MATH MEMBERS

A. Raghuram Debargha Banerjee Neeraj Deshmukh Sneha Jondhale Abhinav Sahani Debarun Ghosh Neha Malik Soumen Maity Advait Phanse Deeksha Adil Neha Prabhu Souptik Chakraborty Ajith Nair Diganta Borah Omkar Manjarekar Steven Spallone Aman Jhinga Dileep Alla Onkar Kale Subham De Amit Hogadi Dilpreet Papia Bera Sudipa Mondal Anindya Goswami Garima Agarwal Prabhat Kushwaha Supriya Pisolkar Anisa Chorwadwala Girish Kulkarni Pralhad Shinde Surajprakash Yadav Anup Biswas Gunja Sachdeva Rabeya Basu Sushil Bhunia Anupam Kumar Singh Jyotirmoy Ganguly Rama Mishra Suvarna Gharat Ayan Mahalanobis Kalpesh Pednekar Ramya Nair Tathagata Mandal Ayesha Fatima Kaneenika Sinha Ratna Pal Tejas Kalelkar Baskar Balasubramanyam Kartik Roy Rijubrata Kundu Tian An Wong Basudev Pattanayak Krishna Kaipa Rohit Holkar Uday Bhaskar Sharma Chaitanya Ambi Krishna Kishore Rohit Joshi Uday Jagadale Chandrasheel Bhagwat Makarand Sarnobat Shane D'Mello Uttara Naik-Nimbalkar Chitrabhanu Chaudhuri Manidipa Pal Shashikant Ghanwat Varun Prasad Chris John Manish Mishra Shipra Kumar Venkata Krishna Debangana Mukherjee Milan Kumar Das Shuvamkant Tripathi Visakh Narayanan Debaprasanna Kar Mousomi Bhakta Sidharth Vivek Mohan Mallick

Compiling, Editing and Design: Shanti Kalipatnapu, Kranthi Thiyyagura, Chandrasheel Bhagwat, Anisa Chorwadwala Art on the Cover: Sinjini Bhattacharjee, Arghya Rakshit Photo Courtesy: IISER Pune Math Members

2017 IISER Pune Math Book

WELCOME MESSAGE BY COORDINATOR

I have now completed over five years at IISER Pune, and am now in my sixth and possibly final year as Coordinator for Mathematics. I am going to resist the temptation of evaluating myself as to what I have or have not accomplished as head of mathematics, although I do believe that it’s always fruitful to take stock, to become aware of one’s strengths, and perhaps even more so of one’s weaknesses, especially when the goal we have set for ourselves is to become a strong force in the international world of Mathematics.

The math personnel are now around 80 strong, including faculty, postdocs, PhD, Integrated PhD, 5th year BS-MS students, teaching assistants, and staff. Our numbers are steadily growing each year. Our post-doc program is getting very strong; this year we went international(!); with some luck, by Fall 2017, we might have around 10 postdocs in Mathematics. We have had continuing success in our BS-MS students getting placed in excellent PhD programs around the world. For example, of our current 5th year students, we have Papia Bera with an offer from Maryland, and Deeksha Adil with several offers from universities such as UC San Diego and Toronto; from our previous batch we had Abhishek Shukla who is now pursuing his PhD at UBC, Vancouver. It’s an absolute pleasure to see our students going places!

In the year 2015-2016, we were selected for a grant under the FIST program that gives us about 1.5 Cr for a high performance computer cluster, a computer lab, and some funds to supplement our library. We have started using these funds in phases; the library has procured many volumes, including several collected works of great mathematicians. Let me add that in a few moments our library will be named as the Srinivasa Ramanujan Library! Our computer lab is getting set-up and very soon we will have a grand inauguration and a series of workshops on computational mathematics are in the offing.

I have had a personal success recently: I was offered a membership at the Institute for Advanced Study at Princeton. This is one of the biggest things that happened to me and I am very excited about my visit to the IAS in the spring and summer of 2018. But, this also goes beyond me, as it bodes well for IISER and also Indian Mathematics to get greater exposure in such exalted centers of Mathematics.

All in all, IISER Pune is doing well and has positioned itself to be an important center for Mathematics in India. But, on a damp note, I feel that we are not yet an international force to reckon with. We have a long way to go in increasing both the quality and quantity of papers in Mathematics. It is my hope that we make it into a place to be well known in international circles as the place that proves great theorems!

A. RAGHURAM COORDINATOR, MATHEMATICS, IISER PUNE

AMIT HOGADI Associate Professor [email protected]

Amit Hogadi received PhD from Princeton University, USA in 2007. He was at Tata Institute of Fundamental Research (TIFR), Mumbai, India before joining IISER Pune in December 2013.

ALGEBRAIC GEOMETRY

One can describe algebraic geometry as the study of varieties, which are spaces defined by vanishing of polynomial equations. I have been interested in moduli spaces which are special types of varieties which parametrize geometric objects.

In the last couple of decades, people have succeeded in applying ideas from topology, especially homotopy theory, to the study of algebraic varieties. This interplay between homotopy theory and algebraic geometry has been one of my latest fascinations.

Currently, I am working on problems which are sometimes clubbed under the title homotopical algebraic geometry and have applications to K-theory, motivic cohomology and also classical topics like Brauer groups.

SELECTED PUBLICATIONS With Indranil Biswas. Fundamental group of quotient singularities. To appear in International Mathematics Research Notices. Esnault, H. and Hogadi, A. (2012). On the algebraic fundamental group of smooth varieties in characteristic p>0. Transactions of the American Mathematical Society 364:2429-2442. Hogadi, A. and Mehta, V. (2011). Birational invariance of the S-fundamental group scheme. Pure and Applied Mathematical Quarterly 7(4):1361- 1370. Hogadi, A. and Pisolkar, P. (2011). On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt. Journal of Number Theory 131(10):1797-1807. Hogadi, A. and Xu, C. (2009). Degenerations of rationally connected varieties. Transactions of the American Mathematical Society 361(7):3931-3394. 2017 IISER Pune Math Book

ANINDYA GOSWAMI Assistant Professor [email protected]

Anindya Goswami received his PhD from Indian Institute of Science (IISc), Bangalore, India in 2008. Following this, he held postdoctoral positions at the Universiteit Twente in Enschede, The Netherlands; INRIA in Rennes, France; and Technion in Haifa, Israel before joining IISER Pune in 2011.

STOCHASTIC CONTROL – GAME THEORY, MATH FINANCE, QUEUING NETWORKS, RENEWAL PROCESSES

I am exploring various topics in Applied Probability. Those include generalization of Black-Sholes-Merton PDE for options in semi-Markov modulated market, Föllmer Schweizer decomposition of an unattainable contingent claim, equilibrium of non-cooperative semi-Markov game under ergodic cost, optimal control under risk sensitive cost, portfolio optimization, large deviation limit, fluid limit in queuing network, PDE techniques in stochastic control and differential games etc.

I use Markov models, filtering techniques, stochastic calculus, infinitesimal generator for semigroup of operators, mild solution technique for parabolic equations, viscosity solution method for HJB/HJI equations, stability analysis of numerical schemes for solving PDE or IE, convergence of value iteration schemes, marginalization technique in rare event simulation for hybrid processes, martingale formulation for Markov processes etc.

SELECTED PUBLICATIONS Goswami, A., Patel, J. and Sevgaonkar, P. (2016). A system of non-local parabolic PDE and application to option pricing. Stoch. Anal. Appl. 34(5):893- 905. Goswami, A. and Nandan, S. (2016). Convergence of estimated option price in a regime switching market. Indian Journal of Pure and Applied Mathematics 47(2), 169-182 Goswami, A. and Saini, R.K. (2014). Volterra equation for pricing and hedging in a regime switching market. Cogent Economics & Finance 1(2):1-11. Atar, R., Goswami, A. and Shwartz, A. (2014). On the risk-sensitive cost for a Markovian multiclass queue with priority. Electronic Communications in Probability 19(11):1-13. Atar, R., Goswami, A. and Shwartz, A. (2013). Risk-sensitive control for the parallel server model. SIAM Journal on Control and Optimization 51:4363- 4386. Basak, G.K., Ghosh, M.K. and Goswami, A. (2011). Risk minimizing option pricing for a class of exotic options in a Markov modulated market. Stochastic Analysis and Applications 29:259-281. Ghosh, M.K., Goswami, A. and Kumar, S.K. (2010). Portfolio optimization in a Markov modulated market: Chapter in Modern Trends In Controlled Stochastic Processes: Theory and Applications. Luniver Press, pp. 181-195. Ghosh, M.K., Goswami, A. and Kumar, S.K. (2009). Portfolio optimization in a semi-Markov modulated market. Applied Mathematics & Optimization 60:275-296. Ghosh, M.K. and Goswami, A. (2009). Risk minimizing option pricing in a semi-Markov modulated market. SIAM Journal on Control and Optimization 48:1519-1541. Ghosh, M.K. and Goswami, A. (2008). Partially observable semi-Markov games with average payoff. Journal of Mathematical Analysis and Applications 345:26-39.

ANISA M H CHORWADWALA Assistant Professor [email protected]

Anisa Chorwadwala received her PhD from University of Mumbai in 2007. Following this, she held postdoctoral positions at the Institute of Mathematical Sciences (IMSc), Chennai; Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy; and Tata Institute of Fundamental Research (TIFR), Mumbai. She has been on the faculty of IISER Pune since April 2011.

SHAPE OPTIMIZATION PROBLEMS

My research work falls mainly in the following two branches of Mathematics: Partial differential equations and Riemannian geometry. I work on shape optimization problems including isoperimetric problems. A typical shape optimization problem is to find a shape which is optimal in the sense that it minimizes a certain cost functional while satisfying given constraints. In many cases, the functional being minimized depends on the solution of a given partial Moving Plane Method for a differential equation defined on the variable domain. punctured ball on the unit sphere

We have solved some shape optimization problems for a class of doubly connected domains over Rank-one symmetric spaces of non-compact type. Among the Rank-one symmetric spaces of compact type, we dealt only with the case of Sn, the unit sphere in ℝn+1 with the induced metric. In fact, we gave a uniform proof for all the three space forms. Here, the functionals to be optimized were the energy functional and the principal frequency associated with the Dirichlet boundary value problem and the Dirichlet Eigenvalue problem respectively, for both linear as well non-linear operators, namely the Laplacian and the p-Laplacian. So far we were solving the shape optimisation problems over a family of punctured balls where the puncture was also spherical and was free to move in the interior of the outer ball. We are now considering solving a shape optimisation problem over a different family of domains. It is again a family of spherical punctured domains; however, this time the puncture is not spherical but has other symmetries and is free to rotate about its `center’. We are also trying to prove analogous results in the area of Spectral Graph Theory.

SELECTED PUBLICATIONS

Anisa, M.H.C. (2017). A glimpse of shape optimization problems. Current Science (Accepted for Publication). Anisa, M.H.C. and Rajesh, M. (2015). An eigenvalue optimisation problem for the p-Laplacian. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145(6):1145-1151. DOI:10.1017/S0308210515000232 Anisa, M.H.C., Rajesh, M. and Francisco, T. (2015). On the Faber-Krahn Inequality for the Dirichlet p-Laplacian, ESAIM: Control, Optimization and Calculus of Variations, 21(1):60-71. Anisa, M.H.C., and Vemuri, M.K. (2013). Two functionals connected to the Laplacian in a class of doubly connected domains on rank one symmetric spaces of non-compact type, Geometriae Dedicata, 167(1):11-21. 2017 IISER Pune Math Book

ANUP BISWAS Assistant Professor [email protected]

Anup Biswas received his PhD from TIFR-Centre for Applicable Mathematics, Bangalore, India in 2011. He held postdoctoral positions at Technion, Israel, and University of Texas, Austin, USA before joining IISER Pune.

STOCHASTIC CONTROLS AND QUEUEING THEORY My broad research area falls under Applied Probability. I am mainly interested in stochastic controls, small noise diffusion, asymptotics of queueing networks and many other related models. In the last couple of years, I have worked on problems from queueing theory that involves measure-valued process. Such processes have proven powerful in modeling queueing networks with general service and reneging distributions. Another important area of queueing theory is scheduling control where one looks for a policy that optimizes certain cost associated to the model. I also work on such control problems.

SELECTED PUBLICATIONS Biswas, A., Hitoshi, I., Subhamay, S. and Lin, W. (2017). On viscosity solution of HJB equations with state constraints and reflection control. SIAM J. of Control and Optimization 55(1):365-396. Arapostathis, A., Biswas, A. and Johnson, C. (2017). On solutions of mean field games with ergodic cost. Journal de Mathematiques Pures et Appliquees 107:205-251. Arapostathis, A., Biswas, A. and Luis, C. (2016). The Dirichlet problem for stable-like operatorsand related probabilistic representations. Communications in PDE 41(9):1472-1511. Arapostathis, A., Biswas, A. and Pang, G. (2015). Ergodic control of multi-class M/M/N+M queues in the Halfin-Whitt regime. Annals of Applied Probability 25(6):3511-3570. Atar, R., Biswas, A. and Kaspi, H. (2015). Fluid limits of G/G/ 1+G queues under non-preemptive Earliest-Deadline-First discipline. Mathematics of Operation Research 40(3):683-702. Biswas, A. (2014). Risk-sensitive control for the multiclass many-server queues in the moderate deviation regime. Mathematics of Operation Research 39(3):908-929. Atar, R. and Biswas, A. (2014). Control of the multiclass G/G/1 queue in the moderate deviation regime. Annals of Applied Probability 24(5):2033- 2069.

ANUPAM KUMAR SINGH Assistant Professor [email protected]

After doing MSc (Mathematics) from Indian Institute of Technology (IIT) Kanpur, Anupam Singh worked for his doctorate at Harish Chandra Research Institute, Allahabad and Indian Statistical Institute (ISI), Bangalore, India and got PhD from ISI in 2007. He held postdoctoral positions at Tata Institute of Fundamental Research (TIFR), Mumbai and Institute of Mathematical Sciences (IMSc), Chennai, India before joining the faculty of IISER Pune in 2008.

