Department Seminars 2015

Abstract:

We know how to multiply two real numbers or two complex numbers. In both cases it is bilinear and norm preserving. It is natural to ask which of the other R^n admits a such multiplication. We will discuss how this question is related to vector fields on sphere and the answer given by famous theorem of J. F. Adams.

Abstract:

Euler system is a powerful machinery in Number theory to bound the size of Selmer groups. We start from introducing a brief history and we will explain the necessity of generalizing this machinery for the framework of deformations as well as the technical difficulty of commutative algebra which happens for such generalizations. If time permits, we talk about ongoing joint work on generalized Euler system with Shimomoto.

Abstract:

Abstract: TBA

Abstract:

Monodromy group of a hypergeometric differential equation is defined as image of the fundamental group G of Riemann sphere minus three points, namely 0, 1, and the point at infinity, under some certain representation of G inside the general linear group GL_n. By a theorem of Levelt, the monodromy groups are the subgroups of GL_n generated by the companion matrices of two monic polynomials f and g of degree n.

If we start with f, g, two integer coefficient monic polynomials of degree n, which satisfy some "conditions" with f(0)=g(0)=1 (resp. f(0)=1, g(0)=-1), then the associated monodromy group preserves a non-degenerate integral symplectic form (resp. quadratic form), that is, the monodromy group is a subgroup of the integral symplectic group (resp. orthogonal group) of the associated symplectic form (resp. quadratic form).

In this talk, we will describe a sufficient condition on a pair of the polynomials that the associated monodromy group is an arithmetic subgroup (a subgroup of finite index) of the integral symplectic group, and show some examples of arithmetic orthogonal monodromy groups.

Abstract: TBA

Abstract:

Systems of Conservation laws which are not strictly hyperbolic appear in many physical applications. Generally for these systems the solution space is larger than the usual BVloc space and classical Glimm-Lax Theory does not apply. We start with the non-strictly hyperbolic system

For n = 1, the above system is the celebrated Bugers equation which is well studied by E. Hopf. For n = 2, the above system describes one dimensional model for large scale structure formation of universe. We study (n = 4) case of the above system, using vanishing viscosity approach for Riemann type initial and boundary data and possible integral formulation, when the solution has nice structure. For certain class of general initial data we construct weak asymptotic solution developed by Panov and Shelkovich.

As an application we study zero pressure gas dynamics system, namely,

where _ and u are density and velocity components respectively.

Abstract:

I will recall basic definitions and facts of algebraic geometry and geometry of quadrics and then i will explain the relation of vector bundles and Hitchin map with these geometric facts.

Abstract:

I will discuss about all the cases in which product of two eigenforms is again an eigenform. This talk is based on one of my works together with my recent work with Soumya Das.

Abstract:

In this talk, we will discuss injectivity sets for the twisted spherical means on $\mathbb C^n.$ Specially, I will explain the following recent result. A complex cone is a set of injectivity for the twisted spherical means for the class of all continuous functions on $\mathbb C^n$ as long as it does not completely lay on the level surface of any bi-graded homogeneous harmonic polynomial on $\mathbb C^n.$

Abstract:

In this talk we will survey recent developments in the analysis of partial differential equations arising out of image processing area with particular emphasis on a forward-backward regularization. We prove a series of existence, uniqueness and regularity results for viscosity, weak and dissipative solutions for general forward-backward diffusion flows.

Abstract:

Fuzzy logic is one of the many generalizations of Classical logic, where the truth values are allowed to lie in the entire unit interval [0, 1], as against just the set {0, 1}. Fuzzy implications are a generalization of classical mplication from two-valued logic to the multivalued setting. In this presentation, we will talk about a novel generative method called the composition, of fuzzy implications that we have proposed. Denoting the set of all fuzzy implications defined on [0, 1], by I, the composition on I can be looked in two different ways, viz.,

(i) a generating method of fuzzy implications, and

(ii) a binary operation on the set I.

