Brochure 2018-19

Mathematics IISER BHOPAL The Indian Institute of Science Education and Research (IISER) Bhopal was established in the year 2008 by the Government of India to promote research and education in basic sciences. The Department of Mathematics at IISER Bhopal has gradually grown to become one of the leading centres in the country for mathematical research and education. It is our constant endeavour to conduct quality research in mathematics, and to train students by providing them a broad and solid foundation in pure and applied mathematics and thereby prepare them for pursuing an academic career in mathematics and a non-academic employment in private and government sectors. The Department offers three degree programs: BS-MS (dual degree), Integrated Ph.D., and Ph.D. These programs attract highly motivated students from all over the country. Our students have secured admissions into the Ph.D. programs at reputed universities and institutes in the country and abroad, and have been recruited by various non-academic organisations. The Department has nineteen faculty members. They represent a wide spectrum of research areas in mathematics. In particular, the Department has strong research groups in the areas of algebra, analysis, geometry, and topology. The Department Dr. Saurabh Shrivastava, aims to strengthen these areas further and expand to diversify into newer areas such as Applied Mathematics, Probability and Statistics in the near future. This will Head of the Department of Mathematics, promote the multidisciplinary research within the department. We have taken up several initiatives in the past to promote and enhance the research IISER Bhopal. and teaching environment in the department. For example, our post-doctoral program and the visitors’ program are the key initiatives in this direction. We are in the process of setting up student exchange and joint degree programs with some of the leading universities in the world. The Department hosts several workshops, schools and conferences aimed at promoting mathematics education and collaborative research. Also, we are proud to have a well-established outreach program through which we reach out to the school children and motivate them by providing exposure to the subject and making them aware of various opportunities in the field. In my opinion it is a significant achievement for a new department to sustain a Prologue balanced growth. I feel the department is progressing well and is set to become a prominent place for mathematical research and teaching. We, of course, need to maintain and improve the research quality. With the current strength and the initiatives that we have taken, we are confident to become a place known for doing great mathematics. Faculty At present, we have nineteen experienced faculty, trained in world class institutes or universities, working in various areas of mathematics including algebra and number theory, topology and geometry, analysis. You may find more information about them on the webpage of the department. Nikita Agarwal Nikita’s broad research interests lie in ergodic theory and pure and applied dynamical systems. The topics include studying the statistical properties of dynamical systems, network theory — dynamics of complex systems as networks of coupled dynamical systems. She is currently interested in the following topics: • Piecewise smooth dynamical systems, known as Open dynamical systems; such systems can be explained using the dynamical billiards model. In particular, she focuses on understanding the ergodic properties of maps with holes using the tools from ergodic theory, symbolic dynamics, and combinatorics. • Network dynamics — interacting discrete/continuous dynamical systems, deterministic as well as networks whose topology varies with time. Her work focuses on obtaining sufficient stability conditions of a continuous-time switched system with some or all unstable subsystems. Selected Publications Nikita obtained her Ph.D. degree from • Nikita Agarwal, Inflation of strongly connected networks, Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 2, The University of Houston, Texas, USA, 367–384. in the year 2011. She joined the Department of Mathematics at IISER • Nikita Agarwal, Alexandre Rodrigues, and Michael Field, Dynamics near the product of planar heteroclinic Bhopal in July 2011 as an Assistant attractors, Dyn. Syst. 26 (2011), no. 4, 447–481. Professor. • Nikita Agarwal, A simple loop dwell time approach for stability of switched systems, SIAM J. Appl. Dyn. Syst. 17 (2018), no. 2, 1377–1394. • Haritha C, and Nikita Agarwal, Product of expansive Markov maps with hole, Discrete Contin. Dyn. Syst. 39 (2019), no. 10, 5743–5774. Kumar Balasubramanian Kumar joined IISER Bhopal as an assistant professor following the completion of his