CONJUGACY QUESTIONS AND REPRESENTATION THEORY OF GROUPS

Conjugacy questions in group theory have been of interest for its connection with the representation theory and they have not been understood very well for Algebraic Groups over arbitrary field. Usually groups are difficult objects and one studies them via their representations to get a better understanding.

Let G (e.g. GLn) be an algebraic group defined over a field k. We denote the k points of G by G(k) (e.g. GLn(k), SLn(k) etc.). An element t of G(k) is said to be real if it is conjugate to its own inverse in G(k). I have been concerned with finding real elements in algebraic groups. Very interestingly often it relates to finding strongly real elements (the elements which are product of two involutions in G(k)).

Apart from studying structure of real elements in Algebraic Groups over k, I also looked at many examples such as linear groups, orthogonal groups, symplectic groups and the groups of type G2 to get better understanding of the problem. Finding real elements helps in the understanding of real characters of the group which in turn give information about those complex representations of the group which are either orthogonal or symplectic.

SELECTED PUBLICATIONS Gates, Z., Singh, A. and Ryan Vinroot, C. Strongly real classes in finite Unitary groups of odd characteristic. To appear in the Journal of Group Theory. Kulshrestha, A. and Singh, A. (2011). Real elements and Schur indices of a group. The Mathematics Student 80:73-84.

Gill, N. and Singh, A. (2011). Real and strongly real classes in PGLn(q) and quasi-simple covers of PSLn(q). Journal of Group Theory 14:461-489.

Gill, N. and Singh, A. (2011). Real and strongly real classes in SLn(q). Journal of Group Theory 14:437-459.

2017 IISER Pune Math Book

AYAN MAHALANOBIS Assistant Professor [email protected]

Ayan Mahalanobis obtained his PhD from Florida Atlantic University, Boca Raton, USA in 2005. He was a Visiting Assistant Professor at the Stevens Institute of Technology, New Jersey, USA for a few years before joining IISER Pune in 2009.

PUBLIC KEY CRYPTOGRAPHY

I work at the intersection of pure mathematics (group theory) and public key cryptography. Cryptography, especially public key cryptography, is the backbone of a modern society. It serves us with the required tools for online transactions and trading, i.e., online commerce.

My research aims to find new cryptograhic primitives and to build secure protocols from that. We look for groups in which the discrete logarithm problem is secure. My recent work has shown that the group of non-singular circulant matrices over a finite field has some properties that make them attractive over the discrete logarithm problem on a finite field. This new finding has opened a new avenue in research of public key cryptography.

SELECTED PUBLICATIONS Mahalanobis, A. The ElGamal cryptosystem over circulant matrices (Submitted). Mahalanobis, A. and Shah, J. (2014). A new guess-and-determine attack on the A5/1 stream cipher. Computer and Information Science 7(1):115- 124. Mahalanobis, A. (2013). Are matrices useful in public-key cryptography? International Mathematical Forum 8(39):1939-1953. Mahalanobis, A. The MOR cryptosystem and extra-special p-groups. Proceedings of WCC 2012, Castro Urdiales, Spain July 9-13, 2012 & To appear in Journal of Discrete Mathematics and Cryptography. Mahalanobis, A. (2013). The automorphism group of the group of unitriangular matrices over a field. International Journal of Algebra 7(15):723- 733. Mahalanobis, A. (2012). A simple generalization of the ElGamal cryptosystem to non-abelian groups II. Communications in Algebra 40:3583-3596. Mahalanobis, A. (2010). The discrete logarithm problem in the group of non-singular circulant matrices. Groups Complexity Cryptology 2:83-89.

BASKAR BALASUBRAMANYAM Assistant Professor [email protected]

Baskar Balasubramanyam completed his PhD from Brandeis University, USA in 2007. In 2007-08, he was a postdoctoral fellow at the Center for Advanced Study in Mathematics at the Ben-Gurion University of the Negev in Israel. Following this, he was a Bateman Instructor at California Institute of Technology, USA during 2008-10. He has been with IISER Pune since September 2010.

MODULAR FORMS AND GALOIS REPRESENTATIONS

My research interests are in Number Theory. A modular form is essentially a function defined on the complex upper half-plane (everything above the real axis) that behaves in a good way under transformations of certain 2x2 matrices with integer entries. The expansion at infinity of such a function gives us a power series whose coefficients have interesting arithmetic properties. An important example of numbers that arise in such a way is the Ramanujan Tau function (n). In order to understand these coefficients, it is useful to consider an object attached to it called the L- function (these are generalizations of the Riemann zeta function). A Galois group is a set of permutations of roots of an irreducible polynomial. For example, complex conjugation permutes the roots of x2+1. It is possible to represent such permutations by matrices. One of the problems in

Number Theory is to try and understand the Galois group by studying its representations. One can also attach L- functions to Galois representations and in some cases modular forms and Galois representations are related through their L-functions.

SELECTED PUBLICATIONS Balasubramanyam, B. and Raghuram, A. Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number

field American Journal of Mathematics (To appear). Balasubramanyam, B., Ghate, E. and Vatsal, V. (2013). On local Galois representations over totally real fields. Manuscripta Mathematica 142:513- 524. Balasubramanyam, B. and Longo, M. (2010). -adic modular symbols over totally real fields. Commenterii Mathematici Helvetici 86:841-865.

2017 IISER Pune Math Book

CHANDRASHEEL BHAGWAT Assistant Professor [email protected]

Chandrasheel Bhagwat completed his PhD in Mathematics from Tata Institute of Fundamental Research (TIFR) Mumbai, India in 2011 and spent a few months as a post- doctoral fellow at IISER Pune. He has received the DST-INSPIRE faculty award and research grant in 2011 and has joined IISER Pune as INSPIRE Faculty in March 2012. From December 1, 2014, he has been working as an Assistant Professor at IISER Pune.

REPRESENTATION THEORY, ANALYSIS & GEOMETRY OF LOCALLY SYMMETRIC SPACES, NUMBER THEORY

From last few years, I have been studying the arithmetic aspects of the spectral theory and geometry of symmetric spaces. My thesis work involved proving various analogues of the classical number theoretic results in the context of symmetric spaces. In a recent joint work with my colleague Dr. Supriya Pisolkar from Mathematics group at IISER Pune, we have proved some results on representation equivalence of discrete subgroups in Lie groups. I am also interested in the study of analytic properties of the Zeta functions associated to a rank one locally symmetric space. My students and me are working on some of the `inverse problems’ on locally symmetric spaces with the aid of Zeta functions and harmonic analysis. Another area in mathematics that fascinates me is special values of L-functions. My earlier joint work with Prof. Raghuram has established period relations for the tensor product of motives. I am currently working on some related problems about special values for automorphic L-functions associated to representations of algebraic groups (GL(n), SO(n) etc.); using techniques like Cohomology of arithmetic groups and Langlands constant term theory. Recently, I have been involved in studying some questions in spectral graph theory and their relation with arithmetic, analysis and geometry. I have worked with two of my colleagues Prof. G. Ambika and Mr. Snehal Shekatkar (from Physics group at IISER Pune) on a project where we established some interesting results for natural number network, viewed as a complex network. In another ongoing project with my colleagues Dr. Anisa MHC and Mr. Pralhad Shinde (from Mathematics group at IISER Pune), we are working on some theorems about graph Laplacians which will be analogues of certain optimization theorems in classical geometry.

SELECTED PUBLICATIONS Bhagwat C. and Raghuram A. (2017). Special values of L-functions for orthogonal groups. Comptes rendus - Mathematics. (To appear). Shekatkar, S.M., Bhagwat, C. and Ambika, G. (2015). Divisibility patterns of natural numbers on a complex network. Scientific Reports 5, Article number: 14280 (2015), doi:10.1038/srep14280 Bhagwat, C. (2015). On Deligne periods for tensor product motives. Comptes rendus - Mathématique 353:191- 195. Bhagwat, C. and Pisolkar, S. (2015). On uniform lattices in real semisimple group. Proceedings of the American Mathematical Society doi: 10.1090/proc/12961 Bhagwat, C. and Raghuram, A. Endoscopy and Cohomology of GL(N). (Accepted for publication in the special issue of the BIMS in honor of Professor Freydoon Shahidi's 70th birthday.)

DEBARGHA BANERJEE Assistant Professor [email protected]

Debargha Banerjee received his PhD degree from Tata Institute of Fundamental Research, Mumbai. He earned his Master’s degree in Mathematics from Indian Statistical Institute and BSc from St Xavier’s College, Kolkata. He got the Australian Research council Discovery postdoctoral fellowship for 2 years at the Australian National University, Canberra, Australia. He was a visiting scientist at the IMSC, Chennai, India and later a guest scientist at the Max Planck institute of Mathematics, Bonn, Germany before joining IISER Pune in November 2013.

AUTOMORPHIC FORMS AND ARITHMETIC GEOMETRY

I n arithmetic geometry, we study the integral solutions of polynomial equations. In arithmetic geometry we usually work over non-algebraically closed fields (like rational numbers), and often in fields of non-zero characteristic (like finite fields), and we may even restrict ourselves even to rings that are not a field (like integers). We use the rich theory of modular forms (more generally, automorphic representations) to find these solutions.

Modular forms/ automorphic forms are generalizations of the periodic functions like sine or cosine. The theory of modular forms made major contributions in several important discoveries in modern mathematics, including the uniform boundedness of torsion point of elliptic curves.

I am interested in understanding objects in geometry called “motives” and objects in analysis called automorphic representations. The overarching bridge/conjecture of Langlands connects these two fairly faraway worlds. I am interested in understanding the integral and p-adic bridges that connect these two beautiful universes.

In the automorphic universe, I wish to focus on modular symbols and special values of L-functions (one of the main objects of study in the Langlands' program). Motives can be described by different cohomology theories. I wish to understand and use different cohomology theories using modular symbols, which in turn give insights into special values of L-functions.

In another work, I developed the theory of differential modular forms for Shimura curves over totally real fields. The differential modular forms were invented by Buium and his collaborators. These work open the p-adic bridge of certain automorpic forms.

SELECTED PUBLICATIONS Banerjee, D. and Krishnamoorthy, S. The Eisenstein elements for level product of two odd primes. Pacific Journal of Mathematics (To appear). Banerjee, D. and Merel, L. The Eisenstein cycles inside the space of modular symbols (Preprint).

Banerjee, D. and Raghuram, A. p-adic L-functions for GLn (preprint, expository article) Banerjee, D. (2014). A note on the Eisenstein elements of prime square level. Proceedings of the American Mathematical Society 142:3675-3686. Banerjee, D. (2014). Differential modular forms on Shimura curves over totally real fields. Journal of Number Theory 135:353-373. Banerjee, D. and Ghate, E. (2013). Adjoint lifts and modular endomorphism algebras. Israel Journal of Mathematics 195(2):507-543. Banerjee, D. and Ghate, E. (2011). Crossed product algebras attached to weight one forms. Mathematical Research Letters 18(1):139-149.

2017 IISER Pune Math Book

DIGANTA BORAH Assistant Professor [email protected]

Diganta Borah received his PhD from the Indian Institute of Science (IISc), Bangalore, India in 2010 and was a Research Associate there until 2011. He joined IISER Pune in January 2012.

SEVERAL COMPLEX VARIABLES

My research interests are in complex analysis, more specifically in several complex variables. Several complex variables is the study of holomorphic functions defined on domains in Cn, or more generally on complex manifolds. The subject established itself as a new independent area of research in the early twentieth century when F. Hartogs discovered a surprising extension phenomenon in the theory of functions of several complex variables which is absent in the theory in one variable. In the complex plane, every domain is a domain of holomorphy, i.e., there exists a holomorphic function on the domain which has no holomorphic extension to a strictly larger domain. Hartogs showed that this is no longer true in higher dimension. Another astonishing feature of several complex variables is the absence of the Riemann mapping theorem which was discovered by Poincare around the same time by showing that the unit ball and the unit polydisc in Cn, n ≥ 2, are not biholomorphic though both of them are simply connected.

In the past, I studied several quantities associated to the Green’s function of multiply connected domain in the complex plane and of a bounded pseudo convex domain in higher dimension. Among them are some intrinsic properties such as geodesics, curvature, L2 –cohomology etc. of the so called Robin metric constructed by N. Levenberg and H. Yamaguchi.

One of my current project with P. Haridas and K. Verma is to understand the asymptotic distribution of the zeros of the Bergman orthogonal polynomials on an archipelago. In another project with my colleagues D. Banarjee and C. Chaudhuri, we are trying to compute the Arakelov self-intersection number of the relative dualizing sheaves for the 2 family of modular curves X0(p ) where p is an odd prime. Another ongoing project with S. Pal is to find the structures of rational inner functions into symmetrized polydisc.

SELECTED PUBLICATIONS Borah, D. (2017). Remarks on the metric induced by the Robin function III. Journal of the Ramanujan Mathematical Society 32:17-42 Borah, D. (2013). Remarks on the metric Induced by the Robin Function II. Michigan Mathematical Journal 62(3):581-630. Borah, D. and Verma, K. (2011). Remarks on the metric induced by the Robin function. Indiana University Mathematics Journal 60:751-802.

KANEENIKA SINHA Assistant Professor [email protected]

Kaneenika Sinha completed her PhD from Queen's University, Canada in 2006. She has held postdoctoral fellowships at University of Toronto, University of Alberta and Mathematical Sciences Research Institute, Berkeley. She was an assistant professor in IISER Kolkata before joining IISER Pune in December 2012.

ANALYTIC NUMBER THEORY, HARMONIC ANALYSIS AND ARITHMETIC GEOMETRY

My primary research interests are in analytic number theory and arithmetic geometry. One of my primary goals is to investigate statistical phenomena in the distribution of sequences that arise from the theory of modular forms, zeta functions of curves over finite fields and eigenvalues of adjacency matrices of certain kinds of graphs.

