## Graduate Student Seminar

### Department of Mathematics and Statistics, IISER Kolkata

### LHC 108

### Organizers

Chandrahas Piduri Monalisa Dutta

If you are interested in presenting any topic, drop an email at dms.gradseminar@iiserkol.ac.in

## 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.

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.