Abstract: The central limit theorem is considered perhaps the most influential theorem of mathematics in the 20th century. It has had significant applications both within mathematics and beyond, energizing literally every other field outside such as medicine, economics and even political theory. After a short history of the evolution of the central limit theorem, we will describe its impact in algebra and number theory and discuss some new applications. The talk will be accessible to a general mathematical audience.

We will present several methods of doing research in mathematics and science and illustrate them through concrete examples. The talk is aimed at a general scientific audience

In 1916 S. Ramanujan discovered three identities involving the Eisenstein series $E_2,E_4,E_6$ and their derivatives. This can be seen as a vector field in the moduli space of an elliptic curve $E$ enhanced with a certain frame of the de Rham cohomology of $E$. For this one needs algebraic de Rham cohomology, cup product and Hodge filtration developed by Grothendieck and Deligne among many others. Viewed in this way, Ramanujan's differential equation can be generalized to an arbitrary projective variety. If time permits I will explain two generalizations of this picture in the case of Abelian varieties and Calabi-Yau threefolds.

The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz (1982). Previously, J. 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 via interspersing subspaces in a purely Banach space theoretic 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, Kozmider 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's space with few operators.

Solitons are solutions of a special class of nonlinear partial differential equations (soliton equations, the best example is the KdV equation). They are waves but behave like particles. The term “soliton” combining the beginning of the word “solitary” with ending “on” means a concept of a fundamental particle like “proton” or “electron”. The events: (1) sighting, by chance, of a great wave of translation, “solitary wave”, in 1834 by ScottRussell, (2) derivation of KdV E by Korteweg de Vries in 1895, (3) observation of a very special type of wave interactions in numerical experiments by Krushkal and Zabusky in 1965, (4) development of the inverse scattering method for solving initial value problems by Gardener, Greene, Kruskal and Miura in 1967, (5) formulation of a general theory in 1968 by P. D. Lax and (5) contributions to deep theories starting from the work by R. Hirota (1971-74) and David Mumford (1978-79), which also gave simple methods of solutions of soliton equations, led to the development of one of most important areas of mathematics in 20th century. This also led to a valuable application of solitons to physics, engineering and technology. There are two aspects soliton theory arising out of KdV Equation • Applied mathematics - analysis of nonlinear PDE leading to dynamics of waves. • Pure mathematics - algebraic geometry. It is surprising that each one of these can inform us of the other in the intersection that is soliton theory, an outcome of KdV equation. The subject too big but I shall try to give some glimpses (1) of the history, (2) of the inverse scattering method and (2) show that algorithm based on algebraic-geometric approach is much easier to derive soliton solutions.

Srinivasa Ramanujan, one of the most inspirational figures in the history of mathematics, was an amateur gifted mathematician from lush south India who left behind three notebooks that engineers, mathematicians, and physicists continue to mine today. Born in 1887, Ramanujan was a two-time college dropout. He could have easily been lost to the world, a thought that scientists cannot begin to absorb. He died in 1920. Prof. Ono will explain why Ramanujan matters today and will share several clips from the film, “The Man Who Knew Infinity,” starring Dev Patel and Jeremy Irons. Professor Ono served as an associate producer and mathematical consultant for the film. About the Speaker: Prof. Ken Ono  is the Thomas Jefferson Professor of Mathematics at the University of Virginia, the Asa Griggs Candler Professor of Mathematics at Emory University and Vice President of the American Mathematical Society. He is considered an expert in number theory. His contributions include several monographs and more than 180 research and popular articles in number theory, combinatorics and algebra. He earned his Ph.D. from UCLA and has received many awards for his research in number theory, including a Guggenheim Fellowship, a Packard Fellowship and a Sloan Research Fellowship. He was awarded a Presidential Career Award for Science and Engineering (PECASE) by Bill Clinton in 2000 and was named a Distinguished Teaching Scholar by the National Science Foundation in 2005. He is also a member of the US National Committee for Mathematics and the National Academy of Sciences. He was an associate producer of the film “The Man Who Knew Infinity” based on the life of Indian mathematician Srinivasa Ramanujan.

In this talk, we shall review models used to describe blood flows in the human brain. We shall give new existence, uniqueness and stability results for some of those models (work in collaboration with D. Maity TIFR-CAM, Bangalore, and A. Roy, Inria-Lorraine). We shall address the issue of estimating the pressure from blood flow measurements, and of the auto regulation phenomenon, which is a natural stabilisation process.

We look at the problem of arranging points in Euclidean space in order to minimize the potential energy of pairwise interactions. We show that the E8 lattice and the Leech lattice are universally optimal in the sense that they have the lowest energy for all potentials that are given by completely monotone potentials of squared distance. The proof uses a new kind of interpolation formula for Fourier eigenfunctions, which is intimately related to the theory of modular forms. The talk is based on a joint work with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Maryna Viazovska.

On smooth k-schemes, it is known that the higher Chow groups of S. Bloch give what is called the motivic cohomology theory on them. When schemes admit singularities, this has been yet unresolved. In this talk, I will give a sketch of a recent new construction, which gives a functorial theory on the category of all schemes of finite type. To motivate the audience, I will begin with basic motivating examples from 1st year graduate course level differential forms, and proceed to build up on it to give a sketch, and discuss some consequences and questions.