Schedule

Abstract:

After reviewing some basic properties of matrices and eigenvalues, we shall introduce several interesting random matrices and discuss some of their spectral properties (as dimension of the matrices increase) that have been studied in statistics, probability, mathematics and other areas of sciences.

Abstract:

Two central themes in holomorphic dynamics are the iteration of rational maps and the action of Kleinian groups on the Riemann sphere. Although these theories developed along largely independent paths, they exhibit striking conceptual parallels and often display similar forms of chaotic behavior. As early as the 1920s, Fatou envisioned that these two classes of dynamical systems could be studied within a unified framework of iterated algebraic correspondences. 

After surveying some basic facts about rational dynamics and Kleinian group actions, we will present concrete evidence for Fatou’s vision by constructing algebraic correspondences that exhibit features of complex polynomials and Fuchsian groups in a single dynamical plane. We will discuss examples of rational Julia set realizations of Kleinian limit sets including classical Apollonian-like gaskets. Time permitting, we will outline the main analytic and algebraic ideas underlying this program and highlight several applications to other areas of mathematics.

Abstract:

The applications of representation theory to number theory may be said to include the whole of the Langlands program. We talk only about four applications that do not pertain to that: (I) A Lower bound for the Partition function p(n); (II) Dedekind Zeta functions and Sunada's Theorem; (III) Kronecker-conjugacy of integer polynomials; (IV) Monodromy groups of polynomials.

As this talk is also a celebration of the anticipated International Mathematics Day, we begin with some very interesting open questions regarding Pi that could be posed even to undergraduate students.

Abstract:

Approximate degree is known to be a lower bound on the quantum query complexity. For a measure $M$, composition question asks if $M(f\circ g) = \Theta (M(f) M(g)$. Even though the composition question for most of the measures based on decision tree (how does  for a measure $M$?) has beenanswered, the complete picture is missing for approximate degree. In this talk we will focus on the composition question of approximate degree.

We will discuss methods (like dual witness) which have been tried to prove composition. This will allow us to prove composition for some interesting classes of functions. Specifically, we will see the composition result when the inner or the outer function is a recursive composition itself. This is a joint work with Sourav Chakraborty, Chandrima Kayal, Manaswi Paraashar and Nitin Saurabh.

Outreach activities for KV School students, such as quizzes and various mathematical activities.