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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.
Below is the pdf version of the tentative schedule for the graduate seminar.
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).
12 and 19 October, 2018
Puja Vacation
5 October, 2018
Speaker: Golam Mostafa Mondal
Title: Equivalent formulations of simply connectedness.
Abstract: An open set is simply connected if the set is connected and every closed curve in the set is homotopic to zero. In this talk we will discuss some equivalent formulations of simply connectedness.
28 September, 2018
Speaker: Nurun Nesha
Title: Boundary value problem in differential inclusions ( continuation of the last talk ).
21 September, 2018
Mid-Semester Examinations
14 September, 2018
Speaker: Nurun Nesha
Title: Boundary value problem in differential inclusions.
Abstract: In this talk we will discuss about necessary and sufficient conditions for the existence of a solution of a gradient inclusion with boundary condition for non-convex case. We will start with the definition of weak differentiability and then will give some examples and discuss some properties of weak differentiability. At the end, we will construct the solution for the above mentioned theorem in R and R^2.
7 September, 2018
Speaker: Sachchidanand Prasad
Title: Orientation Preserving Diffeomorphisms of S^1 is path connected.
Abstract: In this talk we will start by defining the orientation preserving maps on S^1. We will define compact open topology and uniform convergence topology to work on the given space. Then we will prove the space of diffeomorphisms of S^1 is path connected. Moreover, we will show that the space is deformation retract to the space of all rotations of S^1, i.e, SO(2).
31 August, 2018
Speaker: Jiten Kumbhakar
Title: Weierstrass Approximation Theorem.
Abstract: Weierstrass approximation theorem states that every continuous function can be approximated by a sequence of polynomials. In this talk, we will prove this theorem and see some of the applications of it.
17 and 24 August, 2018
Speaker: Amitesh Sarkar.
Title: Equitable Partition.
Abstract: In graph theory an adjacency matrix is a square matrix used to represent a finite graph. In this talk we will define the adjacency matrix of a graph and analyze some important properties of this matrix. Then we will discuss how to compute the eigenvalues of this matrix (named adjacency eigenvalues) for some particular class of graphs using a special type of partition (Equitable partition) of the vertex set of the graph.
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 Gδ 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.
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.
