Mathematics Seminar Series 2019

Year   2018    2019

Title:    Probabilistic Methods in combinatorics (Hypergraph 2-coloring)

Speaker:    Rahul Gangopadhyay

Date:    April 11, 2019 (Thursday)

Abstract:     This talk is a brief survey on the hypergraph 2-coloring problem which is also known as Property-B for a set system. Erdos and Lovasz first proposed this problem for uniform hypergraphs. Lovasz proved that it is NP-complete to decide if an arbitrary hypergraph is 2-colorable. Erdos proved that there exists a $n$-uniform hypergraph with $\Theta((n^2)2^n)$ hyperedges which is not 2-colorable. He used probabilistic methods in the proof. He also conjectured that all the $n$-uniform hypergraphs with $O((n)2^n)$ hyperedges are 2-colorable.This conjecture is not resolved till now. We will discuss three results which prove weaker bounds than Erdos conjectured. All these proofs involve probabilistic methods. Also Erdos result on non-2-colorability is existential in nature. It is still an open problem to create a non-2-colorable $n$-uniform hypergraph with $\Theta((n^2)2^n)$ hyperedges. Gebauer gave the best constructive bound of $\Theta((2^(n^(2/3)))2^n)$ on this quantity. We would also like to discuss this proof. In the end, we will try to show the application of Lovsz's Local Lemma in hypergraph 2-coloring.

Title:    Stochastic Nash games under chance constraints

Speaker:    Vikas Vikram Singh

Date:    March 28, 2019 (Thursday)

Abstract:     The strategic games with stochastic payoffs and constraints have been extensively studied in the literature. The case of risk neutral decision makers has been studied by considering the expectation of random payoffs and constraints. We use chance constraint programming to model the risk averse stochastic Nash games. We show the existence of Nash equilibrium for the case of elliptically symmetric distributions. We also consider the case where underlying probability distribution is only known to belong to a distributional uncertainty set. These cases are modelled by considering the worst case situation using distributionally robust approach. We showed the existence of Nash equilibrium for various distributional uncertainty sets. We also discuss some computational approaches to compute the Nash equilibrium for these games.

Title:    Subordinated Stochastic Processes and Continuous Time Random Walks

Speaker:    Arun Kumar

Date:    March 14, 2019 (Thursday)

Abstract:     Continuous time random walks (CTRWs) are natural to model diffusion phenomena arise in particle physics where the particle on microscopic level can be considered to have random waiting times between random jumps. CTRWs are generalization of classical random walk models. The decoupled CTRWs are stochastic jump processes with arbitrary distributions of jump lengths and waiting times with jumps and waiting times independent. The scaling limit of classical random walk under the conditions of central limit theorem converges to Brownian motion (BM) which models standard diffusion phenomena. However, in many application the condition of finite mean waiting times or finite second moments jumps is not appropriate. In these cases, one get different scaling limit of CTRW which are connected to subordinated stochastic processes.

Title:    The lens of complexity

Speaker:    Maya Saran

Date:    March 13, 2019 (Wednesday)

Abstract:     If two trigonometric series converge everywhere to the same function, must they be the same series? This (old) question leads, in harmonic analysis, to sets called "sets of uniqueness". In the problem of characterising, i.e., recognising, such sets, it turns out that looking through the lens of descriptive set theory is richly fruitful. In this talk I will describe what a result of Kechris-Louveau-Woodin says about the complexity of the collection of sets of uniqueness. While the talk will be mostly expository, time permitting I'll also touch upon some recent work in this area. If you know what the real line is, you should be able to follow most of the talk.

Title:    Endomorphism Rings of Finite Drinfeld Modules

Speaker:    Sumita Garai

Date:    March 8, 2019 (Friday)

Title:    Aspects of Pluralism in Mathematics

Speaker:    Mihir Chakraborty

Date:    March 6, 2019 (Wednesday)

Abstract:     In recent times there has emerged the study of various pluralistic aspects of mathematics and mathematical practices. This has gone so far as the publication of a book named " Pluralism in mathematics : A new position in Philosophy of Mathematics", author, Michele Friend. I have been experiencing this phenomenon as a practicing mathematician for last 30 years or so. Pluralism arises from within mathematics and outside. As examples of 'from within' one can cite various Euclidean and non-Euclidean geometries, Cantorian and non-Cantorian Set theories, various foundations of mathematics, e.g. Logicism, Intuitionism, Formalism, Fictionalism and Pluralism (as mentioned above). These are not all about within. As regards influence from outside we can talk about mathematical practices in various cultures. Role of culture in the shaping of mathematical activity of a particular society, community or group has become an interest of studies of the current era.

Title:    Performance Improvement in Adaptive Control:An Initial Excitation based Framework

Speaker:    Sayan Basu Roy

Date:    February 28, 2019 (Thursday)

Abstract:     This talk would focus on performance issues in adaptive control, which is a class of nonlinear control technique used to tackle structured uncertain dynamical systems. Adaptive control can be visualized as a real-time uncertainty handling mechanism, which uses an online parameter estimator in conjunction with the controller to control a plant with parametric uncertainty.

It is well known that classical adaptive controllers typically guarantee Lyapunov stability in the extended state space (tracking error and parameter estimation error space), and asymptotic convergence of tracking error to zero, however, parameter convergence is only guaranteed if a restrictive condition of persistence of excitation (PE) is satisfied by the regressor signal. The PE condition is stringent since it demands sufficient energy/richness of information content for the entire time duration and, in general, conflicts with the tracking requirement.

The talk would emphasize on transient improvement of adaptive controllers through ensuring parameter convergence without requiring the stringent PE condition. We have developed new class of composite adaptive controllers, which build on two-layers of low-pass filters, and ensure parameter convergence using a milder condition of initial excitation (IE) on the regressor signal. The IE condition, which is shown to be online verifiable, is significantly less stringent than PE, since it demands sufficient energy/richness of the regressor only in the initial time window. The IE-based adaptive control guarantees exponential stability of the tracking and parameter estimation error once the online-verifiable IE condition is satisfied. Unlike conventional adaptive controllers, which only ensure asymptotic tracking, the proposed design is practically more significant due to its capability of exponentially fast tracking performance, which is especially crucial in safety-critical applications.

Title:    The Banach-Tarski paradox

Speaker:    Sankha Basu

Date:    February 1, 2019 (Friday)

Abstract:    The Banach-Tarski paradox is one of the most surprising and counter-intuitive results that follow from the axiom of choice in set theory. It is also commonly called the "pea and the sun paradox'' as it implies that a pea can be cut up into finitely many pieces and reassembled to make the sun. It is not a true paradox, in the sense that no actual contradiction is entailed by it, but is called a paradox since it defies our intuition about solids.

In this talk, I will present the historical genesis of this strange result and then discuss the mathematical proof of it from the axioms of set theory, especially the axiom of choice.