Ph.D. His primary research interest is in the representation theory of p-adic groups. He is currently interested in studying the sign associated to self-dual representations of p-adic groups. When G is a finite group and π is an irreducible self-dual representation of G, a well known classical result of Frobenius and Schur says that the sign of the representation can be computed using its character. The sign of an irreducible self-dual representation also makes sense when G is a p-adic group. He is interested in exploring related problems in this context. Recently, he is also interested in studying the sign problem in the context of coverings of p-adic groups. Selected Publications • Kumar Balasubramanian, Self-dual representations of SL(n, F), Proc. Amer. Math. Soc. 144 (2016), no. 1, 435–444. Kumar obtained his Ph.D. degree • Kumar Balasubramanian, M. Ram Murty, and Karam Deo Shankhadhar, Finite order elements in the integral from the University of Oklahoma, symplectic group, J. Ramanujan Math. Soc. 33 (2018), no. 4, 427—433. USA. He worked on some problems in the Representation theory of p-adic • Kumar Balasubramanian, K. S. Senthilraani, and Brahadeesh Sankarnarayanan, Self-dual representations of groups in his thesis. SL(2, F): an approach using the Iwahori-Hecke algebra, Comm. Algebra 47 (2019), no. 10, 4210—4215. • Kumar Balasubramanian, A note on self-dual representations of Sp(4, F), J. Number Theory 199 (2019), 110 —125. Ajit Bhand The study of reduced equations of motion under symmetries is an important theme in dynamical systems and geometric mechanics which goes back to Euler, Lagrange, Poisson, Liouville, Jacobi, Hamilton, Routh, Noether and Poincaré, among others. The modern reduction theory of mechanical systems was pioneered by Arnold (1966), Smale (1970), Meyer (1973) and Marsden and Weinstein (1974). Much of the work done by these authors was in the context of symplectic, Lagrangian and Hamiltonian frameworks. In his Ph.D. thesis, Ajit worked on an affine connection formulation of reduction theory. He is interested in problems in differential geometry. Specifically, he is interested in studying reduction for nonholonomic systems in which the dynamics are constrained on an invariant, nonintegrable distribution. More recently, he has also gotten interested in analytic number theory, integer-weight modular forms and quasimodular forms. One of the emerging themes in mathematical physics is the interplay of geometry, number theory and string theory. Ajit is interested in questions related to black holes in string theory which may possibly be answered using number theory. Selected Publications Ajit obtained his Ph.D. from Queen's University, Kingston, Canada. As a • Ajit Bhand, Geodesic reduction via frame bundle geometry, SIGMA Symmetry Integrability Geom. Methods postdoctoral researcher, he was at the Appl. 6 (2010), Paper 020, 17 pp. University of Oklahoma, Norman, USA. • Ajit Bhand, and Karam Deo Shankhadhar, On Dirichlet series attached to quasimodular forms, J. Number Theory 202 (2019), 91–106. • Ajit Bhand, and M. Ram Murty, Class numbers of quadratic fields, Hardy-Ramanujan J. (accepted). Angshuman Bhattacharya Anghshuman’s Ph.D. thesis was on aspects of the Weak Expectation Property for C*-algebras and operator systems. His research interest lies in C*-algebraic tensor products and related order theoretic properties. He is also interested in operator algebraic properties of (compact) quantum groups. The Weak Expectation Property was introduced by E. C. Lance in 1973 to characterize maximum tensor product inclusions of C*-algebras and thereby showing the equivalence of amenability and nuclearity of discrete group C*-algebras. This property was put into further relevance in 1993 when E. Kirchberg showed the equivalence of the validity of the Connes Embedding conjecture and the question of the full group C*-algebra of the free group of two generators has the Weak Expectation Property or not. It is therefore of much interest to find out more about the C*-algebras with this property. Operator algebraic properties of quantum groups have been a very active area of research in the recent years. Many properties of discrete group C*-algebras has been successfully studied in the discrete quantum group context using the duality of compact and discrete quantum groups. There are many interesting questions that arise in the noncommutative setting which do not have classical counterparts. Among others, there are questions relating to the non tracial nature of the Haar functional of quantum groups in general, connected to properties like amenability, factorization and other finite dimensional Angshuman obtained his Ph.D. approximation properties which are of particular interest. degree from the

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