Past seminar colloquium

Abstract: Lattice points contained in regions of the plane, and higher dimensional Euclidean spaces, have been a topic of much study, going back at least to the century-old work of Minkowski. In recent decades there have been many developments in the general area, with many new insights brought in. The talk will aim at giving a flavour of the ideas involved, at a rudimentary level.

Tea: 3.30 p.m

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Abstract: Two dimensional polynomial phase signal has uses in modeling black and white texture. We will discuss how to estimate the parameters of the model from observed data set and about the large sample properties of these estimators.

Lecture series abstract: We will survey the von Neumann algebraic approach to locally compact quantum groups in the sense of Kustermans and Vaes, roughly following the treatment by Van Daele. We will begin with an intro to the theory of weights of von Neumann algebras. We will then proceed to describe locally compact quantum groups. If there is any time left, we will present some recent progress in the field. The prerequisite for attending the talks is some basic knowledge of C*-algebras (and perhaps also von Neumann algebras, depending on the audience).

In 1931 Plancherel and Polya observed the following amusing fact: If the averages of a locally integrable function on \mathbb R over balls of radius R with center x converges to f(x) for all x as R goes to infinity then f(x)=ax+b for some real numbers a and b. They went on to show that in higher dimension the limit of ball averages is a harmonic function. We shall talk about a generalization of this result for Riemannian symmetric spaces of noncompact type with rank one.

Abstract: In this talk, we shall introduce the notion of a "quantum channel" and then define a special class of quantum channels called "entanglement breaking channels". For a map in this special class, we shall define what we mean by its entanglement breaking rank. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. The prerequisites for this talk will be minimal with some background in linear algebra, and will be aimed towards students.

Abstract: We discuss approximate Birkhoff-James orthogonality of bounded linear operators defined between normed linear spaces X and Y. As an application of the results obtained, we characterize smoothness of a bounded linear operator T under the condition that K(X, Y), the space of compact linear operators is an M−ideal in L(X,Y), the space of bounded linear operators. 

In his Ph.D thesis, John Tate attached local root numbers with characters of a non-Archimedean local field of characteristic zero. Robert Langlands (later P. Deligne ) proved the existence theorem of non-abelian local root numbers of higher dimensional complex local Galois representations. The local Langlands correspondence preserves thisroot numbers and the global root number is a product of local root numbers. So the explicit computation of the local root numbersis an integral part of the Langlands programs. But for arbitrary higher dimensional Galois representations we do not have any explicit formula for the local root numbers. To give an explicit formula of the local root number of an induced representation of a local Galois group of a non-Archimedean local field F ofcharacteristic zero, first we have to compute the Langlands' lambda function \lambda_{K/F} for a finite extension K/F . The plan of the talk is as follows:

1. Review of local and global root numbers
2. To explain the computations of the local root numbers for a particular local Galois representations (e.g. Heisenberg representations)
3. And the computation of the lambda function \lambda_{K/F} except when K/F is quadratic wildly ramified extensions
4. If time permits, then I will explain some of my ongoing projects regarding root numbers and some open problems.

We will continue with the discussion on (co)tangent space and smooth maps. We will define what is the differential of a smooth map and state the implicit function theorem. We will also work out some concrete examples. 

We consider certain rings of smooth functions on a nice space, like a circle, and try to explore its algebraic nature using the smoothness of these functions. As a motivation for this, recall the ring C[0,1] of continuous real valued function on [0,1]. Recall how we prove that its maximal ideals (algebraic structure) are points using the compactness (topological structure) of [0,1].Only basic knowledge of ring theory and analytic functions is required, however, even these 2 topics will be revised in the beginning of the talk.

Abstract: The goal of the talk is to study complex tori and see some of its properties .We will begin with defining what a complex tori is and show that it is a complex manifold. Then we will  classify complex tori of dimension one  and show that each such tori is  an elliptic curve.

Systems of Conservation laws which are not strictly hyperbolic appear in many physical applications. Generally for these systems the solution spaceis larger than the usual BVl oc space and classical Glimm-Lax Theory does not apply. We start with the non-strictly hyperbolic system(uj) t +j Xi=1( uiuj −i +12) x = 0, j = 1, 2...n.For n = 1, the above system is the celebrated Bugers equation which is well studiedby E. Hopf. For n = 2, the above system describes one dimensional model for largescale structure formation of universe. We study (n = 4) case of the above system,using vanishing viscosity approach for Riemann type initial and boundary data andpossible integral formulation, when the solution has nice structure. For certain classof general initial data we construct weak asymptotic solution developed by Panovand Shelkovich. As an application we study zero pressure gas dynamics system, namely,ut + (u. ∇)u = 0, ρt + ∇.(ρu) = 0,where ρ and u are density and velocity components respectively

 We give an explicit formula for the integral kernel of the twistedKoecher-Maass series associated to a degree two Siegel cusp form F, where the twist is realized by any Maass waveform whose eigenvalue is in the continuum spectrum. From such a kernel we deduce the analytic properties of those twisted Koecher-Maassseries, and show how the later can be expressed in terms of Dirichlet series associated to the Fourier-Jacobi coefficients of $F$. 