In 1916, the German mathematician Hermann Weyl asked the following question: take an irrational number T, look at its multiples T, 2T, 3T, etc. and record the sequence of its decimal parts. While these numbers find a place throughout the interval [0,1), are they likely to cluster around some parts more than others? Weyl discovered that each and every part of the interval [0,1) gets its fair share of elements from the sequence. That is, this sequence is equidistributed in the interval [0,1). In showing this, Weyl discovered and outlined a beautiful technique that was capable of answering generalizations of this question in a wider paradigm. This technique relates the phenomenon of equidistribution to that of studying what are called exponential sums in number theory and places this phenomenon in a wider landscape of harmonic analysis.

Many sequences arising in number theory follow a distribution pattern that can be defined by very elegant functions. In particular, one of the major breakthroughs in recent times is the discovery that certain sequences arising from the Fourier coefficients of modular forms (certain complex-analytic functions with rich inner symmetries and growth conditions) follow the “semi-circle” equidistribution law, also called the Sato-Tate law.

My primary research work focuses on equidistribution of various such families and sequences arising in the context of modular forms, arithmetic geometry and graph theory. I am investigating deeper statistical phenomena associated with such families, for example fluctuations in the distribution and pair correlation.

SELECTED PUBLICATIONS Prabhu, N. and Sinha, K. Fluctuations in the distribution of Hecke eigenvalues about the Sato-Tate measure (submitted). Ram Murty, M. and Sinha, K. (2015). The generalized Dedekind determinant, in SCHOLAR---a Scientific Celebration Highlighting Open Lines of Arithmetic Research, Contemporary Mathematics, 655:153-164. Bucur, A., David, C., Feigon, B., Lalin, M. and Sinha, K. (2012). Distribution of zeta zeroes of Artin-Schreier covers. Mathematical Research Letters 19(6):1329-1356. 2017 IISER Pune Math Book

KRISHNA KAIPA Assistant Professor [email protected]

Krishna Kaipa obtained a BTech from IIT Bombay and an MS degree from University of Maryland College Park, both in Mechanical Engineering. He then pursued a PhD in Mathematics from University of Maryland, College Park. He held assistant professor positions at IIT Bombay and IISER Bhopal before moving to IISER Pune in 2014.

ALGEBRAIC GEOMETRIC CODING THEORY

My recent work is about studying Error Correcting Codes from the viewpoint of projective geometry over finite fields. To give a flavour of the kind of questions I am working on currently, consider k by n matrices over a finite field with q elements (say q is odd) such that all k by k minors of this matrix are nonzero. Let us call such a matrix MDS. Question: What can we say about 3 by q+1 MDS matrices? Segre showed that any such matrix is row equivalent to a matrix whose columns lie on the conic y^2 = xz.

Question: Do there exist k by q+2 MDS matrices when k

SELECTED PUBLICATIONS With P. Beelen, T. Hoholdt and D. Glynn. Counting Generalized Reed-Solomon codes. Advances in the Mathematics of Communication (to appear). With I. Cardinali, L. Giuzzi, and A. Pasini (2016). Line polar Grassmann codes of orthogonal type. Journal of Pure and Applied Algebra, 220(5):1924- 1934. An asymptotic formula in q for the number of [n, k] q-ary MDS codes. IEEE Transactions on Information Theory 60 (2014), no. 11, 7047–7057. With Sudhir Ghorpade (2013). Automorphism groups of Grassmann codes. Finite fields and its Applications 23:80-102. With Harish Pillai (2013). Weight spectrum of codes associated with the Grassmannian G (3,7). IEEE Transactions on Information Theory 59(2):986- 993.

MOUSOMI BHAKTA Assistant Professor [email protected]

Mousomi Bhakta received her PhD from TIFR-Centre for Applicable Mathematics, Bangalore, India in August, 2011. After that she held visiting scientist position in ICTP, Trieste, Italy for two months. Next, she had postdoctoral positions at Technion, Israel for two years and University of New England, NSW, Australia for one year before joining IISER Pune in August 2014.

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

My research interest includes semi linear and quasilinear elliptic partial differential equations arising from geometry and physics. I study existence/nonexistence of positive/sign-changing solutions, qualitative properties of solutions, various estimates of solutions and asymptotic analysis and the profile of solutions using the techniques from nonlinear analysis. One of my main research areas is studying the semi linear and quasilinear elliptic equations with singularity. Such second order equations for example, Hardy-Sobolev type equations arise in astrophysics. Fourth order equations, for example, equations with biharmonic operator arise in continuum mechanics, bio-physics, geometry. In particular, in the modeling of thin elastic plates, clamped plates and in the study of the Paneitz-Branson equation etc. Currently I’m working on non-local equations with fractional Laplace operator/ general integro-differential operator/ p-fractional operator, which have huge applications in physics, engineering, optimization and finance, obstacle problem, conformal geometry and minimal surface problem.

SELECTED PUBLICATIONS Bhakta, M.; Santra, S., On singular equations with critical and supercritical exponents. (submitted) arXiv:1608.00490. Bhakta, M.; Mukherjee, D.; Santra, S., Profile of solutions for nonlocal equations with critical and supercritical nonlinearities. (submitted) arXiv:1612.01759. Bhakta, M.; Mukherjee, D., Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities. Differential and Integral Equations (To appear). Bhakta, M. (2016) Caffarelli-Kohn-Nirenberg type equations of fourth order with the critical exponent and Rellich potential. J. Math. Anal. Appl. 433 (1):681-700. Bhakta, M. (2015). Semi linear elliptic equation with biharmonic operator and multiple critical nonlinearities. Adv. Nonlinear Stud. 15 (4); 835-848. Bhakta, M. and Marcus, M. (2014). Semi linear elliptic equations admitting similarity transformations. J. Funct. Anal. 267 (10): 3894-3930. Bhakta, M. and Marcus, M. (2014). Reduced limit for semi linear boundary value problems with measure data. J. Differential Equations 256 (8):2691-2710. Bhakta, M. and Musina, R. (2012). Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Anal. 75 (9): 3836-3848. Bhakta, M.and Sandeep, K. (2012) Poincar-Sobolev equations in the hyperbolic space. Calc. Var. Partial Differential Equations 44(12) 247-269. Bhakta, M.and Sandeep, K. (2009). Hardy-Sobolev-Maz'ya type equations in bounded domains. J. Differential Equations 247 (1), 11-139. 2017 IISER Pune Math Book

RABEYA BASU Assistant Professor [email protected]

Rabeya Basu received her PhD from Tata Institute of Fundamental Research (TIFR), Mumbai, India in 2007. She then undertook postdoctoral work initially at Harish Chandra Research Institute, Allahabad and later as an NBHM Fellow at the Indian Statistical Institute, Kolkata, India. She was in IISER Kolkata as an assistant professor before joining IISER Pune in 2010.

CLASSICAL ALGEBRAIC K-THEORY

My work is based on problems in classical K-theory which are related to Serre’s problem on projective modules. This famous theorem says that finitely generated projective modules over a polynomial ring over field are free. This involves problems in lower K-theory, in particular study of the Whitehead group K1 due to Hyman Bass, which generalizes the group of units of a ring. Initially, these problems were studied for the general linear groups. Then people started generalizing those results for other classical groups and also for the relative cases. At present, I am working on similar problems for the general quadratic and the general Hermitian groups introduced by Anthony Bak. In addition, I am also trying to study few such problems for the higher K-groups.

I am also interested in commutative algebra, more specifically, in problems related to complete intersection theory, which deals with the number of generators of an ideal. I am trying to learn some computational methods (e.g. use of CoCoA or Macaulay2) to work on such kind of problems with more advanced tools.

Recently, I have developed my interest in a very newly developed subject, viz. Leavitt path algebras (LPA), which is making a bridge between algebra and functional analysis. In the future, I would like to work on the algebraic and K- theoretic aspects of LPA.

SELECTED PUBLICATIONS

Basu, R. Local-global principle for the general quadratic and the general Hermitian groups and the nilpotence of KH1 (Submitted).

Basu, R. (2013). On elementary subgroup of GLn over rings. Ramanujan Mathematical Society Newsletter 24(3).

Basu, R. (2011). Absence of torsion for NK1(R) over associative rings. Journal of Algebra and its Applications 10:793-799. Basu, R. and Rao, R.A. (2010). Injective stability for K1 of regular rings. Journal of Algebra 323:367-377. Bak, A., Basu, R. and Rao, R.A. (2010). Local-global principle for Transvection groups. Proceedings of the American Mathematical Society 138:1191- 1204. Basu, R. and Sridharan, R. (2007). An expository article: On Forster's conjecture and related results. Punjab University Research Journal.

RAMA MISHRA Associate Professor [email protected]

Rama Mishra has obtained her PhD from Indian Institute of Technology (IIT) Bombay, India in 1994. She spent a few years with Harish Chandra Research Institute, Allahabad and Indian Statistical Institute, Delhi as a postdoctoral fellow and then served as a faculty at IIT Kharagpur and IIT Delhi for several years. She worked as a JSPS fellow at Osaka City University, Japan for one year and as a visiting faculty at Boise State University, USA before joining IISER Pune.

LOW DIMENSIONAL TOPOLOGY

Quantum topology is one of the emerging research areas. Many knot invariants naturally arise through nice matrix algebra representation of interesting quantum groups. They have been related to several models in statistical mechanics. I would like to explore some models that are related to Singular knot theory.

I am also interested in classical knot invariants and their application in mathematics and biological sciences. My joint work with Prof Louis Kauffman on nodal parity invariants of knotted rigid vertex graphs discusses the application of this invariant on protein folding classification. I am currently focusing on the following projects dealing with knots and links:

1) Represent knots by rational functions using least possible degree

2) Compute some of the quantum invariants such as Khovanov Homology and Colored Jones polynomials and their categorifications for as many families of knot as possible either by using simple techniques by hand or write effective programs that can perform the computations.

3) Develop some models for DNA as well as RNA/Proteins that can explain folding phenomenon and find a way to design drugs. For (1) and (2), we use the tools from Real algebraic geometry and+ Representation Theory. For (3) besides understanding both the subjects knot theory and biology, we need some interaction with experimental biologists who can show us how to use electron microscopy so that we can observe knots in concrete situations.

SELECTED PUBLICATIONS With Hitesh Raundal (2017). Some spaces of polynomial knots. Topology and its Applications 218:66-69. Knot Theory and its Applications, Contemporary Mathematics, Editor, 2016. With Hitesh Raundal (2015). Spaces of Polynomial knots in low degree. Journal of Knot theory and its Ramifications 24(14). Polynomial Unknotting and Singularity index (2014). Kyungpook Mathematical Journal 54(2):271-292. With Louis H Kauffman (2013). Nodal parity invariants for knotted rigid vertex graphs. Journal of Knot Theory and its Ramifications 22(4). With Shantha Bhushan (2012) Knot theory in understanding proteins. Journal of Mathematical Biology 65(6-7) pp 1187-1213. 2017 IISER Pune Math Book

A RAGHURAM Professor and Coordinator [email protected]

After getting a BTech in Computer Science and Engineering from Indian Institute of Technology (IIT) Kanpur, Raghuram moved to Mathematics and got a PhD in Mathematics in 2001 from the Tata Institute of Fundamental Research (TIFR), Mumbai, India. He had postdoctoral positions at University of Toronto, Canada and TIFR, Mumbai, India and visiting assistant professorships at Purdue University and University of Iowa, USA. He joined Oklahoma State University, USA in 2006 as a tenure-track assistant professor and was promoted to associate professor with tenure in 2011.Raghuram joined IISER Pune in December 2011 as a Professor and the Coordinator of Mathematics. His research has been partially funded by the National Science Foundation, USA, and the Alexander von Humboldt Foundation, Germany.

NUMBER THEORY, REPRESENTATION THEORY AND AUTOMORPHIC FORMS

My group is currently studying the arithmetic properties of special values of automorphic L-functions.

The earliest prototype of a special value of an L-function is the classical formula by Euler which says that the sum  1/n^2 of reciprocals of squares of all positive integers is ^2/6. More generally, suppose M = {a_n} is a sequence of numbers coming from some interesting data, for example, a_p can be the number of solutions of an equation modulo a prime p, then a guiding principle in modern number theory says that to study the sequence M one should study the L-function L(s, M) =  a_n/n^s. One can glean much information about the sequence M by studying first the analytic properties, and second the arithmetic properties of L(s,M).

The Langlands program, considered by many as a grand unifying principle in modern mathematics, bridges different areas of mathematics, like geometry (elliptic curves), number theory (Galois representations) and representation theory (automorphic forms). The central theme making these bridges possible is the notion of an L-function.

Our work uses the results and techniques of the Langlands program to prove theorems about special values of various L- functions. These values encode within them a lot of arithmetic and geometric information of the objects to which the L- functions are attached. In earlier work stemming from my thesis, we have also studied the representation theory and harmonic analysis of p-adic groups.

SELECTED PUBLICATIONS AND EDITORIAL WORK Bhagwat, C., and Raghuram, A. Special values for L-functions for orthogonal groups. To appear in C. R. Math. Acad. Sci. Paris. Balasubramanyam, B., and Raghuram, A. Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field. American Journal of Mathematics (To appear). Bhagwat, C., and Raghuram, A. Endoscopy and Cohomology of GL(n). To appear in special volume of the Bulletin of Iranian Mathematical Society dedicated to Freydoon Shahidi. Raghuram, A. (2016). Critical values for Rankin-Selberg L-functions for GL(n) x GL(n-1) and the symmetric cube L-functions for GL (2). Forum Mathematicum 28(3):457-489. Grobner, H. and Raghuram, A. (2014). On the arithmetic of Shalika models and the critical values of L-functions for GL(2n). With an appendix by Wee Teck Gan. American Journal of Mathematics 136:675-728. Bhagwat, C., and Raghuram, A. (2013). Ratios of periods of tensor product motives. Mathematical Research Letters 20(4):615-628. Gan, W.T. and Raghuram, A. (2013). Arithmeticity for periods of automorphic forms. 187-229, Tata Inst. Fund. Res. Stu. Math. No. 22, TIFR, Mumbai. Harder, G. and Raghuram, A. (2011). Eisenstein cohomology and ratios of critical values of Rankin-Selberg L-functions. C. R. Acad. Sci. Paris Ser. I 349:719-724.

