Colloquia

Abstract:
We shall discuss a sufficient condition for classifying algebraic integers. One such condition is dealing with the trace of powers of algebraic numbers. We start from scratch and introduce the problem. We discuss some elementary results and at the end, we state our results.

Abstract:
This talk is aimed at M.Sc. students in mathematics. However, I hope all mathematicians will find the result interesting and surprising. We shall require only preliminary results from Complex Analysis and General Topology which I am assuming that all of you know. The result has a beautiful and surprising connection with Set theory. I will present the requisite background from Set theory. The problem to be discussed here was asked by Wetzel in Ann Arbor Problem Book and the famous mathematician Paul Erdős gave the surprising and beautiful answer.

Abstract:
For a sequence of reals in [0,1], pair correlation statistics studies the distribution of their gaps. Most arithmetically interesting sequence is expected to have a gap distribution similar to a random sequence. Such results are known in many cases for almost all sequences, although proving for most individual sequences remains challenging. In this lecture, we will survey some of the results including a joint work with Tanmoy Bera and Mithun Das where we look
at these questions in higher dimensions.

Abstract:
In this talk we will discuss the interplay between linear algebra and classical K-theory. The talk will be accessible to graduate students. Basic knowledge in linear algebra, rings and module theory will be enough.

Abstract:
Peg Solitaire is an old board game that has existed in several countries under different names. It’s mathematical understanding at the beginning of the 18th century by a report of Leibniz, and continued later, a most notable step having been achieved by a group led by Conway in the sixties. We will present a unified and gentle introduction to the subject ; on our journey, we will encounter vector spaces and modules in dimension 33 and 37, the finite field F_2, some colourings, the integer points in a polyhedron and the simplex algorithm.

Abstract:
It is a remarkable fact that one can understand properties of numbers using “chaotic” dynamical systems. I will illustrate this principle with several examples and in doing so, try and answer the question in the title.

Abstract:
Birkhoff-James orthogonality is one very successful extension of the orthogonality notion in a Hilbert space. A challenging problem is to understand this in concrete Banach spaces. Motivated by the seminal work of R. Bhatia and P. Semrl, that interprets this orthogonality for individual operators on Hilbert spaces, we derive results in the context of Banach spaces, on operators that are Birkhoff-James orthogonal to a class of operators. This is linked to a hard minimax problem in convex optimization and is partly solved here, using techniques from Geometry of Banach spaces.

Abstract:
I will present an asymptotic formula for the moments of the central values of quartic Dirichlet L-functions. A corollary of the result is a quantitative non-vanishing theorem for these L-values. One of the key inputs is a large sieve type inequality for quartic Dirichlet characters. This is joint work with P. Gao.

Abstract:
In the first half of the last decade, Marcel Filoche and Svitlana Mayboroda observed that a certain solution to an elliptic PDE problem (the landscape function) can be effectively used to numerically predict the localization of eigenfunctions to Schrödinger operators with electric potentials exhibiting randomness. This localization phenomenon through disorder is known as Anderson localization, for which Phillip Anderson won the Nobel prize in physics in 1977. In this talk, we will introduce recent mathematical results showing the exponential decay given by the landscape function, for fundamental solutions, Lax-Milgram solutions, and eigenfunctions for Schrödinger operators; in particular, we will see that the landscape function generalizes previous tools used to study the exponential decay of solutions to Schrödinger operators when the potentials are regular.

Abstract:
The Brouwer’s Fixed Point Theorem and the Borsuk-Ulam Theorem are important theorems in Topology with many applications. Brouwer’s Fixed Point Theorem asserts that a continuous function from a compact convex set to itself has a fixed point. We will describe equivalent versions of the theorem including Sperner’s lemma which is purely combinatorial. The Borsuk-Ulam Theorem states that any continuous function from the n-dimensional sphere to the n-dimensional euclidean space must map two antipodal points to the same point. Lovasz used the theorem to settle a long-standing conjecture about the chromatic number of a certain graph, called the Kneser graph. We will present this proof after introducing the necessary background.

Abstract:
The distribution of values of L-functions plays a central role in analytic number theory. One aspect is a remarkable approximation property, called universality, first discovered by Voronin in the 1970s: appropriate shifts of the Riemann zeta-function approximate any analytic non-vanishing target function. This result has been generalized in various ways. In this talk we shall discuss applications of “joint universality” (simultaneous approximation) to the distributions of zeros.

Abstract:
Moments of central values of families of L-functions are important subjects in analytic number theory. In this talk, I will explain two powerful principles in the study of sharp bounds for these moments: the upper bounds principal of Radziwill and Soundararajan and the lower bounds principal of Heap and Soundararajan. As an application, we will use these principles to obtain sharp bounds for moments of central values of the quadratic family of Dirichlet L-functions.

Abstract:
The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz(1982) . Previously, Lindenstrauss (1966) studied similar versions of this notion in the context of non-separable reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the existence of linear Hahn-Banach extension operators in a purely Banach space theoretical set up. In this paper, we study similar questions in the context of Banach modules and module homomorphisms, in particular, Banach algebras of operators on Banach spaces. Based on Dales, Kania, Kochanek, Koszmider and Laustsen (2013), and also Kania and Laustsen(2017) , we give complete answers for reflexive Banach spaces and the non-reflexive space constructed by Kania and Laustsen from the celebrated Argyros-Haydon space with very few operators.

Abstract:
Isometries are maps between metric spaces which preserve distance between elements. In this talk we first recall the structure of (surjective) linear isometries on some important Banach spaces. Basic introductory material will be followed by recent results related to the following problem: when is a given finite set of modulus one complex numbers the spectrum of a linear isometry on a complex Banach space? Necessary conditions on such a set will be presented. Since sufficient conditions are related to the structure of specific Banach spaces, some particular cases will be considered and insightful examples will be given.