The rest of the talk will be a discussion of the composition on I along these two aspects. Firstly, we will discuss the closures of fuzzy implications with respect to some desirable properties. Then the effect of the composition on fuzzy implications that can be obtained from other generating methods of fuzzy implications, namely, (S,N)-, R-, f-, g- implications will also be discussed.

Secondly, looking at the composition as a binary operation on the set I, we will show that it forms I a lattice ordered monoid. Since it cannot be made a group, we determine the largest subgroup, denoted by S, obtained in I and we propose some group actions on I employing S. Finally, we demonstrate that, using one such group action, we have obtained, for the first time, representations of the Yager’s families of fuzzy implications.

Abstract:

Fuzzy logic is one of the many generalizations of Classical logic, where the truth values are allowed to lie in the entire unit interval [0, 1], as against just the set {0, 1}. Fuzzy implications are a generalization of classical implication from two-valued logic to the multivalued setting. In this presentation, we will talk about a novel generative method called the composition, of fuzzy implications that we have proposed. Denoting the set of all fuzzy implications defined on [0, 1], by , the composition on can be looked in two different ways, viz.,

a generating method of fuzzy implications, and
a binary operation on the set .

The rest of the talk will be a discussion of the composition on along these two aspects. Firstly, we will discuss the closures of fuzzy implications with respect to some desirable properties. Then the effect of the composition on fuzzy implications that can be obtained from other generating methods of fuzzy implications, namely, (S, N) -, R-, f -, g- implications will also be discussed.

Secondly, looking at the composition as a binary operation on the set , we will show that it forms a lattice ordered monoid. Since it cannot be made a group, we determine the largest subgroup, denoted by , contained in and we propose some group actions on employing . Finally, we demonstrate that, using one such group action, we have obtained, for the first time, representations of the Yager’s families of fuzzy implications.

Abstract:

In the first part of the talk, we study infection spread in random geometric graphs where n nodes are distributed uniformly in the unit square W centred at the origin and two nodes are joined by an edge if the Euclidean distance between them is less than . Assuming edge passage times are exponentially distributed with unit mean, we obtain upper and lower bounds for speed of infection spread in the sub-connectivity regime,

In the second part of the talk, we discuss convergence rate of sums of locally determinable functionals of Poisson processes. Denoting the Poisson process as N, the functional as f and Lebesgue measure as , we establish corresponding bounds for

in terms of the decay rate of the radius of determinability.

About the speaker:

J. Michael Dunn is Oscar Ewing Professor Emeritus of Philosophy, Professor Emeritus of Computer Science and of Informatics, at the Indiana University-Bloomington. Dunn's research focuses on information based logics and relations between logic and computer science. He is particularly interested in so-called "sub-structural logics" including intuitionistic logic, relevance logic, linear logic, BCK-logic, and the Lambek Calculus. He has developed an algebraic approach to these and many other logics under the heading of "gaggle theory" (for generalized galois logics). He has done recent work on the relationship of quantum logic to quantum computation and on subjective probability in the context of incomplete and conflicting information. He has a general interest in cognitive science and the philosophy of mind.

Abstract:

I will begin by discussing the history of quantum logic, dividing it into three eras or lives. The first life has to do with Birkhoff and von Neumann's algebraic approach in the 1930's. The second life has to do with the attempt to understand quantum logic as logic that began in the late 1950's and blossomed in the 1970's. And the third life has to do with recent developments in quantum logic coming from its connections to quantum computation. I shall review the structure and potential advantages of quantum computing and then discuss my own recent work with Lawrence Moss, Obias Hagge, and Zhenghan Wang connecting quantum logic to quantum computation by viewing quantum logic as the logic of quantum registers storing qubits, i.e., "quantum bits.". A qubit is a quantum bit, and unlike classical bits, the two values 0 and 1 are just two of infinitely many possible states of a qubit. Given sufficient time I will mention some earlier work of mine about mathematics based on quantum logic.