SOUMEN MAITY Associate Professor [email protected]

Soumen Maity received a PhD from the Theoretical Statistics & Mathematics Unit at Indian Statistical Institute, Kolkata, India in 2002. He has postdoctoral experience from Lund University Sweden; Indian Institute of Management (IIM) Kolkata, India; and University of Ottawa, Canada. Prior to joining IISER Pune in 2009, he was at Indian Institute of Technology (IIT) Guwahati and Indian Institute of Technology (IIT) Kharagpur, India.

DISCRETE MATHEMATICS - COVERING ARRAYS AND SOFTWARE TESTING

Covering arrays are combinatorial objects that have been successfully applied in the design of test suites for testing systems such as software, circuits and networks, where failures can be caused by the interaction between their parameters. The columns of a covering array provide a test suite for software testing. For most applications, it is desirable to construct covering arrays with minimum number of columns. We give an algebraic construction that improves some of the best known covering arrays of strength four with three symbols. We also construct strength three and strength four testing arrays with high coverage.

We perform a new generalization of covering arrays called covering arrays on 3-uniform hypergraphs. This is useful in situations in which some combinations of parameters do not interact; in these cases, we do not insist that these interactions to be tested, which allows reductions in the number of required test cases.

Finally, we consider four most extensively studied graph products in literature and give upper and lower bounds on the size of a covering array on product graph. We find families of graphs for which the size of a covering array on product graph achieves the lower bound.

SELECTED PUBLICATIONS Akhtar, Y., Maity, S., and Chandrasekharan, R.C. (2016) Generating test suites with high 3-way coverage for software testing, 2016 IEEE International Conference on Computer and Information Technology, IEEE Computer Society, 10-17. Akhtar, Y., Maity, S. and Chandrashekharan, R.C. (2015). Covering Arrays of Strength Four and Software Testing, Book Name: Mathematics and Computing, Springer Proceedings in Mathematics and Statistics 139: 391-398. Maity, S., Akhtar, Y., Chandrashekharan, R.C. and Colbourn J.C. (2017). Improved strength four covering arrays with three symbols (Submitted). Akhtar, Y. and Maity, S. (2017). Mixed covering arrays on 3-uniform hypergraphs. (Submitted) Maity, S. (2012). 3-Way software testing with budget constraints. IEICE Trans. Information & Systems E95-D (9): 2227-2231. Maity, S. and Nayak, A. and Ramsundar, S. (2007). Characterization, Testing and Reconfiguration of Faults in Mesh Networks, INTEGRATION, the VLSI Journal Vol. 40: 525-535.

2017 IISER Pune Math Book

STEVEN SPALLONE Associate Professor [email protected]

Steven Spallone received his PhD from the University of Chicago, USA in 2004. Afterwards he did postdoctoral work at the Max-Planck Institute in Bonn, at Purdue University, and at the University of Oklahoma, USA. He also visited the Tata Institute of Fundamental Research (TIFR) Mumbai, India. He joined the faculty of IISER Pune in July 2012.

PARITY QUESTIONS IN REPRESENTATION THEORY

Recently my research group has been studying determinants of representations, orthogonality of self-dual representations, and the question of which orthogonal representations have a spin structure.

SELECTED PUBLICATIONS With Arvind Ayyer and Amritanshu Prasad, Representations of symmetric groups with non-trivial determinant. J. Combin. Theory Ser. A, to appear. With Arvind Ayyer and Amritanshu Prasad (2015). Odd partitions in Young's Lattice, Sem. Lothar. Combin. 75:Art. B75g, 13pp. With Amritanshu Prasad and Pooja Singla (2015). Similarity of matrices over local rings of length two. Indiana Univ. Math. J. 64 (2):471-514.

SUPRIYA PISOLKAR Assistant Professor [email protected]

Supriya Pisolkar obtained her PhD degree from Harish Chandra Research Institute, Allahabad, India in 2010. She was a postdoctoral fellow at Tata Institute of Fundamental Research, Mumbai for about three years before joining IISER Pune in December 2013.

NUMBER THEORY, GALOIS COHOMOLOGY, ARITHMETIC ASPECTS OF SYMMETRIC SPACES

Broadly speaking, my area of research is number theory and I have been working on problems mostly related to local fields. My thesis deals with questions having a common theme of understanding behavior of norm maps in Galois extensions of local fields. As an application, I computed the Chow group of Zero-cycles of degree zero on Châtelet surfaces over local fields. This work relies on the work of Bloch, Swinnerton-Dyer, Colliot-Téllène, Sansuc.

In joint work with Dr A Hogadi, we proved a conjecture of Hesselholt which predicts the vanishing of the cohomology 1 group H (G(L/K), W(OL)) for a Galois extension of local fields L/K. This can be seen as an analogue of Hilbert theorem- 90. In the future, I would like to work on generalizations of this result.

Over the last couple of years, I have also been fascinated by arithmetic aspects of locally symmetric spaces which are special type of manifolds. The theory of locally symmetric spaces is a beautiful amalgamation of theory of Lie groups, algebraic groups, analysis, differential geometry, representation theory. In a joint work with Prof CS Rajan and Dr C Bhagwat we have studied questions related to commensurability type problems of these spaces.

Currently, I am interested in understanding the work of Gopal Prasad and Rapinchuk which establishes a connection between arithmetic and geometric aspects of these spaces, giving rise to series of questions in this area.

SELECTED PUBLICATIONS Pisolkar, S. and Rajan, C.S. (2016). On the splitting fields of generic elements in Zariski dense subgroups. Journal of Algebra 457:106-128. Bhagwat, C.S. and Pisolkar, S. (2016). On uniform lattices in real semisimple groups. Proceedings of the American Mathematical Society 144, no. 7, 3151–3156. Bhagwat, C., Pisolkar, S. and Rajan, C.S. (2014). Commensurability and representation equivalent arithmetic lattices. International Mathematical Research Notices International Mathematical Research Notices No. 8:2017–2036. 2017 IISER Pune Math Book

MANISH MISHRA Assistant Professor [email protected]

Manish Mishra did his B.Tech. in Materials & Metallurgical Engineering from Indian Institute of Technology (IIT) Kanpur. He then moved to mathematics. He got his Ph.D. in mathematics in 2013 from Purdue University, USA. He then held postdoctoral positions at the Hebrew University of Jerusalem, Isreal and the Heidelberg University, Germany. He joined IISER Pune in August 2016.

LOCAL LANGLANDS PROGRAM, REPRESENTATION

THEORY OF REDUCTIVE GROUPS OVER LOCAL FIELDS

One way of defining number theory is that it is the study of the existence of rational solutions of polynomials equations

defined over integers. This amounts to studying the absolute Galois group Γ of the field of rational numbers. The group

Γ is very large and is difficult even to write down particular elements of it. One studies it by studying its action on linear spaces, i.e., by studying its space of representations. The Langlands program is a scheme of organizing this vast information. It relates Galois representations with certain, more concretely defined, analytic objects called automorphic representations from which arithmetic information can be extracted. Important applications of the Langlands program to some of the famous problems in number theory include the Artin’s conjecture on L-functions, Fermat’s Last Theorem, the Sato-Tate conjecture, and the behavior of Hasse-Weil zeta functions.

The conjectures within the Langlands program are quite precise and technical. My current research is about the local Langlands program. The projects I am working on are about/related to – study of Hecke algebras associated to Bernstein blocks and the centers of these blocks, describing Langlands-Shelstad transfer via Hecke algebra morphisms, ABPS conjecture and the stable Bernstein center conjecture.

SELECTED PUBLICATIONS Manish, M. (2016). Signs of self-dual depth-zero supercuspidal representations. arXiv:1610.04149 Manish, M. (2016). Bernstein center of supercuspidal blocks. Journal für die reine und angewandte Mathematik doi: 10.1515/crelle-2016-0041. Manish, M., Mirko, R. (2016). Genericity under parahoric restriction, Manuscripta Mathematica doi: 10.1007/s00229-016-0864-9. Manish, M. (2015). Langlands parameters associated to special maximal parahoric spherical representations. Proceedings of the American Mathematical Society,143:1933-1941. Manish, M. (2015). A Galois side analogue of a theorem of Bernstein. Journal of the Ramanujan Mathematical Society 30(4):397-402. Manish, M. (2016). Generic representations in L-Packets, International Journal of Number Theory 12(6):1613-1624.

TEJAS KALELKAR Assistant Professor [email protected]

After completing his MSc from Indian Institute of Technology, Bombay, Tejas Kalelkar got his PhD from Indian Statistical Institute, Bangalore in 2010. He was then a postdoctoral fellow at Institute of Mathematical Sciences, Chennai and a Chauvenet Postdoctoral Fellow at the Washington University in St Louis, USA. He joined IISER Pune in December 2013.

LOW DIMENSIONAL TOPOLOGY

My area of research is low-dimensional topology. This is a very active area of research with several longstanding conjectures proved fairly recently, like Thurston’s Geometrization conjecture (which implies the Poincare conjecture) and the Virtual Fibering conjecture. Within low-dimensional topology I focus mainly on foliations, triangulations and Heegaard splittings of 3-dimensional manifolds.

A closed book looks like a solid 3-dimensional object, but on closer analysis is observed to consist of 2-dimensional pages stacked tightly together. Similarly, every 3-manifold can be built by stacking 2-dimensional surfaces tightly together into what is called a foliation. I am studying a special class of foliations called taut foliations which tell us useful topological properties of the 3-manifold.

On cutting open a 3-manifold along a special embedded surface, called the Heegaard-splitting surface, we end up with two simple pieces called handlebodies. Every closed 3-manifold has such a splitting surface which may not be unique. I am currently working on a structure for these splitting surfaces when a 3-manifold has infinitely many of them.

Every 3-manifold can be built by suitably sticking tetraheda together. Normal surfaces are surfaces embedded particularly `nicely' with respect to such a triangulation. I have proved several results using normal surfaces, such as a weak converse of Haken’s well-known result about normality of incompressible surfaces with respect to every triangulation of the 3-manifold.

SELECTED PUBLICATIONS Kalelkar, T. and Roberts, R. (2015). Taut foliations in surface bundles with multiple boundary components. Pacific Journal of Mathematics 273(2):257-275. Gadgil, S. and Kalelkar, T. (2013). A Chain complex and Quadrilaterals for normal surfaces. Rocky Mountain Journal of Mathematics, 43(2):479-487. Kalelkar, T. (2009). Incompressibility and normal minimal surfaces. Geometriae Dedicata 142(1):61-70. Kalelkar, T. (2008). Euler characteristic and quadrilaterals of normal surfaces. Proceedings of the Indian Academy of Sciences Mathematical Sciences 118(2):227-233.

2017 IISER Pune Math Book

UTTARA NAIK-NIMBALKAR Visiting Faculty [email protected]

Uttara Naik-Nimbalkar joined IISER in February 2015. She has about thirty years of teaching and research experience at the University of Pune with visiting positions at the Michigan State University, the University of Edinburgh and University of Waterloo.

STATISTICS

My research interests are in the areas of inference in stochastic processes, survival analysis reliability theory and statistics in finance. Currently I am working on two problems. (i) Modeling and analysis of statistical dependence: for example, in load sharing systems the failure of a component may reduce (increase) the lifetimes of the surviving components resulting in dependence between their lifetimes. (ii) Hypothesis testing problems in competing risks with missing causes of failure: the failure of a unit is caused by one of the several risks acting on it. It is of interest to know whether some risk dominates the others but sometimes the actual cause of failure for some units is missing.

SELECTED PUBLICATIONS Dewan, I. and Naik-Nimbalkar, U. On competing risks with masked failures. IMBIC volume "Mathematical and Statistical Applications in Biology, Engineering, Environment and Information Science" edited by Y. P. Chaubey et al., Springer (Accepted for Publication). Naik Nimbalkar, U. (2016). Likelihood, estimating functions and method of moments. Mathematics Student 85(1-2): 63-78. Sutar, S.S. and Naik-Nimbalkar, U.V. (2016). A model for k-out-of-m load sharing systems. Communications in Statistics - Theory and Methods 45(20):5946-5960. Sutar, S.S. and Naik-Nimbalkar, U.V. (2014). Accelerated failure time models for loadsharing systems. IEEE Transactions on Reliability 63(3):706- 714. Deshpande, J.V., Dewan, I. and Naik-Nimbalkar, U.V. (2010). A family of distributions to model load sharing systems. Journal of Statistical Planning and Inference 140(6):1441-1451. Lam, K.F., Deshpande, J.V., Lau, E.H., Naik-Nimbalkar, U.V., Yip, P.S. and Xu, Y. (2009). A test for constant fatality rate of an emerging epidemic, with applications to the SARS syndrome in Hong Kong and Beijing. Biometrics 64(3):869-876.

VIVEK MOHAN MALLICK Assistant Professor [email protected]

After completing BStat (1999) and MStat (2001) from Indian Statistical Institute, Vivek Mallick got his PhD from Tata Institute of Fundamental Research (TIFR) Mumbai, India in 2008. He has joined IISER Pune in 2012 after completing his postdoctoral research at the Institute of Mathematical Sciences in Chennai and Centre de Recerca Matematica in Barcelona, Spain.

ALGEBRAIC GEOMETRY

My research can be divided into three categories: intersection theory, derived categories, and T-varieties.

Intersection theory In intersection theory, one defines invariants of algebraically defined geometric spaces in terms of how the spaces lying on the given space intersect each other. One can study those invariants to determine deep geometric properties of the space.

Derived category Given a variety (a type of algebraically define geometric space), one can define another algebraic object called derived category. It is known that a lot of the geometric properties of the space translate to algebraic properties of the derived category.

T-variety While studying an object one many times first determine its group of symmetries. The same holds for varieties. When the group of symmetries contain a torus, one can, with some additional hypothesis describe a variety purely in terms of some combinatorial data. Currently I am working on such varieties.

