Spring 2022

15 April, 2022

Title: The Atiyah-Jänich Theorem

Speaker: Satwata Hans

Abstract: The space of Fredholm operators over a separable Hilbert space $\mathcal{H}$ has been an important area of study in Functional Analysis. It has gained more relevance over time due to the significant works of Sir Michael Atiyah and multiple other mathematicians, which have focused on various topological and geometric connections to the Fredholm operators. In this talk, we will see a famous theorem called the Atiyah-Jänich Theorem, which states that the space of Fredholm operators gives an alternate characterization of the K-Theory of any compact Hausdorff space.

8 April, 2022

Title: The Sensitivity Conjecture and Fourier Analysis of Boolean functions

Speaker: Debmalya Bandyopadhyay

Abstract: One of the most celebrated unsolved problems in theoretical computer science over the last three decades has been the sensitivity conjecture. After about 30 years since it was conjectured, in 2019, Hao Huang gave a 3-page proof of the conjecture using spectral techniques. The talk will introduce the conjecture and go on to explain Huang's simple proof. It will then touch upon some basics of Fourier Analysis of Boolean functions to introduce a corollary of another unsolved problem: The Fourier Entropy Influence (FEI) conjecture. Towards the end, I would be talking about the classification of flat, homogeneous, and Boolean polynomials, that I have been working on for the last few months, in connection to the FEI conjecture.

1 April, 2022

Title: Dimensionality Reduction for Multivariate and Functional Data

Speaker: Avishek Chatterjee, PhD Student, IISER Kolkata

Abstract: Dimensionality reduction as a preprocessing step to machine learning is effective in removing irrelevant and redundant information, increasing learning accuracy, and improving result comprehensibility. The complexity of any classifier depends on the number of inputs. This determines both time and space complexity and the necessary number of training instances to train such a classifier. In this talk, we discuss Principal Component Analysis (PCA) and Functional Principal Component Analysis (FPCA), the most popular and widely used dimensionality reduction techniques for multivariate and functional data respectively. Further, we talk about a local FPCA based classifier, which is especially suited for hard classification framework.

Autumn 2019

Below is the pdf version of the tentative schedule for the graduate seminar.

Click me.

14 November, 2019

Title: Sieve Theory

Speaker: Rachita Guria

Abstract: In this talk, we will discuss very basic sieve theory i.e., Legendre–Eratosthenes sieve. Sieve theory is a set of general techniques in number theory, designed to count or to estimate the size of sifted sets of integers. Following the introduction to exclusion-inclusion and the formula of Eratosthenes-Legendre, we give a number of examples of arithmetic problems that can be attacked by the sieve.

31 October, 2019

Speaker: Bidesh Das

Title: Weak Solution of Elliptic PDEs

Abstract:

3-10 October, 2019

Puja vacation.

26 September, 2019

Speaker: Samiron Parui

Title: Tournament Matrix

Abstract: In this talk, we will start with a round-robin tournament of n teams and show how can the results of a tournament transform a complete graph to a directed graph. The tournament matrix is the adjacency matrix of that directed graph. We will discuss some interesting properties of the graphs and the matrix. We will first consider a game which ends with a win and hence with a loss and there will be no draw.

Prerequisites: Basic Linear Algebra and Basic Graph Theory .

19 September, 2019

Speaker: Nurun Nesha

Title: Sobolev Inequalities

Abstract: In this talk, we will discuss the embeddings of various Sobolev spaces into others. Assuming the density of smooth functions in relevant Sobolev spaces, we will prove the Gagliardo-Nirenberg-Sobolev inequality, Morrey's inequality for smooth functions and then we will establish the estimates for arbitrary functions in the various relevant Sobolev spaces.

12 September, 2019

Speaker: Avishek Chatterjee

Title: Dimensionality Reduction Problem

Abstract: In machine learning classification problems, there are often too many factors on the basis of which the final classification is done. These factors are basically variables called features. The higher the number of features, the harder it gets to visualize the training set and then work on it. Sometimes, most of these features are correlated, and hence redundant. This is where dimensionality reduction algorithms come into play. Dimensionality reduction is the process of reducing the number of random variables under consideration, by obtaining a set of principal variables. One of the most useful methods used for dimensionality reduction is Principal Component Analysis (PCA). This method was introduced by Karl Pearson. It works on a condition that while the data in a higher dimensional space is mapped to data in a lower dimension space, the variance of the data in the lower dimensional space should be maximum.

5 September, 2019

Teachers' Day

29 August, 2019

Speaker: Dr. Somnath Hazra

Title: Homomorphism between C(X) and C(Y)

Abstract: In this talk, we will be discussing all homomorphisms between C(X) and C(Y).

Spring 2018

13 and 20 April, 2018

Speaker: Prahllad Deb.

Title: Transversal intersection and Jordan-Brouwer separation theorem.

Abstract: The classical Jordan curve theorem states that every simple closed curve divides the Euclidean plane, into two open connected sets, the "inside" and the "outside" of the curve. In this couple of talks we shall try to understand the generalisation of this phenomenon to higher dimension, namely, the Jordan-Brouwer separation theorem. This theorem states that a connected, compact hypersurface in a Euclidean space divides it into two connected components, "inside" and "outside" of the hypersurface where the inside one is a compact manifold with the given hypersurface as the boundary. For example, the unit sphere (set of all vectors of unit norm) in a Euclidean space divides it into the set of all vectors with norm less than or equal to one and the set of all vectors with norm strictly greater than one. In order to accomplish our goal, in our first presentation we develop one of the key ingredients - the transversal intersection of two submanifolds of a Euclidean space. In the second lecture we discuss the proof of the Jordan-Brouwer separation theorem. These lectures will be self-contained.