Economic inequality arises due to the inequality in the distribution of income and assets among individuals or groups within a society, or region or even between countries. For continuous evaluation of different economic policies taken by the government, computation of Gini index periodically for the whole country or state or region is very important. But not all countries can afford or do not collect data from households in a relatively large scale periodically.In order to compute a confidence interval of Gini index for a particular country or a region at given time, there exist fixed-sample size methods. However, for achieving a level of accuracy of estimation within some pre-specified error bound i.e. for constructing a fixed-width confidence intervals for Gini Index, no fixed sample size methodology can be used. This problem falls in the domain of sequential methodology. To date there does not exist any multi-stage or sequential procedure for constructing fixed-width confidence intervals for Gini Index.In this presentation, a fixed-width confidence interval estimation procedure of Gini index will be presented under simple random sampling scenario along with several asymptotic properties like convergence results on final sample size and also the coverage probability which are proved without any specific distributional assumption.A discussion will be made on use of other sampling schemes as well. 

We will describe the structure of isometries on Hilbert spaces, following H. Wold and J. von Neumann. Then we will proceed to represent (concrete and not so concrete) tuples of commuting isometries. We will draw a list of observations on subtlety of this object and link it up with some deep problems in operator theory and function theory. We will also report on some recent results concerning invariant subspaces of the Hardy space over the unit polydisc.  

Abstract :     Here we study the Marshall-Olkin formulation of bivariate Pareto distribution, which includes both location and scale parameters and find efficient estimation techniques of the parameters of corresponding distribution. We use Maximum likelihood Estimation through the EM algorithm for the parameter estimation. Pareto distribution is heavy-tail in nature. It plays an important role in the Extreme Value Theory. So these distributions can be very useful in modeling the data related to finance, insurance, climate and network-security etc. These distributions can be used to analyze data related to any bivariate-component systems, e.g. axial length of two eyes of a diabetic patient. A numerical simulation is performed to verify the performance of different proposed algorithms

We shall discuss covering spaces of graphs, leading to facts about subgroups of free groups.We shall then construct geometric models, inspired by these graphs,and analyze some concrete examples. 

The study of of degree d curves in $\mathbb{CP}^2$ is one of the interesting topic which arises in the context of Enumerative Geometry. The specific problem, which I will talk about, can be stated as follows:

Let $\mathbb{CP}^2$ be a compact complex surface and $L \rightarrow \mathbb{CP}^2$ a holomorphic line bundle that is sufficiently ample. Let $\mathcal{D} := \mathbb{P}^\frac{d(d+3)}{2}$ be the space of all degree d complex curves in $\mathbb{CP}^2$ - What is $\mathcal{N}(\mathfrak{X}_{k})$, the number of curves in $\mathbb{CP}^2$, passing through $\frac{d(d+3)}{2} -k$ generic points, having singularity of type $\mathfrak{X}_{k}$, where $k$ is the codimension of the singularity $\mathfrak{X}_{k}$?

In this talk we will describe what is a zero set of vector bundle over $\mathbb{CP}^2$ and then calculate $\mathcal{N}({A_1})$ for degree d complex curves in $\mathbb{CP}^2$. Here
$\mathcal{N}({A_1})$ represents number of degree d curves in $\mathbb{CP}^2$ passing through $\frac{d(d+3)}{2} -1$ points having $A_1$ singularity. 

Abstract: We show that the solution  of the equation   in  and  in  is radial by using maximum principle

Abstract: We will describe: (i) What is Controllability problem (ii) Examples and known results: ODE (finite dim), transport equation, Heat equation (infinite dim ). Then we consider compressible Navier-Stokes equations in one dimension, linearized around a constant steady state $(Q_0,V_0)$, with $Q_0 > 0,V_0\geq 0$. It is a coupled system involving both transport and parabolic effects. We study the controllability of this linearized system in bounded interval $(0,L)$. We find that the properties of the two semigroups $(e^{tA})_{t\geq0} $ (the one when $V_0 = 0$ and the one when $V_0> 0$) and the spectrum of $A$ are completely different where $A$ is the corresponding linearized operator. We obtain several interesting positive and negative results for the null controllability and approximate controllability of the system using interior or boundary control in both the cases $V_0 =0$ and $V_0>0$.

