SCHOOL OF NATURAL SCIENCES
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Mathematics

The Department of Mathematics carries out research in a range of pure and applied mathematics. The department has a close relationship with the Institute for Innovations and Inventions with Mathematics and IT (IIIMIT), a research centre at SNU dedicated to applications of mathematics and computing to real-world problems. We have regular research seminars, and have already hosted national conferences such as the 27th Annual Conference of the Ramanujan Mathematical Society (October 2012) and the Northern Regional Conference of the National Initiative in Mathematics Education (November 2011, co-hosted with Ambedkar University, Delhi). We support collaboration with other institutions through grants for travel and workshops.

Research is further supported by facilities such as individual laptops/desktops for faculty, a 30-PC computer lab with Mathematica and Matlab, a generous library budget for books, and subscriptions to individual journals as well as collections such as JSTOR, Springer Link and SIAM.

Current research areas of our faculty include:

Graph Theory Ordinary and Partial Differential Equations
Network Theory Dynamical Systems
Number Theory Numerical Analysis
Functional Analysis & Operator Theory Game Theory
Operator Algebras Bioinformatics
Harmonic Analysis Mathematical Finance
Representation Theory Mathematics Education
Quantum Groups  

We are seeking to expand our department’s research activities into areas such as Probability, Statistics, Optimization, etc.


Research Interests: Finite Element Method

I am currently involved in the error analysis of the Finite Element Model recently developed for the mathematical modeling and numerical simulation of the motion and deformation of red blood cells (RBC) and vesicles subject to an external incompressible flow in a microchannel.


Research Interests: Representation Theory, Mathematical Finance, Mathematics Education

I am currently engaged in developing a research program in Mathematical and Computational Finance in collaboration with IIIMIT. The group is headed by Prof Sunil Bowry of the School of Management and Entrepreneurship at SNU. It brings together faculty from Mathematics, Finance and Computer Science. We have installed a high performance IBM server for our computational work and are starting work on high-frequency data from the National Stock Exchange.


Research Interests: Bioinformatics, Computational Finance

I am interested in studying networks in DNA molecules based on SNP patterns, copy number variations and gene expression patterns. 

I am also involved with the Computational Finance group at IIIMIT which is interested in carrying out different kinds of analysis on high frequency data of the financial market in India.


Research Interests: Hopf Algebras, Quantum Groups, Topological and Homotopy Quantum Field Theory

Category theory has already started occupying a central position in contemporary mathematics and mathematical physics, and is also applied to theoretical computer science. I am interested in categorifying various objects and studying the underlying algebraic structures hidden in them. For example, I am interested in Quantum groups and Homotopy Quantum Field Theory (HQFT) in dimension two and more specifically on the underlying algebraic structures in a purely categorical setup.


Further plans: For a finite dimensional Hopf algebra H, there are these beautiful isomorphisms of braided categories between the centre of a category of finite dimensional representations of H, the category of Yetter-Drinfeld modules over H (left H-modules and right H-comodules with some compatibility conditions) and the category of finite dimensional representations of the quantum double of H. I want to explore these results without assumptions on dimensionality of the Hopf algebra in a general categorical set up.


Research Interests: Functional Analysis, Operator Theory, Dynamical Systems

Under Functional Analysis and Operator Theory, we are looking at characterizing closed subspaces of Hardy spaces that are invariant under the action of some special isometries. We are also solving problems related to obtaining common invariant subspaces of pairs of isometries. In addition, far more general situations are being explored that involve obtaining Hilbert spaces contained in Hardy spaces under some natural assumptions. The methods developed in the course of the above explorations have been applied to the dual space BMOA of the Hardy space H1 to obtain invariant subspaces and common invariant subspaces under certain isometries.

I am also collaborating with Prof L M Saha of IIIMIT in work on Dynamical systems, where we are studying a variety of non-linear systems. These non-linear systems typically arise in Ecology, Biology, Medicine, Optics, Acoustics, and Astronomy. We are looking at new ways of measuring chaos in these systems, using both discrete and continuous models. We aim to explore neural networks in light of chaos as well in the near future.

Research Interests: Functional Analysis, Operator Theory, Error Correcting Codes, Encryption, Mathematics Education

I work on various problems related to spaces of analytic functions, for example generalizing results of de Branges on invariant subspaces. Over the last few years, I have worked extensively on incorporating research into undergraduate education using problems from areas such as cryptography and error-correcting codes. I am also involved with the use of technology in mathematics education.


Research Interests: Algebraic Graph Theory, Discrete Mathematics, Algebraic Number Theory

I am interested in finding applications of Linear Algebra and Abstract Algebra to Graph Theory. One of the problems I am presently working is finding a necessary and sufficient condition for the graph such that its adjacency algebra is a coherent algebra. I am also exploring graph theoretical and algebraic properties of graphs whose adjacency algebra is a coherent algebra and their connections to Statistical Design Theory.

Other problems of interest are representations of cyclotomic fields and their subfields by adjacency matrices of circulant graphs and 0-1 companion matrices, and finding necessary and sufficient conditions for circulant graphs/digraphs to be singular.


Research Interests: Frame theory, Operator Theory and Function theory.

I am interested in the Feichtinger Conjecture which has been proven to be equivalent to the Kadison–Singer Problem and hence to many open problems in pure and applied areas of mathematics. This is a problem in frame theory, and I am investigating it using operator theory and function theory. It has been known for long that the conjecture is true for the Hardy space H2 on the unit disk, but nothing is known about the weighted Hardy spaces. I am currently exploring the conjecture for this later family of Hardy spaces. 

Research Interests: Complexity, Game theory, Formal language and automata theory, Mathematical ecology, Mathematical modeling, Mathematics education
My essential research focus is on understanding complexity in natural and man-made systems. I prefer a game theoretic perspective and adopt a computational network approach via finite automata and related structures to develop an insight into the essential dynamical processes that occur in these systems, so as to understand the emergent phenomena.

Another core aspect of my current research concerns Graph Theory and its applications in analyzing a variety of networks that arise in the context of natural and social systems, in particular analyzing the various graph centrality indices and modeling networks as automatic structures form.

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