Seminar on Jan 9 - Subhashree Mohapatra

Title: Non-conforming Least-squares Spectral Element Method for Stokes Equations

Speaker: Dr Subhashree Mohapatra (IIT Bhubaneswar)

Time: 3:30 to 4:30pm, Thursday, January 9, 2014
Venue: TBA 

Abstract: In the first part we propose a non-conforming hp spectral element method to solve two dimensional Stokes equations with primitive variable formulation on distributed memory based parallel computers. The method is essentially a least-squares method and applicable to both smooth and non-smooth domains. We use geometrical meshes near the corners to overcome the singularities. The transformation \(P^k=e^{τ_k} p\) is used along the sectoral region to resolve the singularity of pressure variable. A suitable parallelizable block diagonal preconditioner is constructed so that the condition number of the preconditioned system is \(O((\ln W)^2)\), where W is the degree of polynomial.

We prove our method to be exponentially accurate. Preconditioned conjugate gradient method is used to obtain the numerical solution. The method is applicable to Dirichlet, Neumann, mixed boundary conditions. Finally we present numerical results on smooth and non-smooth domains with different boundary conditions and discuss computational complexity of the method.

In the second part of the work a three dimensional Stokes problem on smooth domains with Dirichlet boundary conditions is discussed. Numerical results on a cube are presented.

Seminar on Nov 27 - Pradip Kumar

Darboux chart on projective limit of weak symplectic Banach manifolds

Pradip Kumar

Harish-Chandra Research Institute, Allahabad

Time: 2 to 3 pm, Wednesday, November 27, 2013
Venue: CR 207

Abstract: In this talk, I will explain the meaning of Darboux chart on finite dimensional symplectic manifold. I will extend the definition of Darboux chart to infinite dimensional manifolds. I will prove an existence theorem for Darboux charts on weak symplectic PLB manifolds.

Seminar on October 29 - Tirthankar Bhattacharyya

Spectral and Complete Spectral Sets

Prof. Tirthankar Bhattacharyya

Indian Institute of Science, Bangalore

Time: 12 to 1 pm, Tuesday, October 29, 2013
Venue: CR 201

Abstract: A compact set \(K\) in \({\Bbb C}^n\) is a spectral set for a commuting tuple \(T=(T_1,\dots,T_n)\) of bounded Hilbert space operators if the joint spectrum of \(T\) is contained in \(K\) and if the von Neumann-type inequality \[ ||r(T_1,\dots,T_n)|| \le \sup\{ |r(z)| : z\in K\} \] holds for all rational functions \(r\) in \(Rat(K)\). If the corresponding inequality holds even for every \(m\times m\) matrix \(r=(r_{ij})\) with entries in \(Rat(K)\) then \(K\) is called a complete spectral set for \(T\). By a result of Arveson, \(K\) is a complete spectral set for \(T\) if and only if \(T\) possesses a normal boundary dilation over \(K\). For nice domains in \(\Bbb C\), such as the unit disc or annulus, the conditions of being spectral and completely spectral are equivalent. In the lecture we start with the classical theory and try to present some more recent positive and negative results on one and several variable spectral sets.

Seminar on October 10 - Sutanu Roy

Title: Associative twisted tensor product of C*-algebras

Speaker: Sutanu Roy, University of Goettingen

Date: October 10


Time & Venue: 12-1 pm, CR 204 (2nd Floor, B Wing)


Abstract: In this talk, we shall carry forward the notion of quasitriangular quantum groups, introduced by Drinfeld, in Hopf algebraic to C*-algebraic framework. Then we show that the coaction category of C*-algebras over a quasitriangular quantum group is monoidal.

New Maths Books in the Library

Lots of new books have just arrived at the SNU library, and especially ones related to algebra!