Randomly evolving systems with components that interact with each other have been of great interest in research. In particular, in this talk, we will discuss interacting urn processes with reinforcement. We consider N interacting two-colour Friedman urns and discuss the synchronization limit and fluctuation theorems. Theinteraction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fractionof balls of that colour in all urns combined. Further extensions and applications to understanding opinion dynamics on networks etc. will also be discussed.

Abstract: In this talk, our focus will be on certain classical results due to Ingham, Levinson and Paley-Wiener which find optimal decay of the Fourier transform of nonzero functions vanishing on `large sets'. We will talk about these theorems in details and their generalizations on the $n$- dimensional Euclidean space, the $n$-dimensional torus and certain non-commutative Lie groups. 

In this talk, we study the configuration of systoles (minimum length geodesics) on closed hyperbolic surfaces. The set of all systoles forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surfaces, we call these admissible. There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first result characterizes admissibility. It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, to the classical result that there are only two minimal non-planar graphs).

I will present interactions among 2-dimensional conformal field theory, which is a kind of quantum field theory in physics, theory of operator algebras and the Moonshine conjecture which predicted mysterious relations between the finite simple group Monster and the elliptic modular function. I will emphasize representation theoretic aspects and do not assume anyknowledge of these theories. 

 In Quantum Information Theory, quantum states and quantum channels are central objects of study. Mathematically, a quantum state is a positive semidefinite matrix of unit trace, and a quantum channel is formalized by a linear map which is completely positive and trace preserving (CPTP).  

In the first part of this talk, we describe a rank function, that we call the ``entanglement breaking rank", of a special class of quantum channels called ``entanglement breaking channels". We show how this rank parameter for a particular channel links to one of the most celebrated problems in frame theory, commonly referred to as Zauner's conjecture. This helps us present an analytic, perturbative approach to the conjecture rather than algebraic. This part of the talk is based on a joint work with Vern Paulsen, Jitendra Prakash (NISER alumnus), and Mizanur Rahaman.

In the second part of this talk, we discuss an ordering of quantum states referred to as ``quantum majorization'', which is a natural generalization of the concept of matrix majorization in the quantum mechanical setting. We shall briefly revisit the work by Gour et al. (Nature Communications, 2018); they established a characterization for majorization of quantum states in the finite-dimensional setting via the notion of conditional min-entropy. We then outline our work where we extend the characterization by Gour et al. to the context of semifinite von Neumann algebras. Our method relies on a connection between the conditional min-entropy and the operator space projective tensor norm for injective von Neumann algebras. This part of the talk is based on a joint work with Priyanga Ganesan (NISER alumnus), Li Gao, and Sarah Plosker.

This is the third lecture on this topic with the following:

Abstact: Radon-Nikodym theorem of measure theory was proved in 1930 and A. N. Kolmogorov laid the foundation of modern probability in 1933 where the formal played a crucial role. In this Short Lecture Series in Mathematics (SLSM), we discuss the conditioning of probability theory in the light of Radon-Nikodym theorem. Upon the interest of the audience, we may introduce the notion of martingals and their convergence theorems. We have scheduled these lectures in the F-slot of Thursday and planned to go in accordance with the pace of the audience. Basic understanding of measure theory and functional analysis will be the prerequisite.

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In this talk, we discuss our attempts to solve an important open problem regarding Terrain Guarding: is the state-of-the-art polynomial-time approximation scheme for the problem [Gibson2014], a simple local search construct, optimal? In this regard, we first sketch a proof of a recent advancement in this domain by Mustafa and Jartoux [Jartoux2018]. This work proves, for geometric hitting set problems which give rise to arbitrarily large grid-like exchange graphs, that the local search algorithm is indeed optimal. In light of this result, we question if the Terrain Guarding problem can give rise to such grids. We will conclude the talk by laying out a possible plan for future work on this problem.

• Title: The sign of the Gauss sum  
• Abstract: One way to understand the connections between two different structures is to mix the two and to observe the resulting object! This is what Gauss did when he started his study of what is now called a Gauss sum. A Gauss sum is a mix of an additive map with a multiplicative map, where both the operations are done modulo a fixed prime number. It is not hard to see that the value of the most interesting case of the Gauss sum, appropriately normalized, is either $\pm 1$ or $\pm \imath$. However, Gauss proved in 1805, apparently it took four years for him to do this (!), that it is always $1$ or $\imath$. We'll discuss this result of Gauss and some of its generalizations.

