta function at odd arguments still remain a mystery. However, in hiscelebrated notebooks, Ramanujan discovered a marvelous formula for these values that linksthem to the theory of modular forms. We will give an overview of the contributions of Eulerand Ramanujan and report on some recent advances in the theory. The talk will be accessibleto a general audience.


Time : 17:15 - 18:30Venue : Institute Auditorium, SoM, IIT Bombay

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