	Title	Author	Publisher
1	An Introduction to Abstract Algebra	Robinson, Derek J S	Hindustan Book Agency
2	Basic Abstract Algebra	Bhattacharya,P  Bet al	Cambridge University Press
3	Basic Algebra	Cohn, P M	Springer India
4	Algebra, Vols I to IV	Luthar and Passi	Narosa
5	Basic Quadratic Forms	Gerstein, Larry J	American Mathematical Society
6	Class Field Theory	Artin, Emil	W.A. Benjamin
7	Commutative Algebra with a View toward Algebraic Geometry	Eisenbud, David	Springer
8	Exercises in Modules and Rings	Lam, T Y	Springer
9	Exercises in Classical Ring Theory	Lam, T Y	Springer
10	Introduction to Ring Theory	Cohn, P M	Springer
11	The Classical Groups	Weyl, Hermann	Hindustan Book Agency
12	Undergraduate Algebra	Lang, Serge	Springer
13	Abstract Algebra	Dummit, David S	Wiley
14	An Introduction to Mathematical Cryptography	Hoffstein, Jeffrey	Springer
15	Finite Fields and Applications	Mullen, Gary L	American Mathematical Society
16	Further Algebra and Applications	Cohn, P M	Springer
17	Combinatorial Techniques	Sane,Sharad S	Hindustan Book Agency
18	A Classical Introduction to Modern Number Theory	Ireland,Kenneth F	Springer India
19	A Course in Number Theory and Cryptography	Koblitz, Neal	Springer India
20	Elementary Number theory	Jones, Gareth A	Springer
21	Introduction to Analytic Number Theory	Apostol	Narosa
22	Elementary  Number Theory	Krishnan, V K	Universities Press
23	An Introduction to Laplace Transforms and Fourier Series	Dyke, P P G	Springer
24	A Course in Calculus and Real analysis	Ghorpade, Sudhir	Springer
25	A Course in Multivariable Calculus and Analysis	Ghorpade, Sudhir	Springer
26	Complex Made Simple	Ullrich, David C,	American Mathematical Society
27	Complex Numbers from A to …Z	Andreescu and Andrica	Birkhauser
28	Differential Equations	Ross, Shepley L	Wiley India
29	Introduction to Calculus and Classical Analysis	Hijab, O	Springer
30	Real Mathematical Analysis	Pugh, C C	Springer
31	Introduction to Measure and Integration	Rana, I K	Narosa
32	System Dynamics	Palm, William J	Tata McGraw Hill
33	Introduction to the Mathematics of Finance	Williams, R J	American Mathematical Society
34	Geometry for College Students	Isaacs, I Martin	Brooks/Cole
35	Morse Theory	Milnor, John W	Hindustan Book Agency
36	Graph Theory	Diestel, Reinhard	Springer
37	Connected at Infinity	Bhatia, Rajendra	Hindustan Book Agency
38	Connected at Infinity II	Bhatia, Rajendra	Hindustan Book Agency
39	Mathematics and its History	Stillwell, John	Springer
40	An Introduction to Numerical Analysis	Atkinson, Kendall E	Wiley India
41	Numerical Methods for Scientific and Engineering Computation	Jain, Iyengar and Jain	New Age International
42	Linear Programming	Hadley	Narosa
43	Operations Research	Taha, Hamdy A	Pearson
44	Differential Equations with Mathematica	Abell and Braselton	Morgan Kauffmann
45	Mathematical Modelling with Case Studies	Barnes and Fulford	CRC Press

Seminar on September 25 - Abhishek Ranjan

Title:              Arbitrage Structure and Finite Date Model with Financial Restriction
Speaker:       Dr Abhishek Ranjan

Department of Applied Mathematics
Université Paris 1 Panthéon Sorbonne, France

Time:               2 pm, Wednesday, September 25, 2013.

Venue:             TBA

Abstract:         We consider a (T+1)-date model of a financial exchange economy with finitely many agents having non-ordered preferences and portfolio constraints. There is a market for physical commodities for every state today and in the future, and financial transfers across time and states are allowed by means of finitely many nominal or numeraire assets. We examine the properties of the financial structure F and the set of its (limited) arbitrage-free prices Q_F. The set of arbitrage-free prices is shown to be a convex cone under a sufficient condition that holds in particular for short lived assets. Furthermore, we provide examples of equivalent financial structures F and F’ such that Q_F is a convex cone, but Q_F’ is neither convex nor a cone. At the end, we provide several existence results of equilibrium in a financial exchange economy for which portfolios are either defined by linear constraints or a convex set extending the framework of unconstrained case by Cass (1984, 2006), Werner (1985), Duffie (1987), Gaenakopolos and Polemarchakis (1997), framework of linear equality constraints by Balasko et al. (1990) and framework of 2-date by Cornet and Gopalan (2007), Aouani and Cornet (2011, 2013).

About the Speaker:   Dr Ranjan completed his PhD from the Paris School of Economics and Université Paris 1 Panthéon Sorbonne in 2012. His research interests are in Applied Mathematics, General Equilibrium Models, Financial Economics, and Decision Under Uncertainty.

Seminar on September 24 - Vijay Patankar

Congruences between Modular Forms and p-Adic Families of Modular Forms

(from Ramanujan to Hida)

Speaker:       Dr Vijay Patankar

                         International Institute of Information Technology, Bangalore

Time:               3 pm, Tuesday, September 24, 2013.

Venue:            TBA

Abstract:  In this expository talk, we will give an introduction to congruences between modular forms as first observed by Ramanujan and how that has led to p-adic families of modular forms. 

Among many other things, Ramanujan studied certain natural arithmetic functions such as the Partition function and the Sum of Divisors function.  In 1916, Ramanujan in his paper On certain arithmetical functions, observed certain congruences between distinct arithmetic functions and hence between the generating functions associated to them (which are in fact modular forms). In 1967, Serre interpreted these congruences in terms of Galois representations and conjectured the existence of Galois representations associated to modular forms (proved by Deligne 1968). In 1972, Serre constructed a p-adic family of Eisenstein forms. In 1986, Hida constructed p-adic families of cusp-forms and the associated p-adic families of Galois representations.

All these developments were essential tools for Andrew Wiles' proof of Fermat's Last Theorem. 

About the Speaker: Dr Patankar obtained his PhD from the University of Toronto in 2005. He has held positions at the Cold Spring Harbor Laboratory (New York), Microsoft Research India (Bangalore), Bhaskaracharya Pratishthana (Pune) and Indian Statistical Institute (Chennai). His research interests are in Number Theory, Algebraic Complexity Theory and Cryptography.