22 March and 6 April, 2018

Speaker: Ashis Pati.

Title: Calculus of Variations.

Abstract: We have learned how to optimize real valued functions in our graduation courses. In calculus of variations we will find ways of optimizing functionals defined on a space of functions. Functionals are often expressed as definite integrals involving functions and their derivatives, say, for instance the energy functional. We are interested in finding an extremal function(s) that makes the functional attain a maximum or minimum value. We will briefly discuss classical and direct methods of solving these problems.

15 March, 2018

Speaker: Debmalya Basak.

Title: Visualizing Lattice Points from the Origin.

Abstract: I will begin by motivating our topic through some definitions and computational results. We will show how this problem is deeply related to the Riemann zeta Function using properties of the Euler totient function. Finally, we will see how to find arbitrarily large spaces which are completely 'invisible' from the origin by applying the Chinese remainder theorem.

8 March, 2018

International Women's Day. Celebrations at TRC auditorium from 3:15 pm. Refreshments at 6 pm.

1 March, 2018

Holi vacation.

22 February, 2018

Mid-semester examinations.

15 February, 2018

Speaker: Arnab Char.

Title: Spectral Theory of Hypergraphs.

Abstract: A hypergraph is a higher dimensional generalization of graphs. This talk will give a brief idea about how to construct connectivity hypermatrices and explore a few results using this construction. Also, results from spectral graph theory and hypergraph theory will be correlated.

1 and 8 February, 2018

Speaker: Sachchidanand Prasad.

Title: Convex functions on a real Banach space.

Abstract: A continuous convex function of a real variable is differentiable except perhaps at countable points of its interval of continuity. In this talk we will start with the concept of derivatives on real Banach spaces, namely Gâteaux and Fréchet derivatives. Hence we will see that a continuous convex function on a real Banach space is Fréchet (Gâteaux) differentiable on a dense G_δ subset of its domain.

25 January, 2018

Speaker: Sweta Mahajan.

Title: Functions of bounded variation.

Abstract: Functions of bounded variation arise in the study of Riemann-Stieltjes integral and rectifiable curves. We will define the total variation of a function of one variable. We shall consider the space of functions of bounded variation, its algebraic properties and its relation with other classes of functions. Then we shall consider analytic properties such as realizing a function of bounded variation as difference of two increasing functions.

Autumn 2017

8 and 15 November, 2017

Speaker: Monalisa Dutta.

Title: Bézout's theorem.

Abstract: Bezout's theorem concerns with counting the number of points at which two curves intersect. This will be a two part talk. In the first part, algebraic preliminaries required to state and prove Bezout's theorem will be introduced.

25 October and 1 November, 2017

Speaker: Sanjoy Chatterjee.

Title: The Jordan canonical form.

Abstract: When an operator on a finite-dimensional complex inner product space is not diagonalizable, the next best thing that one can hope for is a Jordan canonical form. We will define a Jordan canonical form and show that the matrix of every operator on a finite-dimensional complex inner product space is similar to a Jordan canonical form in some basis.

11 and 18 October, 2017

Speaker: Sachchidanand Prasad.

Title: The inverse and implicit function theorems.

Abstract: The inverse function theorem lists sufficient local conditions on a vector-valued multivariable function to conclude that it is a local diffeomorphism. We will prove the inverse function theorem and use it to prove the implicit function theorem for multi-dimensional real euclidean spaces​. We will closely look at the inverse function theorem in one dimension and a holomorphic version of it in the complex field. Finally, we will define regular values and show that the special linear group ​of degree n over the real field​ is a submanifold of general linear ​group of degree n over the real field.

4 October, 2017

Speaker: Sanjoy Chatterjee.

Title: Topological properties of the orthogonal matrix group and the unitary matrix group.

Abstract: We will show that the orthogonal matrix group and the unitary matrix group are compact spaces. We will also show that the unitary matrix group is a connected space. Finally, we will look at the connected components of the orthogonal matrix group.

27 September, 2017

Puja vacation.

20 September, 2017

Midsemester examinations.

6 and 13 September, 2017

Speaker: Nurun Nesha.

Title: Rademacher's theorem.

Abstract: Rademacher's theorem states that if a function from an open set of n-dimensional real space into the reals is almost everywhere differentiable in that open set. We will first prove the case for 1-dimensional real space and then we will prove the general case.

30 August, 2017

Speaker: Monalisa Dutta.

Title: Hilbert's Nullstellensatz.

Abstract: Hilbert's Nullstellensatz establishes a fundamental relationship between geometry and algebra. In its simplest form, it states that any maximal ideal in a polynomial ring in n-many variables over an algebraically closed field is generated by n-many polynomials of degree one. We will also show that this statement is equivalent to the fact that any family of polynomials in this polynomial ring, whose generating ideal is not the whole ring, has a common zero. We will further show, using a method called "Rabinowitsch trick ", how Hilbert's Nullstellensatz relates "algebraic sets" to ideals in polynomial rings over algebraically closed fields. Here, the concept of an "algebraic set" will be defined in the talk. We finish with some concrete applications of the ideas developed.

16 and 23 August, 2017

Speaker: Chandrahas Piduri.

Title: CW Complexes.

Absract: CW complexes will be introduced with examples. In the second talk, products of CW complexes will be discussed. We will also show that CW complexes are normal spaces.