SELECTED PUBLICATIONS Dubey, U.V. and Mallick, V.M. (2012). Reconstruction of a superscheme from its derived category. Journal of the Ramanujan Mathematical Society 27(4). Dubey, U.V., Mallick, V.M. (2012). Spectrum of some triangulated categories. Journal of Algebra 364:90-118. Mallick, V.M. (2009). Roitman's theorem for singular projective varieties in arbitrary characteristic. Journal of K-Theory 3:501-531.

2017 IISER Pune Math Book

POSTDOCTORAL FELLOWS

CHITRABHANU CHAUDHURI

Chitrabhanu did his PhD in Mathematics from Northwestern University 2007-2013 and was a postdoctoral fellow at Max Plank Institute for Mathematics (2013-2014) before joining IISER Pune. His research revolves around the topology and geometry of the Moduli of Curves over Complex numbers. The Moduli Space of Curves parameterize algebraic curves or Riemann surfaces, up to isomorphisms. It has been a central topic in mathematical research for several decades and brings various fields of mathematics together, for example Algebraic Geometry, Geometric Group theory and Teichmüller theory. It has important connections with physics as well. There are mainly two directions of research that Chitrabhanu has pursued: one roughly concerns finding some bounds on the topological complexity of certain partial compactifications on the Moduli of genus g curves. This involves computing certain cohomology groups; and the other line of research is about certain geometrically relevant subrings of the cohomology ring of the Moduli space known as the Tautological ring.

Chaudhuri, C. (2015). Equivariant cohomology of certain moduli of weighted pointed rational curves. Manuscripta Mathematica The Cohomological Excess of Certain Moduli Spaces of Curves of Genus g. International Mathematical Research Notices IMRN (2013). Topological Bounds on Certain open Subvarieties of the Compactified Moduli space of Curves; PhD thesis, Northwestern University 2013.

DILPREET KAUR

Dilpreet obtained PhD from IISER Mohali in April 2015. She served as Asisstant Professor in Symbiosis Institute of Technology, Pune for one year after completing her PhD and joined IISER Pune in June 2016 as NBHM- Post Doc Fellow. Her area of interest is Group Theory; in particular, conjugacy classes and representations of finite groups. In her PhD she studied properties of conjugacy classes and representations of special 2-groups. Currently, she is working on classifying z- classes of some well known groups. She is also interested in classifying groups whose all irreducible represenatations of degree greater then 1 are of symplectic type.

Dilpreet, K. and Amit K. (2015). strongly real special 2-groups. Communications in Algebra 43(3). Dilpreet, K. and Amit K. (2015). Character table of real special 2-groups, J. Ramanujan Math. Soc. 30(4). Dilpreet, K. and Amit K. (2015). Quadratic forms in characteristic 2 and a Wedderburn decomposition over rationals. (submitted).

POSTDOCTORAL FELLOWS

DIPRAMIT MAJUMDAR

Dipramit Majumdar obtained his PhD from Brandeis University, USA in 2013. He was a visiting scientist in ISI Bangalore from 2013-2014 before joining IISER Pune in 2014. Dipramit's research interest lies in number theory. Even though he is sometimes intrigued by elementary number theoretic problems such as Class Groups and Diophantine equations, his main research interest lies in p-adic families of modular forms.

Dipramit was at IISER Pune till August 2016 and is currently a faculty member at the Math department of IIT Madras.

Majumdar, D. Endoscopic transfer between Eigenvarieties for definite Unitary groups. Journal of Ramanujan Matematical Society (To appear). Baskar, B. and Majumdar, D. Ordinary families of automorphic forms on definite unitary group Conference Proceedings of p- adic Aspects of Modular Forms (To appear) Jha, S. and Majumdar, D. Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms. Asian Journal of Mathematics (To appear) Majumdar, D. (2015). Geometry of the eigencurve at critical Eisenstein series of weight 2. Journal de Théorie des Nombres de Bordeaux 27(1):183-197. Kulkarni, M., Majumdar, D. and Sury, B. (2015). l-Class groups of cyclic extensions of prime degree l. Journal of Ramanujan Matematical Society 30(4):413-454.

KRISHNA KISHORE

Krishna Kishore obtained his PhD from Indiana University-Bloomington, USA, in December 2014. After his graduation he joined IISER Pune from August 2015. His research interests include algebraic groups, and representation theory. In particular, he works on the questions related to the notion of representation variety, which is the variety of homomorphisms from discrete subgroups of isometries of hyperbolic plane also referred to as Fuchsian groups. Algebraic invariants associated to such an object are of interest in the study of moduli spaces. His work is focused on estimating the dimension of representation variety of Fuchsian groups in semisimple real algebraic groups using the deformation theory of Weil.

Krishna Kishore, Representation variety of Fuchsian groups in SO (p, q) (submitted). 2017 IISER Pune Math Book

POSTDOCTORAL FELLOWS

RATNA PAL

Ratna completed her MSc in Mathematics from IIT Kanpur in 2010 and completed PhD from IISc, Bangalore in 2016. She has joined IISER Pune as post-doctoral research fellow in December, 2016. Her research interest lies in Dynamics in Several Complex Variables.

With Kaushal Verma: Dynamical properties of families of holomorphic mappings. Conform. Geom. Dyn. 19 (2015), 323–350. With Sayani Bera (2016). Dynamics of semigroups of entire maps of $mathbb{C}^k$. Complex Anal. Synerg. 2(1): Paper No. 2, 22 pp. 32H02 (32H50).

ROHIT DILIP HOLKAR

Rohit worked in the University of Gottingen in Germany during 2010-2014 to receive his PhD. Then, before joining IISER Pune, during 2014-16, he was a postdoctoral fellow in the Federal University of Santa Caterina, Florianopolis, Brazil. Rohit works in operator algebras and C*- dynamical systems. The equivalence of the category of compact Hausdorff spaces and commutative unital C*-algebras is a well-known result in modern analysis, and so is the one that establishes the equivalence between commutative von Neumann algebras and nice measure spaces. This analysis-geometry interplay is taken further by the “Noncommutative Geometry” of Alain Connes. Rohit's interest is the interplay of analysis and topology. He uses C*-algebras and topological groupoids to study the interplay. Presently, he is investigating the topological analogue of an algebraic notion called proper C*-correspondence.

Holkar, R.D. (2017). Composition of topological correspondences. Journal of Operator Theory 77(23-24):217-241. Holkar, R.D. (2015). Topological construction of C*- correspondences for groupoid C*- algebras. Journal of Operator Theory 77:1:217-241. Holkar, R.D. and Renault, J. (2013). Hypergroupoids and C*- algebras.C. R. Math. Acad. Sci. Paris 3513(22-24):911-914. Alcides, B., Holkar, R.D. and Ralf, M. (2016). A universal property for groupoid C*- algebras. arxiv:1612.04963 (Submitted).

SHANE D’MELLO

Shane did his PhD in Mathematics at Stony Brook University from 2008 to the end of 2013, under the guidance of Prof Oleg Viro. Subsequently, he was a post-doctorate at the Tata Institute of Fundamental Research until he joined IISER Pune as a post-doctorate in August 2015. Shane is interested in the topology of real algebraic varieties, which is the study of the topology of objects that can be described real algebraically, for example as the real zero set of a real polynomial. Such objects may arise as surfaces, planar curves, curves in \mathbb{RP}^3$, or curves in some other real algebraic variety.

POSTDOCTORAL FELLOWS

The algebraic structure imposes restrictions on the topology and classifying such objects for low degrees has been the main motivating problem, but in attempting to solve them, many other interesting questions have emerged along the way. Shane's main focus has been on curves, especially M-curves, rational real algebraic plane curves, and real algebraic knots. He is currently exploring some of its connections with classical knot theory. He has also worked a bit on enumerative geometry.

D'Mello, S. Classification of real rational knots of low degree in the 3-sphere. Journal of Knot Theory and its Ramifications (Accepted for Publication). D'Mello, S.; Mishra, R. Constructing Real Algebraic Knots via Gluing (Submitted). Biswas, I., D'Mello, S., Mukherjee, R., Pingali, V. Rational cuspidal curves on del-Pezzo surfaces (Submitted). D'Mello, S. (2015) Chord Diagrams and Generic Real Rational Planar Curves of Degree 4. International Mathematics Research Notices 22:1485-11507. Biswas, I., D'Mello, S.(2015) A criterion for M-curves. Journal of the Ramanujan Mathematical Society30(4):403-411. Biswas, I., D'Mello, S. M-curves and symmetric product. Proceedings - Mathematical Sciences, Indian Academy of Sciences (Accepted).

TIAN AN WONG

Tian An was raised in Malaysia, obtained his PhD from the City University of New York, Graduate Center in 2016, and was a postdoctoral fellow at the Max Planck Institute for Mathematics, Bonn before joining IISER Pune in February 2017. He is generally interested in problems that are number theoretic in nature, but draw from various other fields of research, for example homotopy theory, derived algebraic geometry, symmetric spaces, knot theory, and so on. Come by and talk to him about anything!

On his own, he likes to think about problems 'Beyond Endoscopy', a kind of wild west in the Langlands Program. The Langlands program is a major research area relating various aspects of arithmetic geometry, number theory, and representation theory. The theme of Beyond Endoscopy is a method proposed by Langlands of solving the Functoriality Conjecture, one of the two central problems in the field. It involves a blend of analytic theory of L-functions and harmonic analysis on real and p-adic groups, the tool of choice being the Arthur-Selberg trace formula.

Flórez, J., and Karabulut, C., Eisenstein cocycles for GL(n) and special values of L-functions in imaginary quadratic fields (Submitted). González, O., Kwan, C.H., Miller, S.J., van Peski, R., and Wong, T.A., On smoothing singularities of elliptic orbital integrals and beyond endoscopy (Submitted). Wong, T.A., Lower bounds for Weil's explicit formula via Selberg's trace formula (Submitted).

2017 IISER Pune Math Book

POSTDOCTORAL FELLOWS

UDAY BHASKAR SHARMA

Uday completed his PhD from the Institute of Mathematical Sciences in August 2016 and joined IISER Pune in January 2017 as a Postdoctoral Research Fellow. His areas of interest are combinatorics and representation theory of finite groups. In his PhD he studied matrices over finite fields in particular and enumerated the simultaneous similarity classes of tuples of commuting matrices over a finite field, which involves using similarity class types (also known as z-classes). Currently, he is working on the problem of enumerating classes of tuples of commuting matrices of some well know matrix groups, and also on two-generator problems of some well known groups. He is also interested in character tables of groups of invertible 2x2 matrices over local rings of finite length.

Uday, B. S. (2016). Simultaneous similarity classes of commuting matrices over a finite field. Linear Algebra and its Applications 501. Uday, B. S. (2016). Asymptotic of Number of Similarity Classes of Commuting Tuples. Journal of the Ramanujan Mathematical Society 31(4).

GRADUATE STUDENTS

ADVAIT PHANSE GIRISH KULKARNI PhD Student PhD Student

Advait has completed MSc from Sir Girish completed MSc Mathematics from IIT Parshurambhau College in July 2014 and Madras in 2010 and MSc from University of joined IISER Pune in August 2014. His Western Ontario, Canada in 2012. Girish research interest lies in manifold theory. He joined IISER Pune in 2013. He presented a is currently working toward PhD with the result in The International Conference of guidance of Dr Tejas Kalelkar. the Indian Mathematical Consortium, Varnasi in December 2016, this is a joint work with his advisor Dr. Amit Hogadi. AYESHA FATIMA PhD Student GUNJA SACHDEVA Ayesha completed her BS-MS from IISER PhD Student Pune in 2012 before joining the PhD program in August 2013. She is currently Gunja completed MPhil in Mathematics working toward PhD with the guidance of from Dayalbagh Educational Institute, Dr Chandrasheel Bhagwat. Her interest lies Deemed University, Agra in December 2011 in Number Theory and Representation and joined IISER Pune in August 2012. She Theory. is currently working toward PhD with the guidance of Prof A Raghuram in the area of Number Theory. CHAITANYA AMBI PhD Student JYOTIRMOY GANGULY Chaitanya completed his MSc from IIT PhD Student Kharagpur in 2009. He joined IISER Pune in August 2013 as a Project Jyotirmoy completed MSc in Mathematics Fellow. Currently, he is a PhD student from University of Hyderabad in 2013 and studying with Prof A Raghuram. joined IISER Pune soon after. He is currently working with Dr Steven Spallone in the area of Representation Theory of DEBANGANA MUKHERJEE Symmetric groups. PhD Student

Debangana completed MSc in Mathematics MAKARAND SARNOBAT from Mount Carmel College, Bangalore in PhD Student 2013 and joined IISER Pune soon after. She is currently working with Dr Mousomi Makarand completed MSc in Mathematics Bhakta in the area of semilinear elliptic from Sir Parshurambhau College, Pune in partial differential equations. 2012 and joined IISER Pune immediately after that. He is working with Prof Raghuram toward PhD in the area of algebraic number theory and representations theory.

2017 IISER Pune Math Book

GRADUATE STUDENTS

Makarand has been a teaching assistant for these courses: Introduction to Proofs; NEHA PRABHU Linear Algebra; Multivariable Calculus; and PhD Student Complex Analysis. Neha Prabhu joined IISER Pune in August 2012. She will soon be completing her PhD MANIDIPA PAL under the guidance of Dr. Kaneenika Sinha. PhD Student Her interests lie in the theory of modular forms, elliptic curves and related areas in Manidipa joined IISER Pune in January 2012. analytic number theory. In her thesis she She has completed course work, applies techniques from harmonic analysis comprehensive exams and minor thesis, to study the asymptotic distribution of which are in partial fulfillment of the PhD eigenvalues of Hecke operators acting on program. Her minor thesis was carried out spaces of modular cusp forms and Maass with guidance from Dr Chandrasheel cusp forms. Bhagwat. Manidipa is presently working with Dr Baskar Balasubramanyam in the area of algebraic number theory. PRABHAT KUSHWAHA PhD Student

MILAN KUMAR DAS Prabhat completed MSc from the PhD Student Department of Mathematics, Banaras Hindu University, Varanasi and joined IISER Pune Milan completed MSc from Vidyasagar in January 2012. He has successfully University, West Bengal in 2012. He joined completed all the requirements by the IISER Pune in August 2013. Milan is working Institute viz. course work, comprehensive with Dr Anindya Goswami on stochastic exams and minor-thesis. Prabhat is control. currently working with Dr Ayan Mahalanobis in the research area of public key cryptography. NEERAJ DESHMUKH PhD Student PRALHAD SHINDE Neeraj completed his BS-MS from IISER PhD Student Mohali in 2015 and, subsequently, joined

IISER Pune. Broadly, his interests lie in Pralhad completed Masters in Mathematics Geometry - algebraic and differential. He is in 2011 from University of Pune, Pune and working with Dr. Amit Hogadi towards a joined IISER Pune in January 2012. He is PhD in the area of Algebraic Stacks and working with Dr Ayan Mahalanobis and Dr Spaces. Anupam Singh in the areas of cryptography

and computational group theory.

