1st CMI Alumni Conference

Chennai Mathematical Institute, January 7-10, 2015

Talks

  • S Akshay, Reachability problems for Markov chains.

    In this talk, we consider a basic problem for Markov chains which lies at the heart of many model checking questions: given a finite Markov chain with distinguished source and target states, and a rational number r, does there exist an integer n such that the probability to reach the target from the source in n steps is r? Surprisingly, it turns out that the decidability status of this problem is still open. We provide evidence of the hardness of the problem by relating it to the Skolem Problem: a number-theoretic decision problem whose decidability has been open for decades. We also explore links between these problems and other classically hard problems such as program termination.


  • C S Aravinda, Dynamics of geodesic conjugacies.

    The question of whether a time-preserving geodesic conjugacy determines a closed, negatively curved Riemannian manifold up to an isometry is one of the central problems in Riemannian geometry. While an answer to the question in this generality has yet remained elusive, we review some of the available affirmative results.


  • Baskar Balasubramanyam, Special values of adjoint L-functions and congruences between automorphic forms.

    Let f and g be primitive cusp forms with Fourier coefficients a_n(f) and a_n(g). f and g are said to be congruent modulo a prime p if a_n(f) = a_n(g) mod p for all. I will give an overview of results that relate this phenomenon to the special value at s=1 of certain L-functions attached to cusp forms. I will then talk about my joint work with A. Raghuram in this direction.


  • Anandam Banerjee, Equivalence relations on algebraic cobordism cycles.

    Algebraic cobordism gives a lifting of the topological theory of complex cobordism to the setting of algebraic geometry. As in the case of complex cobordism, algebraic cobordism is closely related to formal group laws, which allows one to construct many new theories from algebraic cobordism. In particular, algebraic K-theory and the Chow ring can be obtained in this way. It is an interesting question to ask whether various adequate equivalences for algebraic cycles can be lifted up to the level of algebraic cobordism cycles. After briefly describing the well-studied equivalence relations on algebraic cycles and introducing the definition of algebraic cobordism, I will talk about analogues of these equivalence relations at the level of algebraic cobordism cycles and compare them with each other. In particular, I will also discuss the behaviour of algebraic cobordism modulo numerical equivalence as a cohomology theory. This work is done jointly with Jinhyun Park.


  • Kuntal Banerjee, Widths of Arnold tongues and Herman rings.

    The Arnold tongues are found in the two-parameter family of circle homeomorphisms. Herman rings occur naturally when we have an analytic circle diffeomorphism whose rotation number is irrational and Diophantine. In this talk we will discuss about the widths of the Arnold tongues and some possible results on the Herman rings.


  • Shiladitya Banerjee, Geometry, mechanics and fluctuations in Life.

    Living cells come in an astounding variety of shapes and sizes ranging from robust shaped bacteria to irregular shaped animal cells. The inherent softness in cellular matter couples its geometry to the emergent mechanical response. Furthermore, all defining processes of life rely on irreversible energy consumption and are driven far from equilibrium. My talk will use a highly interdisciplinary approach combining soft matter physics, statistical mechanics, physical chemistry and differential geometry to address three questions of fundamental importance in our understanding of cell biology:

    - What determines the stability of biological shapes?
    - Can geometric constraints be used to sculpt the mechanical response and molecular scale architecture?
    - Are there any defining laws of fluctuations governing biological matter?

    The talk should be accessible to non-experts.


  • Pabitra Barik, Hitchin pairs on a singular curve.

    In this talk, we present the moduli problem of rank 2 torsion free Hitchin pairs of fixed Euler characteristic χ on a reducible nodal curve. We describe the moduli space of the Hitchin pairs. We define the analogue of the classical Hitchin map and describe the geometry of general Hitchin fibre.


  • Rajesh Chitnis, Coping with Big Data and Big Time.


  • Amit Deshpande, The geometry of semidefinite programs.

    I'll present some geometric ideas in rounding semidefinite relaxations for graph cuts. After a brief review of the rich history of convex geometry and metric embeddings applied to algorithms, I'll present a recent result (with Rakesh Venkat) on finding sparse cuts in graphs of low threshold-rank. I'll try to keep it accessible to everyone, assuming only basic linear algebra.