Abstract:Message Authentication Code  (MAC) is a symmetric key cryptographic primitive to ensure data authentication. Traditionally, MACs are constructed using block ciphers. In this talk, we review how MACs can be constructed using Algebraic Structures and analyze their strength against related-key attacks. The talk will be self explanatory and no formal cryptographic background will be assumed.

This will be the second lecture of a series of five on Finite Fields.

Local Index formula is at the heart of the so called hard Riemannian aspects of Noncommutative Geometry. We will try to see why thisis so important. However so far we have only one computation of the local index formula. We will discuss two more computations. 

In this talk, we will see how the notion of graph homomorphism generalizes the concept of coloring and help defining parameters such as clique number, independence number, chromatic number etc. for different types of graphs, such as, oriented graphs, signed graphs, colored mixed graphs etc.More importantly, we will see how these new concepts and works relate itself to the popularly known theories.

Abstract: In this talk, we try to understand elements of chaos theory in dynamical systems numerically and graphically. We also try to understand the the various properties like period doubling and chaos of the famous Logistic Map. Along with it some other less known maps, the Tent Map and the Gauss Map will also be analyzed. In particular Gauss Map shows very interesting properties like coexisting attractors and reverse period doubling.The principal aim is to explore the deep relationship among dynamical systems, chaos and fractals, and to uncover structure even when order seems to be absent.Prerequisite :Mathematical Background till class 12.

Google Meet link: meet.google.com/bvx-swhw-epg

Abstract: Enumerative geometry of curves in the complex projective plane is very classical. In this talk, we consider two generalisations of classical enumerative geometric problems of curves in the plane. We consider curves in  that lie inside a  in ; such curves are called planar curves. We discuss the enumeration of degree  planar curves in  having  nodes and another singularity of codimension , and that intersect  generic lines and pass through  generic points in , where  with . We also count the number of elliptic () curves of degree  in a del Pezzo surface  that pass through  generic points of .

Fractal Interpolation Function is a novel method to construct irregular functions frominterpolation data. The talk will begin with a brief introduction of Fractals and differentkinds of Fractal Interpolation Functions. Then, I will intro duce a special kind of FractalInterpolation Function called “Super Fractal Interpolation Function” (SFIF) for finersimulation of the objects of the nature or outcomes of scientific experiments that revealone or more structures embedded into another (i.e. hybrid structures). We will lookat construction of SFIF wherein, an Iterated Function System (IFS) is chosen from apool of several IFS at each level of iteration leading to implementation of the desiredrandomness and variability in fractal interpolation of the given data. Further, I willdiscuss properties of integrability and differentiability of an SFIF. Finally, I will showa result on convergence of a Cubic Spline SFIF.

Abstract: Independence is a fundamental idea in probability theory. This is the same as product measures with total measure 1.
However, for non-commutative structures, the classical definition of independence does not work. This gave rise to the concept
of free independence as initiated by Voiculescu. We shall present an easy introduction to this idea using partition theory, Mobius function, moments and free cumulants. We shall also explain the deep and interesting connection between free independence and large dimensional random matrices. If time permits, we shall also see how it becomes useful in statistical analysis of high dimensional time series.  

We will discuss jet spaces and symmetries of partial differential equations.

Let $Z(t):=\zeta\left(\half+it\right)\chi^{-\half}\left(\half+it\right)$ be Hardy's function,where the Riemann zeta function $\zeta(s)$ has the functional equation $\zeta(s)=\chi(s)\zeta(1-s)$.Hardy's function has been used to compute the zeros of $\zeta\left(\half+it\right)$ and plays a crucial role in the theory of the Riemann zeta function.In this talk we will compute the large values of $Z(t)$ and $-Z(t)$ using the resonance method. 

Abstract: Employing solution of heat equation, we prove Taylor's theorem with Peano form of the remainder. In addition, we derive the Taylor series of an infinitely differentiable function under the additional assumption that the n'th derivative does not grow faster than the n'th power of some fixed positive constant.

Googlemeets link: https://meet.google.com/pdg-ymra-wkv

All are cordially invited. 