GRADUATE STUDENTS

of the graduate program. Sushil is presently ROHIT JOSHI working with Dr Anupam Singh on algebraic PhD Student groups.

Rohit has a Master’s degree from IIT Kanpur. He has completed course work and TATHAGATA MANDAL cleared comprehensive exams toward IISER PhD Student Pune’s PhD program in Mathematics and is currently working with Dr Steven Spallone. Tathagata completed his MSc in He was the winner of silver medal in Mathematics from Visva-Bharati University, International Mathematics Olympiad in Santiniketan, India in 2012. He joined IISER 2004 while in his 12th standard and was Pune in August 2013. He is interested in chosen as the IISER Pune Mathematics Number Theory and Algebraic Geometry Student of the Year in 2013. and is presently trying to understand the endomorphism algebras of modular motives under the supervision of Dr SOUMYADIP THANDAR Debargha Banerjee. PhD Student

Soumyadip has completed his Msc (in PHD STUDENTS WHO JOINED Mathematics) from University of Pune in 2015 and joined IISER Pune the same year. IN AUGUST 2016 Soumyadip was part of the IISER Pune Math PhD program until Fall 2016. BASUDEV PATTANAYAK PhD Student

SUPRITA TALNIKAR KARTIK ROY PhD Student PhD Student

Suprita joined the PhD program at IISER RAMYA NAIR Pune in August 2015. She has a gold medal PhD Student in MSc (Mathematics) from Visvesvaraya National Institute of Technology, Nagpur in Ramya completed her Master’s degree from 2015, and has completed BSc from RTM Mumbai University in 2015 (was the gold Nagpur University in 2013. Suprita was a medalist of her batch) and joined IISER part of the Mathematics discipline until July Pune in 2016. She is currently doing her 2016. coursework and is interested in topology and algebra.

SUSHIL BHUNIA RIJUBRATA KUNDU PhD Student PhD Student

Sushil completed MSc (Mathematics) in Rijubrata completed his MSc from IIT 2010 from Jadavpur University, Kolkata, Guwahati and joined IISER Pune as a India. He joined IISER Pune in January 2012 research fellow in August 2016. He is and completed coursework along with currently doing coursework and a reading comprehensive examinations that are part 2017 IISER Pune Math Book

GRADUATE STUDENTS

project under the guidance of Dr. Anupam fulfilling the coursework requirements that Singh. He is interested in the research areas are a part of the graduate program. research of Group theory and Representation theory. GARIMA AGARWAL Integrated PhD Student SOUPTIK CHAKRABORTY Garima completed her BSc (Hons) PhD Student Mathematics from University of Delhi in 2016 and joined IISER Pune as an Integrated PhD student the same year. SUDIPA MONDAL She is currently completing the PhD Student coursework requirements as part of Sudipa joined IISER Pune in August, 2016 the Integrated PhD program. after completing MSc from IIT Kanpur. She is a doing reading project under the supervision Dr. Chandrasheel Bhagwat. She SUBHAM DE is interested in Representation theory. Integrated PhD Student

SURAJPRAKASH YADAV SHUVAMKANT TRIPATHI PhD Student Integrated PhD Student

DEBAPRASANNA KAR Integrated PhD Student

Debaprasanna has joined IISER Pune Integrated PhD program in Mathematics in August 2015 after graduating from the Institute of Mathematics & Applications under Utkal University, Odisha. He is presently going through the coursework of Integrated PhD program and is interested in the research area of algebra.

NEHA MALIK Integrated PhD Student

Neha completed her BSc (Hons) in Mathematics from University of Delhi in 2015 and joined IISER Pune as an Integrated PhD student same year. She has completed her Masters' project on representation theory under the guidance of Dr. Steven Spallone. Currently, she is

BS-MS FIFTH YEAR STUDENTS

DILEEP ALLA AJITH NAIR BS-MS Student BS-MS Student Dileep is working for his Master’s thesis Ajith is a fifth-year BS-MS student working under the guidance of Dr. Kaneenika on harmonic analysis and representation Sinha. His interests include Number theory under the guidance of Dr. Theory and Discrete Mathematics. The Chandrasheel Bhagwat. goal of his 5th year project is to study the distribution of eigenvalues of adjacency matrices of regular graphs using techniques in equidistribution CHRIS JOHN theory. BS-MS Student OMKAR MANJAREKAR Chris is working with Prof. Tejas Kalelkar on BS-MS Student geometry and topology of 3-manifolds. Omkar is working under the guidance of Dr. Anindya Goswami on the problem of option pricing in a Regime Switching Jump Diffusion Model.

DEBARUN GHOSH BS-MS Student PAPIA BERA BS-MS Student Debarun is workingunder the guidance of Dr. Steven Spallone on the Determinants Papia is currently working with Prof. A. of Representations of Hyperoctahedral Raghuram on Representation Theory for groups. her fifth year thesis project. The aim of the project is to explicitly write the Plancherel Formula for GL (2) over a p- adic field. DEEKSHA ADIL BS-MS Student SHIPRA KUMAR Deeksha is doing her 5th year thesis BS-MS Student under Prof. Saket Saurabh from the Institute of Mathematical Sciences, Shipra is a BS-MS student working with Chennai. Her thesis involves finding Dr. Amit Mitra at IIT, Kanpur. Her work efficient parameterized algorithms for involves forecasting exchange rates some NP-hard variants of the classical using concepts in time series analysis and machine learning like neural Stable Marriage problem. networks and deep learning algorithms. Her aim to is to build a hybrid model combining the methods learns for a superior forecast in comparison to the existing methods.

2017 IISER Pune Math Book

BS-MS FIFTH YEAR STUDENTS

SIDHARTH VISAKH NARAYANAN BS-MS Student BS-MS Student

Visakh is working with Dr. Rama Mishra VARUN PRASAD on Functorial knot theory, broadly is the BS-MS Student study of a "categorification" of knot theory (Khovanov homology) and a knot Varun is currently working with Dr. Amit theoretic characterization of certain Hogadi on a computational foundation coherence theorems (for braided of mathematics called the Univalent monoidal categories). He is interested Foundations. It is a new emerging field in Topology, Homology theory and at the intersection of mathematics, logic foundations of mathematics. and philosophy and theoretical computer science. One of the key aims of this project is to develop a software which checks mathematical proofs for correctness. He is also pursuing an extended project at the interface of mathematics and neurobiology in collaboration with Prof. Semir Zeki, who is a neurobiologist at the University College, London.

TEACHING ASSISTANTS AND PROJECT FELLOWS

AMAN JHINGA SUCHETA DAWN Teaching Assistant Teaching Assistant

Aman has completed BSc (Hons) Maths Sucheta completed her BSc in and Computing and MSc (Mathematics) Mathematics from St Xavier's College, in June 2012 from Panjab University, Kolkata in 2013 and MSc in Chandigarh and is presently assisting Mathematics from University of Pune in with teaching at IISER Pune. 2015. Sucheta was a part of the Mathematics discipline till Summer 2016. ONKAR KALE Teaching Assistant PALLAVI KHARADE Onkar has completed B.Sc. in 2014 from Project Fellow Fergusson College and M.Sc.in 2016 from Department of Mathematics, Pallavi did her BSc (Computer University of Pune. Science) from Fergusson college and MSc (Scientific Computing) from University of Pune. She joined as a SNEHA JONDHALE Project Fellow working with Ayan Teaching Assistant Mahalanobis and was part of the Math discipline till August 2016. Sneha has completed BSc from PVP College, Pravaranagar in 2013 and MSc from University of Pune in 2015. She is SHASHIKANT GHANWAT presently assisting with teaching at IISER Teaching Assistant Pune.

Shashikant completed BSc from T.C. College Baramati in 2013 and MSc UDAY JAGADALE from University of Pune in 2016. He Teaching Assistant is presently assisting with teaching at IISER Pune.

ABHINAV SAHANI

Teaching Assistant 2017 IISER Pune Math Book

THE FIRST SET OF PHD STUDENTS TO GRADUATE FROM IISER PUNE MATH DISCIPLINE

HITESH RAUNDAL PhD Student

Hitesh obtained MSc in Applied Mathematics from University of Hyderabad and joined IISER Pune in January 2011. Here, he worked with Dr Rama Mishra in the area of knot theory and defended his thesis on January 25, 2017. Hitesh is currently in a postdoctoral position at HRI, Allahabad.

SUDHIR PUJAHARI PhD Student

Sudhir’s research interests include modular forms, Eichler-Selberg trace formula, Maass Forms, Kujnetsov trace formula, L-functions, equidistribution theory, extremal functions in fourier analysis. He worked with Dr Kaneenika Sinha towards his PhD and is currently holding a postdoctoral position at HRI, Allahabad. Sudhir defended his PhD thesis on November 11, 2016.

YASMEEN AKHTAR PhD Student

Yasmeen’s primary field of interest is combinatorics and graph theory. She worked with Dr Soumen Maity and the title of her thesis is Covering Arrays and Generalizations. Yasmeen defended her thesis on October 3, 2016 and is currently holding a postdoctoral position at Academia Sinica(ISSAS) in Taiwan.

SUPPORT PERSONNEL

KALPESH PEDNEKAR Office Assistant

Kalpesh Pednekar completed his Bachelor Degree in Commerce from University of Mumbai. He joined IISER Pune in March 2013 and is presently assisting Mathematics group in various administrative responsibilities in addition to assisting the Physics Group at IISER Pune.

SUVARNA GHARAT Office Assistant

Suvarna completed her BSc in Physics from Fergusson College and her post-graduation from Modern College, Pune. She joined IISER Pune in July 2011 and is presently assisting the Mathematics group in various administrative responsibilities.

2017 IISER Pune Math Book

PUBLICATIONS

2016

1. Akhtar, Y., Maity, S., Chandrasekharan, R.C., and Colbourn, C.J., Improved strength four covering arrays with three symbols (submitted) 2. Akhtar, Y. and Maity, S., Mixed covering arrays on 3-uniform hypergraphs. https://arxiv.org/abs/1508.07393 (submitted) 3. Akhtar, Y. and Maity, S., Covering arrays on product graphs. https://arxiv.org/abs/1512.06966 (submitted) 4. Akhtar, Y., Maity, S., and Chandrasekharan, R.C., Generating test suites with high 3-way coverage for software testing, 2016 IEEE International Conference on Computer and Information Technology, IEEE Computer Society, 10-17, 2016. 5. Akhtar, Y. and Maity, S., Covering arrays on product graphs. In 47th Southeastern International Conference on Combinatorics, Graph theory and Computing. Boca Raton, FL. March 2016. 6. Baskar Balasubramanyam, Dipramit Majumdar. Ordinary families of automorphic forms on definite unitary groups, “p- adic aspects of modular forms”, World Scientific. Proceedings of a workshop held in IISER Pune in June 2014. 7. Baskar Balasubramanyam, p-adic distributions attached to twisted tensor L-functions for GL_2 over CM fields. To appear in J. Number Theory. 8. Banerjee, Debargha and Srilakshmi Krishnamoorty (2016). The Eisenstein elements of modular symbols for level product of two distinct odd primes. Pacific Journal of Mathematics. 281(2):257-285. 9. Banerjee, D. and Merel L., The Eisnestein cycles inside the space of modular symbols (submitted) 10. Banerjee, D. and Raghuram, A., p-adic L-functions for GL_n, “p-adic aspects of modular forms”, World Scientific, Proceedings of the workshop held at IISER Pune in June, 2014. 11. R. Basu Local-global principle for the general quadratic and the generalHermitian groups and the nilpotence of KH1. (To appear in the Journal of Mathematical Sciences, Springer). 12. Bhagwat, Chandrasheel; Pisolkar, Supriya On uniform lattices in real semisimple groups. Proc. Amer. Math. Soc. 144 (2016), no. 7, 3151–3156. 13. Bhagwat, C. and Raghuram, A. Endoscopy and Cohomology of GL(N). (Accepted for publication in the special issue of the BIMS in honor of Professor Freydoon Shahidi’s 70th birthday.) 14. Bhakta, M; Caffarelli-Kohn-Nirenberg type equations of fourth order with the critical exponent and Rellich potential. J. Math. Anal. Appl. 433 (2016), no. 1, 681–700. 15. Bhakta, M; Semilinear elliptic PDE’s with biharmonic operator and a singular potential, Electron. J. Differential Equations, Vol. 2016 (2016), No. 261. 16. Bhakta, M; Infinitely many sign changing solutions of an elliptic problem involving critical Sobolev and Hardy-Sobolev exponents, accepted in Proc. Indian Acad. Sci. Math. Sci. 17. Bhakta, M; Mukherjee, D; Multiplicity results and sign changing solutions of non local equations with concave-convex nonlinearities, accepted in Differential and Integral Equations. 18. Bhakta, M; Santra, S; On singular equations with critical and supercritical exponents (submitted). 19. Bhakta, M; Mukherjee, D; Semilinear nonlocal elliptic equations with critical and supercritical exponents (submitted). 20. Bhakta, M; Mukherjee, D; Santra, S; Profile of solutions for nonlocal equations with critical and supercritical nonlinearities (submitted). 21. Sushil Bhunia, Ayan Mahalanobis and Anupam Singh Gaussian Elimination in Symplectic and split orthogonal groups