  • Deepak D'Souza, Refinement-Based Verification of Functional Correctness.

    The notion of refinement of Abstract Data Types (ADT's) gives us a natural way of defining a notion of functional correctness of ADT implementations. In turn, refinement conditions can be phrased as Floyd-Hoare style annotations in the program text. We describe our experience with this approach while applying it to prove the correctness of the scheduler of a small real-time operating system called FreeROS.


  • Sushmita Gupta, Online computation with advice.

    The talk will be on the area of online computation. We will discuss competitive analysis, the established analytical tool and the emergence of new techniques that seek to improve on it. In particular, we will discuss the new paradigm of advice complexity that has led to a plethora of results and the potential for a new direction of research.


  • Nagarajan Krishnamurthy, Stochastic Social Cloud.

    Social Clouds have been gaining importance because of their potential for efficient and stable resource sharing without any (monetary) cost implications. There is a need, however, to look at how a social structure or relationship evolves to build a Social Cloud (by identifying factors that affect the social structure) and how social structure impacts individual resource sharing behaviour.

    This talk is based on joint work with Pramod Mane and Kapil Ahuja, IIT Indore, and is an extension of our work "Externalities and Stability in Social Cloud", International Conference on Game Theory for Networks (GameNets) 2014, Beijing.

    We shall briefly discuss the paper presented at GameNets 2014 which looks at a pairwise resource (or pairwise service) sharing social network model to explore the interdependence between social structure and resource (service) availability for an individual user or player. The paper also investigates effects of social structure on individual resource availability, analyses positive and negative externalities, and aims to characterize stable social clouds.

    Further, in this talk, we shall discuss our current research on a stochastic game model for social clouds and some results. This model takes into consideration the evolution of the network over time, as the shared resources (shared services) move across different users (players). These dynamic changes lead to a stochastic game where the states incorporate the position of the resources (services) at different points in time. We consider discounted accumulated payoffs and look at stability and externalities.


  • Raghav Kulkarni, Decision Trees, Solvable Groups, and Music of Primes.

    In this talk we present some recent results on the long-standing (around 40 years) Evasiveness Conjecture in the domain of decision tree complexity that asserts that any non-trivial monotone transitive Boolean function is evasive, i.e., any decision tree for it must query all the variables in order to determine the function. We will briefly survey the beautiful topological approach of Kahn, Saks, and Sturtevant (1984) that derives evasiveness by constructing certain constrained actions of solvable group. Then we briefly present the further connection discovered by Babai, Bannerjee, Kulkarni, and Naik (2010) to some well-known theorems and conjectures in number theory such as: Goldbach Conjecture, Fermat primes, and Weil character sum estimates. Finally we conclude with some more recent results on this topic and several open questions.


  • Debapriyo Majumdar, Query Suggestions without Query Logs.

    As we type in the Google search box, Google keeps giving us suggestions. How does Google know the possible queries? Well, more than 5 billion search queries are hitting Google everyday, so most things that we can search for has already been searched by someone before. It is the same for other web-scale search engines. But what about a smaller scale search engine which does not have such a huge and exhaustive query log? Say for example - a custom site search, or even Google scholar? As of today, interactive query suggestion for such systems are either not present, or very primitive.

    In this talk, we will see a systematic first-of-a-kind method for suggesting queries without using query log. We will also discuss the challenges in evaluating such a system and present a new evaluation approach.

    The talk will be simple and easy for CMI audience.


  • Swarnava Mukhopadhyay, Strange Duality of G-theta functions.

    The space of G-theta functions are non abelian versions of classical theta functions. They are also known as Verlinde spaces or the space of Conformal Blocks. Strange duality is a duality between the space of G-theta functions for two different groups. Strange duality for vector bundles with fixed determinant was proved by P. Belkale and by A. Marian- D. Oprea. For symplectic bundles, it was proved by T. Abe. Strange duality for some maximal subgroups of E_8 are also known due to the works of A. Boysal-C. Pauly.

    In this talk, we discuss a conjectural strange duality for odd Spin and orthogonal bundles. This is a joint work with R. Wentworth. If time permits, we will also discuss a strange duality between G_2 and F_4-theta functions at level one.


