Lagrange's Four Square Theorem
Abstract: Most problems in Number theory are easy to state. But unlike most, Langrange's four square is easy to prove. It states that every natural number is a sum of four squares. In the talk, we will sketch the proof and then look at ways in which we can generalize the result. This talk will be short (30 to 45 minutes) and made accessible to all (including freshmen) by the speaker.
February 5, 2020
Smooth dependence of trajectories on initial conditions/parameters
Abstract: I will present the proof of smooth dependence of solutions to initial value problem in Ordinary Differential Equations (ODEs) on parameters, justify the Variational Equation and reinterpret the problem to give another independent proof using Implicit Function theorem.
November 7, 2019
Discrete Valuation Rings
Abstract: Examples, definition, properties, characterization of discrete valuation rings. Requirements- Knowledge of the definitions of principal ideal domain, local ring, field of fractions of a domain.
October 25, 2019
Posets
Abstract: We start with an introduction about lattices. We then move our attention to distributive lattices and prove the fundamental theorem of finite distributive lattices.
October 17, 2019
Topics in Combinatorics
Abstract: The involution principle ( generalized idea of bijection ) and it's various applications. Then I'll take most elegant theorem, Lemma of Gessel-Viennot, one that reveals via involutions an astounding connection between lattice paths and determinants. And discuss a number it's interesting applications - 1) proving some theorems on determinants “at a glance" purely combinatorially. 2) an unexpected and beautiful characterization of the Catalan numbers. 3) enumeration of plane partitions. 4) most beautiful application - number of spanning trees in graph comes as determinant of a certain matrix. Note-This had to be cancelled due to unavailability of the speaker.
October 10, 2019
Posets
Abstract: A partially ordered set (also poset) formalises and generalises the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. In this seminar, we introduce posets and notions of graded posets, rank generating polynomials, antichains and ideals.
October 3, 2019
Differentiable Manifolds
Abstract: Differentiable Manifolds form the most basic and natural objects in advanced calculus as is seen by the natural form that Stokes Theorem takes in the manifold setup. In this talk we will give an overview of Differentiable Manifolds including basic definitions and examples of submanifolds as well as abstract manifolds with applications to Lie Groups, Riemannian Geometry, Dynamical Systems and so on.
September 26, 2019