PUBLICATIONS

(preprint) http://arxiv.org/abs/1504.03794 (submitted) 22. Bhunia, Sushil and Mahalanobis, Ayan and Shinde, Pralhad and Singh, Anupam TheMOR Cryptosystem in orthogonal and symplectic groups in odd characteristic (submitted) 23. Ari Arapostathis, Anup Biswas, Luis Caffarelli. The Dirichlet problem for stable-like operators and related probabilistic representations, Comm. Partial Differential Equations 41 (2016), No. 9, 1472--1511. 24. Borah, Diganta, Haridas, Pranav, Verma, Kaushal Comments on the Green's function of a planar domain (submitted) 25. Biswas, I.; D'Mello, S. “M-curves and symmetric product” Proceedings - Mathematical Sciences, Indian Academy of Sciences (Accepted) 26. D'Mello, S. Classification of real rational knots of low degree in the 3-sphere. Journal of Knot Theory and its Ramifications (Accepted) 27. D'Mello, S.; R. Mishra. Constructing Real Algebraic Knots via Gluing (submitted) 28. Biswas, I.; D'Mello, S; Mukherjee, R; Pingali, V. “Rational cuspidal curves on del-Pezzo surfaces” (submitted) 29. Milan Kumar Das, Anindya Goswami, Tanmay Patankar, Pricing derivatives in a regime switching market with time inhomogeneous volatility (submitted) 30. Milan Kumar Das, Anindya Goswami, Nimit Rana Risk Sensitive Portfolio Optimization in a Jump Diffusion Model with Regimes (submitted) 31. Anindya Goswami, Sanket Nandan Convergence of estimated option price in a regime switching market, Indian J. Pure Appl. Math. 47, 169 - 182 (2016) DOI: 10.1007/s13226-016-0182-7 32. Anindya Goswami, Jeeten Patel, Poorva Shevgaonkar A system of non-local parabolic PDE and application to option pricing Stoch. Ann. App. 34, 893 - 905 (2016) DOI: 10.1080/07362994.2016.1189340 33. Rohit Dilip Holkar, Topological construction of C*-correspondences for groupoid C*-algebras, available at arxiv:1510.07534. Journal of Operator Theory (Accepted for Publication). 34. Alcides Buss, Rohit Dilip Holkar, and Ralf Meyer, A universal property for groupoid C*-algebras, available at arxiv:1612.04963 (submitted) 35. Rohit Dilip Holkar., Composition of topological correspondences, to appear in the Journal of Operator Theory, record nr. 2129, (2015), Preprint- arxiv:1510.08581. 36. Cardinali Ilaria, Giuzzi Luca, Kaipa Krishna, Pasini Antonio, Line polar Grassmann codes of orthogonal type, J. Pure Appl. Algebra 220 (2016), no. 5, 1924–1934. 37. Kaipa Krishna, Deep holes and MDS extensions of Reed-Solomon codes (submitted). 38. Krishna Kishore Representation Variety of Fuchsian Groups in SO (p; q) Geometriae Dedicata, Volume 184 (2016), Issue 1, pp 193–209 39. Prabhat Kushwaha and Ayan Mahalanobis A probabilistic baby-step giant-stepalgorithm https://arxiv.org/abs/1701.07172 (submitted) 40. Mahalanobis, Ayan and Singh Anupam Gaussian elimination in unitary groups with an application to cryptography (to appear) Journal of algebra, combinatorics, discrete structures and applications http://arxiv.org/abs/1409.6136 41. Ansari Abdullah, Hardik Gajera and Ayan Mahalanobis On improvements of the r-adding walk in a finite field of characteristic 2, Journal of Discrete MathematicalSciences and Cryptography 19(1)13-38, 2016.

2017 IISER Pune Math Book

PUBLICATIONS

42. Majumdar, D. Endoscopic transfer between Eigenvarieties for definite Unitary groups. Journal of Ramanujan Mathematical Society (To appear). 43. Manish Mishra Signs of self-dual depth-zero supercuspidal representations, arXiv:1610.04149 (Submitted). 44. Manish Mishra Generic representations in L-Packets, International Journal of Number Theory, Vol. 12, No. 6 (2016) 1613-1624. 45. Manish Mishra Bernstein center of supercuspidal blocks, Journal für die reine und angewandte Mathematik (Crelle’s Journal), (2016), DOI: 10.1515/crelle-2016-0041. 46. Manish Mishra and Mirko Rösner Genericity under parahoric restriction, Manuscripta Mathematica, (2016), DOI: 10.1007/s00229-016-0864-9. 47. Rama Mishra and Hitesh Raundal Spaces of Polynomial knots in low degree, Journal of knot theory and its Ramifications, Volume 24, Issue 14, December 2016 48. Rama Mishra Jones polynomial and Khovanov Homology of Weaving Knots W (3, n), Joint with Ross Stafeldt (submitted) 49. Uttara Naik-Nimbalkar (2016). Likelihood, estimating functions and method of moments. Mathematics Student 85(1-2): 63-78. 50. Santosh Sutar and Uttara Naik-Nimbalkar (2016). A model for k-out-of-m load sharing systems. Communications in Statistics - Theory and Methods 45(20):5946-5960. 51. Pisolkar, S. and Rajan, C.S. (2016). On the splitting fields of generic elements in Zariski dense subgroups. Journal of Algebra 457:106-128. 52. Prabhu, Neha Density of solutions to quadratic congruences, Czechoslovak Mathematical Journal (accepted) 53. Prabhu, Neha and Sinha, Kaneenika, Fluctuations in the distribution of Hecke eigenvalues about the Sato-Tate measure (submitted). 54. Pujahari, S. Distribution of gaps of equidistributed sequences (Submitted). 55. Spallone, Steven Residues of intertwining operators for Sp (6): the hyperbolic term vanishes, Prime Numbers and Representation Theory, Lecture Series of Modern Number Theory vol. 2, Science Press, Beijing, 2016. 56. Spallone, Steven Representations of Symmetric Groups with Non-trivial Determinant (with Arvind Ayyer and Amritanshu Prasad), J. Combin. Theory Ser. A (To appear).

PUBLICATIONS

2017 57. Bhagwat, C. and Raghuram, A. Special Values of L-functions for Orthogonal Groups, Comptes rendus - Mathématique, 10.1016/j.crma.2017.01.016 58. Anisa M H Chorwadwala, “A glimpse of Shape Optimization Problems” Current Science (Accepted for Publication). 59. Ari Arapostathis, Anup Biswas, Johnson Carroll. On solutions of mean field games with ergodic cost, Journal de Mathematiques Pures et Appliqees 107 (2017), 205-251. 60. Anup Biswas, Hitoshi Ishii, Subhamay Saha, Lin Wang. On viscosity solution of HJB equations with state constraints and reflection control, SIAM J. Cont. and Optim. (To appear). 61. Rami Atar, Anup Biswas, Haya Kaspi, Kavita Ramanan, A Skorokhod Map on Measure-Valued Paths with Applications to Priority Queues, Annals of Applied Probability (accepted) 62. Borah, Diganta Remarks on the metric induced by the Robin function III JRMS 32(2017), 17-42 63. Beelen Peter, Glynn David, Høholdt Tom, Kaipa Krishna, Counting generalized Reed-Solomon codes, Accepted in January 2017 for publication in Advances in the Mathematics of Communication. 64. Kushwaha, Prabhat, Towards the Equivalence of Diffie-Hellman Problem and Discrete Logarithm Problem for Important Elliptic Curves Used in Practice, IEEE ISEA Asia Security & Privacy Conference 2017, 9--12 65. Rama Mishra and Hitesh Raundal Some spaces of polynomial knots, Topology and its Applications, Vol. 218, 2017, Pages 66–92. 66. Isha Dewan, and Uttara Naik-Nimbalkar. On competing risks with masked failures. IMBIC volume "Mathematical and Statistical Applications in Biology, Engineering, Environment and Information Science" edited by Y. P. Chaubey et al., Springer (Accepted for Publication).

2017 IISER Pune Math Book

COLLOQUIA AND SEMINARS

COLLOQUIA

1. Srilakshmi Krishnamoorthy (IIT, Chennai) March 4, 13. Souvik Goswami (ICMAT, Madrid, Spain) 2016, Lifting Congruencies to weight 3/2 December 5, 2016, Story of height pairings 2. V. G. Narasimha Kumar Cheruku (IIT, Hyderabad) 14. Louis H. Kauffman (University of Illinois, Chicago, May 20, 2016, On the Gaps Between Non-Zero Fourier USA) December 9, 2016, Introduction to Knots, Coefficients of Cusp Forms Knotoids and detecting the UnKnot. 3. Shailesh Tipnis (Illinois State University, USA) August 15. Louis H. Kauffman (University of Illinois, Chicago, 8, 2016, A card trick: Protocol, Graph model and USA) December 9, 2016, Quantum computing and extensions Majorana Fermions [Institute colloquium] 4. Rajendra Bhatia (Indian Statistical Institute, Delhi) 16. Dinakar Ramakrishnan (California Institute of August 18, 2016, A brief history of Fourier series [Institute Technology, USA) January 9, 2017, Rational points Colloquium] 17. Sophie Morel (Princeton University, USA) January 5. Rajendra Bhatia (Indian Statistical Institute, Delhi) 13, 2017, Global Langlands parameters August 19, 2016, Riemannian geometry and matrix means 18. Denis Benois (University of Bordeaux, France) 6. Krishna B. Athreya (Iowa State University, USA) January 20, 2017, On the p-adic height pairing August 26, 2016, General Glivenko Cantelli theorems 19. Luis Lomeli (Pontificia Universidad Catolica De 7. Sourav Pal (IIT, Mumbai) September 9, 2016, Valparaiso, Chile) January 27, 2017, On Langlands Rational dilation and its connection with geometry of functoriality and the Ramanujan conjecture over underlying complex domain function fields 8. Jonathan Dawes (University of Bath, UK) September 20. Agnid Banerjee (TIFR CAM, Bangalore) January 15, 2016, Unpublished patterns of thought: Alan Turing's 27, 2017, Gradient bounds of Modica type and a later work on morphogenesis [Institute Colloquium symmetry type result for parabolic reaction diffusion organised by Mathematics along with Humanities and equations Social Sciences] 21. K. Sandeep (TIFR-CAM, Bangalore), February 10, 9. Manish Mishra (IISER, Pune) September 23, 2016, 2017, Moser-Trudinger and Adams Inequalities Hecke algebras and the Langlands program 22. Deepak Dhar (IISER Pune), February 17, 2017, 10. Hari Sahasrabuddhe (IIT, Mumbai) October 21, The Super-Zeno effect: a problem from quantum 2016, An overview of artificial intelligence control theory 11. Kalyan Chakraborty (HRI, Allahabad) November 11, 23. Gadadhar Misra (IISc, Bangalore), March 11, 2016, Class numbers of certain number fields 2017, The Grothendieck inequality 12. Shailesh Tipnis (Illinois State University, USA) November 18, 2016, Path decompositions of regular graphs and multigraphs

COLLOQUIA AND SEMINARS

VISITORS’ SEMINARS

1. Safdar Qudoos (NISER, Bhubaneswar) March 7, Northern Cyprus Campus, Turkey) August 26, 2016, 2016, Hochschild and cyclic (co)homology of certain Group actions and non-Kahler complex manifolds algebras 16. Stephan Baier (TIFR, Mumbai) August 26, 2016, 2. Anirban Bose (IMSc, Chennai) April 18, 2016, Real On gaps between zeros of Epstein's zeta function elements in groups of type F4 17. Akshaa Vatwani (University of Waterloo, Canada) 3. Dilpreet Kaur (Symbiosis Institute of Technology, September 1, 2016, Twin primes and the parity Pune), April 19, 2016, Real special 2-groups problem 4. Gugan Thoppe (Technion, Israel) April 22, 2016, 18. Mathew Joseph (University of Sheffield, UK) Random $d$ - complexes: minimal spanning acycles and September 9, 2016, Longest increasing path within the persistence diagrams critical strip 5. Neha Gupta (Shiv Nadar University, Uttar Pradesh) 19. Subhamoy Maitra (Indian Statistical Institute, June 14, 2016, Frobenius algebras and 2-d topological Kolkata) October 3, 2016, On Boolean functions with quantum field nonlinearity greater than bent concatenation bound 6. Arunangshu Biswas (Presidency University, Kolkata) 20. Debdip Ganguly (Technion, Israel) October 18, June 16, 2016, Statistical simulation using the Adaptive 2016, Sharp functional inequalities of Hardy type and Markov Chain Monte Carlo techniques involving curvature on Riemannian manifolds 7. Krishnendu Gongopadhyay (IISER, Mohali) July 1, 21. Jacques Giacomoni (University of Pau, France) 2016, Reversible elements of complex hyperbolic October 20, 2016, Diaz-Saa inequality for variable isometries exponent problems 8. Vivekananda Roy (Iowa State University, USA) July 1, 22. Rohit D. Holkar (IISER, Pune) November 10, 2016, 2016, Standard errors for importance sampling An invitation to C*-algebras and groupoids estimators with multiple Markov chains 23. K. N. Raghavan (IMSc, Chennai) November 16, 9. Ashish Srivastava (Saint Louis University, USA) July 5, 2016, Singularities of Schubert varieties---a selective 2016, A question of Kaplansky and some recent survey developments 24. Jose Ignacio Burgos Gil (ICMAT, Madrid, Spain) 10. Umesh Dubey (IISc, Bangalore) July 22, 2016, November 18, 2016, Where do little elliptic curves go? Homological projective duality 25. Matteo Longo (University of Padova, Italy) 11. Mamta Balodi (IMSc, Chennai) July 28, 2016, A dual November 23, 2016, The Iwasawa main conjectures for version of Ore's theorem for Boolean intervals elliptic curves 12. Eshita Mazumdar (IIT, Mumbai) July 29, 2016, 26. Ludger Oberbeck (Justus-Liebig-Universität Modification of Griffiths theorem Giessen, Germany), November 23, 2016, Feynman Kac 13. Jeremy Eckhause (RAND Corporation) August 16, representation of path dependent PDEs 2016, Using Dynamic Programming to Solve Sequential 27. Sandeep Rakshit (D Y Patil Institute of Decision Models Management, Pune), November 23, 2016, Big Idea- 14. Krishna B. Athreya (Iowa State University, USA) Data and Analytics August 26, 2016, Weierstrass approximation theorem for 28. Umesh Dubey (HRI, Allahabad) November 24, continuous functions on unit interval by polynomials via 2016, Tensor triangular Chow group the weak law for coin tossing 29. Matteo Longo (University of Padova, Italy) 15. Mainak Poddar (Middle East Technical University, November 25, 2016, Variation of anticyclotomic 2017 IISER Pune Math Book