  • Debajyoti Nandi, Interplay between partitions, representation theory and VOAs.

    In this talk, I will present a brief overview of the interplay between partition identities and representation theory of infinite-dimensional Lie algebras using vertex-operator-theoretic techniques. Vertex operator algebraic theory was developed in conjunction with string theory in physics and the theory of "monstrous moonshine" and infinite-dimensional Lie algebra theory in Mathematics. This "non-classical" theory has deep connections with various diverse areas of mathematics and theoretical physics such as modular functions, finite groups, Lie algebras and string theory.

    This colloquium-style talk will not assume any prior knowledge of these specialized fields. We will only explore the algebraic-combinatorial aspects of the application of the vertex operator theory to representation theory of affine Kac-Moody Lie algebras.


  • Yashonidhi Pandey, Brauer Group of moduli of torsors under parahoric group scheme G over a curve.

    We compute the Brauer group of the moduli stack and smooth locus of moduli space of semi-stable parahoric G-torsors over a curve X.


  • Vimala Ramani, On some fractional differential equations in electro chemistry.

    In this talk, we will consider some fractional differential equations that arise in electro chemistry and derive their analytical solutions using homotopy analysis method. The analytical solutions are in terms of special functions. We compare the analytical solutions with the numerical solutions obtained using Adam-Bashforth type method. We will also discuss the results regarding the numerical solution of a system of FDE's of non-commensurate order.

    This talk is based on the joint work with S.Ramanathan, S.N.Victoria and R.Garappa.


  • Rajarshi Ray, Parallelizing State Space Exploration Algorithm of Hybrid Systems using Support Functions.

    In my talk, I will introduce hybrid automata and an algorithm for computing reachable states of linear hybrid systems with bounded inputs using support functions. The properties of support functions as a representation of symbolic states will be presented. Simple parallel variants of the algorithm will be presented. The obtained speed up in computation in time multicore systems will also be shown on some hybrid automata models.


  • Arnab Saha, Arithmetic Jet Spaces.

    In this talk, we will look at the arithmetic jet spaces which are constructed in analogue with differential algebra. The arithmetic jet spaces attached to any scheme X defined over ℤ contains information of lifts of Frobenius of X and has found various applications in Diophantine geometry and modular forms. We will give an overview of the main results as well as some recent progresses.


  • Parameswaran Sankaran, Twisted conjugacy in PL homeomorphism groups of the interval.

    Let G be a finitely generated (infinite) group and let φ:GG be an automorphism of G. One has the φ-twisted conjugacy action of G on itself, defined as x → gxφ(g-1). The orbits of this action are called the φ- twisted conjugacy classes. One says that G has the R-property if, for every automorphism φ of G, there are infinitely many φ-twisted conjugacy classes.

    In my talk, based on joint work with D. L. Gonçalves and R. Strebel, I will discuss the R-property for certain PL-homeomorphisms groups of the interval [0,b]. The talk will be made accessible to the non-specialists.


  • Saket Saurabh, Algorithmic Applications of Two Families Theorem.

    The Two-Families Theorem of Bollobás for extremal set systems and its generalization to subspaces of a vector space of Lovász are the corner-stones in extremal set theory with numerous applications in graph and hypergraph theory, combinatorial geometry and theoretical computer science. We refer to Gil Kalai's blog for more information on these theorems. In this talk we will survey these two results and show its recent algorithmic applications. The talk will be accessible to everyone.


  • Raghunath Tewari, Simultaneous Time-Space Bounds for the Graph Reachability Problem

    Graph reachability is a fundamental problem in computer science and understanding it's computational complexity is of utmost importance. There exist linear time algorithms for this problem such as BFS and DFS however both these algorithms also require a linear amount of space. In 1970 Savitch showed that directed graphs reachability can be decided in O(log2 n) space. However his algorithm requires Ω(nlog n) time. This gave rise to the following question -- can we decide reachability in polynomial time and polylogarithmic space simultaneously? In 1992, Barnes, Buss, Ruzzo and Schieber made some progress by giving a polynomial time and O(n/2 log n  ) space algorithm for this problem.

    In this talk, I will present some recent progress on this problem, particularly for certain classes of topologically restrictive graphs such as planar graphs, bounded genus graphs and minor-free graphs.