COLLOQUIA AND SEMINARS

Iwasawa invariants in Hida families Chandra on enhanced local Langlands parameters 30. H. Ananthnarayan (IIT, Mumbai) December 5, 2016, 38. B. Rajeev (Indian Statistical Institute, Bangalore) Idealizations and connected sums January 19, 2017, A re-look at Ito's Stochastic 31. Debraj Chakrabarti (Central Michigan University) differential equations December 12, 2016, The \overline{\partial}-problem in 39. Luis Lomeli (Pontificia Universidad Catolica De pseudoconcave annuli Valparaiso, Chile) January 23, 2017, L-functions via the 32. Anand Deopurkar (University of Georgia Athens, Langlands-Shahidi method USA) December 20, 2016, Vector bundles and finite 40. Luis Lomeli (Pontificia Universidad Catolica De covers Valparaiso, Chile) January 25, 2017, On globalization 33. Asilata Bapat (University of Georgia Athens, USA) methods and a look at twisted symmetric square L- December 20, 2016, Towards compactifications of functions Calogero–Moser space 41. Mladen Dimitrov (L’Université Lille 1), February 8, 34. Dr. Jacob Matherne (University of Massachusetts 2017, On the exceptional zeros of p-adic L-functions of Amherst, USA) January 2, 2017, A combinatorial Fourier Hilbert modular forms transform for quiver representation varieties in type A 42. Sangeeta Bhatia (Western Sydney University), 35. Amod Agashe (Florida State University, USA) January February 21, 2017, Algebraic Models of Large Scale 3, 2017, The cohomology of certain quotients of products Genome Rearrangement Events of upper half planes and upper half spaces 43. Tian An Wong (IISER, Pune), March 1,2017, 36. Aditya Karnataki (TIFR, Mumbai) January 11, 2017, p- Eisenstein cocycles and p-adic L-functions over adic uniformization of some locally symmetric spaces imaginary quadratic fields 37. Anne Marie Aubert (Institute of Mathematics at Jussieu, Paris, France) January 13, 2017, A view à la Harish-

WEEKLY SEMINARS

1) CODING THEORY Organizer: Krishna Kaipa

The seminar had two themes:

A) The MDS conjecture Participants: Mukul Rai Choudhury, Sukanya Pandey and Komal Muluk (Third year, IISER Pune) Objectives: To understand the state of the art on the problem of resolving the MDS conjecture in finite geometry and coding theory. Special focus is on a simpler question: is the degree k rational normal curve in a finite projective space a complete arc.

B) Bounds on the rate of a code Participants: Rutuja Kshirsagar (5th year, BITS Pilani Goa campus), Amit Bhati, Sunil Choudhari (3d year, IISER Pune). Objectives: To understand the best known upper bounds on the asymptotic rate of error correcting codes, with a view to improving the McEliece-Rodemich-Rumsey-Welch bound.

2) ALGEBRAIC GEOMETRY STUDENTS’ SEMINAR Organizer: Amit Hogadi

The algebraic geometry students’ seminar was started with the goal to study Ravi Vakil's book on Algebraic geometry called the "The Rising Sea". It is activity and completely managed by the students. The students take turns to lecture on topics from the book. Regular participants include 5 Ph.D. students (Girish Kulkarni, Neeraj Deshmukh, Suraj Yadav, Basudev Pattanaik and Kartik Roy) and 2 BS-MS students (Arpith Shanbhag and S. Sidharth). The seminar is held once every week.

3) RESEARCH SEMINAR Organizer: Rohit D. Holkar

The aim of this seminar is to bring physicists and mathematicians together to discuss their research and find collaborators. Speakers are expected to discuss their research with the seminar group targeting a broad audience. In addition to this, the aim is to encourage the audience to relate their work with that of the speaker, and widen the horizons of research. The planned sequence of speakers is as follows: 1. Chaitanya Ambi; 2. Prof. Deepak Dhar; 3. Bhooshan Gadre; 4. Rohit Diip Holkar; 5. Girish Kulkarni; 6. Advait Phanse; 7. Varun Prasad; and 8. Makarand Sarnobat.

Out of this, four talks have already taken place. Please visit https://sites.google.com/site/homerdholkar/teaching/sping2017semi for more details.

2017 IISER Pune Math Book

MINI COURSES

ROUGH PATH THEORY Instructor: Atul Shekhar, ISI Bangalore 5 - 7 September, 2016

Rough Path theory was introduced by T. Lyons in order to make sense of calculus based on paths which are very irregular and where classical theory fails to apply, for example, Brownian motion. A pathwise notion of integration, called rough integration, will be defined and it will be shown that it matches with classical integration theories. The theory applies to many other (non-martingale) stochastic processes, e.g., fractional Brownian motion. Such pathwise understanding comes with many applications such as stability results for SDE under smooth approximations of noise. In a recent work by M. Hairer, the idea of rough path theory was used to give natural solution concepts to many degenerate SPDEs like the Kardar-Parisi-Zhang (KPZ) equation. This mini course intends to give a mild introduction to the subject.

SYMPOSIA AND CONFERENCES

AUTOMORPHIC FORMS ON METAPLECTIC GROUPS AND RELATED TOPICS July 4-9, 2017

Organizers: Prof. A. Raghuram (IISER Pune), Dr. Chandrasheel Bhagwat (IISER Pune), Prof. Solomon Friedberg (Boston College) and Prof. Erez Lapid (Weizmann Institute of Science).

One of the most prestigious Mathematics event of its kind, this conference was organized jointly by IISER Pune and Boston College. Eminent researchers from across India and elsewhere gave research talks. Around 40 mathematicians participated including 17 speakers. This conference was helpful in facilitating stimulating academic interactions among the participants and taking IISER Pune to the world map of mathematical research. The conference was financially supported by NBHM and Boston College.

2017 IISER Pune Math Book

SYMPOSIA AND CONFERENCES

MATHEMATICS SYMPOSIUM August 11-12, 2016

Organizers: Math Events Committee consisting of Anisa Chorwadwala, Chandrasheel Bhagwat, Kaneenika Sinha, Diganta Borah

IISER Pune Mathematics Symposium, an annual departmental activity, is a platform for the mathematicians, at all stages of careers, in the department to showcase their research via short talks. The symposium intends to stimulate collaborations and to expose the younger mathematicians to the research being done in the department.

This year the symposium was organised by the Two eminent mathematicians from TIFR-CAM, Professor M.S. Narasimhan and Professor Adimurthi A., were present as external experts during the event.

Twenty-two talks on diverse topics from many broad areas of mathematics were given during this two-days event. The areas covered included number theory, representation theory, algebraic geometry, knot theory, differential equations, control theory, and cryptography. An unprecedented number of graduate students, nine to be precise, gave an overview of their research. The discussions that ensued were an indication of the success of the symposium. The high number of talks by women mathematicians of the department also contributed to its success.

MATH CLUB ACTIVITIES

1) ABEL PRIZE SYMPOSIUM The Abel prize was established by the Government of Norway in honor of Niels Henrik Abel, a Norwegian mathematician, who made pioneering contributions in the fields of algebraic equations and elliptic functions. In 2016, the prize was awarded to Sir Andrew Wiles who, in 1995, finally solved the centuries old unsolved problem of Fermat's Last Theorem. As part of the symposium, the club organised two talks:

On Sir Andrew Wiles' work Dr. Debargha Banerjee, IISER Pune This was an informal talk where Dr. Debargha discussed the history and initial breakthroughs made towards the proof of Fermat's last theorem and how Sir Andrew Wiles won a 358-year war fought by some finest mathematicians of our era. He also talked about the technical problem that came out in the original proof of Wiles and how he circumvents the problem leading to a rich theory of p-adic Galois representations. He mentioned some of the other remarkable works of Sir Andrew John Wiles including Coates-Wiles derivatives and the main conjecture of Iwasawa theory.

On Pierre Deligne's work Dr. Vivek Mohan Mallick In this talk, Dr. Mallick gave a glimpse of Deligne's work.

2) NONSTANDARD ANALYSIS: CALCULUS WITHOUT ε AND δ Talk by Prof. Jeffrey Adler Prof. Adler is from the Department of Mathematics and Statistics at the American University. In this talk the familiar notions of “ε-δ” definitions were explored. Analysis was taken up from a different perspective, that of without the ε and δ!

3) STUDENT TALKS AND SEMINARS Construction of Numbers Visakh Narayanan Numbers can be regarded as the first example of abstraction. Using just sets one can construct a 'model' for numbers. In this talk, a way of constructing numbers, starting from the empty set was discussed. Construction of natural numbers, integers, rationals and finally real numbers was taken up.

Category Theory Seminar Series Arun Kumar, Abhishek Shukla, Varun Prasad, Visakh Narayanan In this series, a few basic concepts and definitions in category theory were explored. 2017 IISER Pune Math Book

MATH CLUB ACTIVITIES

Mathematical Logic Seminar Series Varun Prasad, Visakh Narayanan This was a series, in which we attempted to understand the current foundations of mathematics. The framework of propositional logic, and a few concepts from first order logic were explored.

An Introduction to Modern Knot Theory Visakh Narayanan This series was an introduction to an invariant called the Khovanov homology, which is a homology theory for knots.

4) TALK BY DR. JEREMY ECKHAUSE Using Dynamic Programming to Solve Sequential Decision Models Jeremy Eckhause is an Operations Researcher at the RAND Corporation.This talk was an introduction to the topic of Dynamical Programming, and also included some interactive exercises and illustrative examples.

5) TALK BY PROF. LOUIS KAUFFMAN Louis Kauffman is a mathematician, topologist at the University of Illinois at Chicago. We had an interactive session where we explored the connections of knots theory with areas of physics, especially statistical mechanics. The session ended with a very thought provoking discussion on the halting problem, and the foundations of mathematics.

The 2016 Math Day started a week ahead with tag line of ‘’ Math day is coming ‘’ spanning a week-long series of events like posters with poses, puzzles, Math everywhere photo contest. By unveiling a mural designed by a few math members, Math museum was inaugurated by Padma Bhushan M.S. Raghunathan, chair National Center for Mathematics and a distinguished professor at IIT Bombay. The inauguration was followed by the release of Mathematics Profile booklet that records our yearly departmental activities. The later part of the event encompassed various games and activities such as Math Pictionary, Rubik-mania, Human Knot Theory, Pi-thon; python programming skill contest, Prime number Bee and so on conducted by game of thrones themed Math club house. The main attraction of the day was the world’s largest Hexaflexagon designed by IISERites. The post lunch session began with the Math Quiz Finals followed by talks by Prof M.S. Raghunathan on “Mathematics: Art that would rather be Science?” and Prof. Manish Jain of the Science Center at IUCAA. The highlight of the evening was the Math Play ‘Partition’, based on the life of the great mathematician Dr. S. Ramanujan. Post dinner was the screening of a documentary based on the life of Ramanujan starring our very own Prof. A. Raghuram, Head of the math department. And last but not the least, the cherry on the top, was the Math Treasure Hunt for mystery-lovers which began at midnight and went on till the wee hours of the morning.

An original and complete version of this article by Sayantika Mondal has appeared on www.iiserpune.ac.in

MATH DAY 2016 March 12 2017 IISER Pune Math Book

OUTREACH ACTIVITIES BY MATH MEMBERS

Conducting a Math workshop at Ishan Vikas Shibir

At DST-INSPIRE Camp

WITH LEND-A-HAND-INDIA

This year Math discipline participated in an outreach programme with Lend-A-Hand-India, an international NGO which has arranged a 2-days evaluation and guidance programme for selected math secondary and higher secondary school teachers from Satara district in IISER Pune. This programme was in coordination with the outreach programme of IISER Pune, DISHA of IISER Pune and assisted by Rohit Dilip Holkar. Tejas Kalelkar and Chandrasheel Bhagwat were one of the main speakers at this programme. Kalelkar took this opportunity to discuss ‘what research in pure mathematics means’ and some celebrated open problems in mathematics. He classified the present day mathematics into three parts, namely, analysis, geometry or topology, and algebra and elaborated on each part. Bhagwat gave the teachers puzzles to solve and explained the mathematics behind these puzzles and their solutions. He talked about how math teaching becomes joyful when it includes math puzzles and real life based problems. DISHA volunteers discussed the career opportunities in IISER Pune and other government institutes.

LIGHT MOMENTS

Sunday morning hike to Panchvati Hill

Raghuram with Anil Kapoor and Dev Patel at the premiere of The Man Who Knew Infinity at the International Film Festival of India (IFFI) in Goa 2017 IISER Pune Math Book

ART WITH MATH

One of the entries for the Math Convocation Flag competition. This entry titled “Ideal-tri-o- circle” was designed by fourth year BS MS student Tanushree Shah.

Mladen Dimitrov from (Université Lille 1, France) who gave a talk on p-adic L-functions at the Math discipline is an expert at Rubik’s cube. Seen here is his doodling with a cube while working on some serious mathematics. (Dimitrov has been named in the Hall of Fame of International Mathematics Olympiad for scoring a perfect score in the 1994 Olympiad.)

INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH (IISER) PUNE Dr. Homi Bhabha Road, Pashan Pune-411008, India Phone: +91 20 25908001 www.iiserpune.ac.in

2017 IISER Pune Math Book

Indian Institute of Science Education and Research (IISER) Pune Dr. Homi Bhabha Road, Pashan Pune-411008, India Phone: +91 20 25908001

www.iiserpune.ac.in