This talk consists of two parts; the first one dealing with a study of certain positive curvatures under the Ricci flow and the second one dealing with a rigidity question in geodesic flows.In 2010 Buchard Wilking proved that for every Ad_SO(n;C)-invariant subset S of the Lie algebraso(n;C) one can attach a notion positive curvature, which we call positive S-curvature, which is preserved by the Ricci flow. We study the properties of positive S-curvatures in reference to the Ricci flow. This part of work is in collaboration with Harish Seshadri and Soma Maity. We shall also discuss a problem motivated by blow-up considerations coming from a conjecture of Richard Schoen.In the second part of the talk we will study a rigidity question in Riemannian geometry, viz, when isRiemannian manifold determined by its geodesic flow? After a brief overview of this problem, We will present our work on the rigidity of the at cylinder. This is a joint work with C. S. Aravinda.

The well-known Pythagorean theorem which relates the lengths of sides of a right triangle to the length of the hypotenuse dates as far back in time as the Babylonians. Later in school when one gets a flavour of trigonometry, this result is neatly packaged in the equation $cos^2 \theta + sin^2 \theta = 1$. With further mathematical sophistication in terms of analytic geometry, one starts talking about $cos \theta, sin \theta$ as representing projections of a unit vector onto the x-axis and y-axis. In this talk, we will discuss the manifestations of the Pythagorean theorem and its converse in higher dimensions. In $n$ dimensions, we will see how this can be viewed as the problem of characterizing diagonal entries of a $n \times n$ projection matrix. Somewhat surprisingly, the set of vectors representing diagonal entries of rank $k$ projections in $M_n(\mathbb{C})$ forms a convex set with a simple description of its extreme points. In more generality, these results reveal connections between various parts of mathematics like combinatorics, representation theory, symplectic geometry which we will briefly touch upon.

We will discuss the general framework of quantum probability as the non-commutative generalization of the classical probability, operations on that and some of the quantum impossibilities. 

Abstract: In this talk, we define normal number and state Borel conjecture. In 1973, Mahler first proved a result towards Borel conjecture. Here, we discuss a conditional quantitative version of Mahler result.

In the talk, we discuss certain lifting between ellipticcusp forms and Hilbert cusp forms over real quadratic field which isknown as Doi-Naganuma lifting. We also discuss its close relationship to theShimura lifting which is a lifting from elliptic modular forms tomodular forms of half-integral weight.

In this talk, we shall discuss the theta series associated with positive definite integral quadratic forms and some of its applications in getting formulas for the number of representations of positive integers by quadratic forms, using the theory of modular forms. The second part of this talk is about the construction of the Shimura and Shintani mappings between certain subspaces of modular forms of half-integral weights and integral weight respectively. 

Abstract: In this talk we will discuss the w*-complete metric approximation property for  mixed q-Araki-Woods von Neumann algebras for any Q=(q_i,j)_i,j  where q_i,j real numbers with max |q_i,j|<1. By adapting an ultraproduct technique of Junge and Zeng, we prove that the mixed q-Araki-Woods von Neumann algebras is isomorphic to ultraproducts of some mixed q-Gaussian and some q-Araki-Woods von Neumann algebras. As a consequence, we discuss its applicability for  the completely bounded norm of radial multipliers on mixed q-Gaussian von Neumann algebras to mixed q-Araki-Woods von Nuemann algebras, hence establishing the w*-cmap for mixed q-Araki-Woods von Neumann algebras.

Classical distribution theory in higher dimensions is largely focused on the Gaussian distribution. However, for the Gaussian distribution it is well known that every marginal distribution, every conditional distribution and all linear transformations are also Gaussian. Besides, it is also obvious that these properties chosen individually are not sufficient conditions to characterize the Gaussian distribution. There are many non-Gaussian distributions which share some of these features with the Gaussian distribution. In the present talk we introduce couple of new theorems which characterizes the Gaussian distribution. Besides, one of the extensive application of the Gaussian distribution will be in classical classification theory. In the current note we also discusses the extensions of classical classification results to the theory of extreme value analysis. Problem in action: Let P and Q be a two sets of probability measures defined on the same sigma field. Let T be a transformation which maps every measure from P to one and only measure in Q. Now, given Q and T can we characterize the set P ? 

Abstract: Hilbert noted that the polynomial x^4 - 10 x^2 + 1 is irreducible over the integers whereas it is reducible modulo all primes. What is behind this? If two polynomials f,g with integer coefficients take the same set of values modulo all primes, what is the relation between f and g? What proportion of primes divide numbers of the form 2^n + 1? How many of these are of the form 4m+3? What about primes dividing 7^n + 12^n in some arithmetic progression? Are there infinitely many prime numbers such that the decimal expansion of 1/p recurs with period p-1? Given an integer a, if every prime dividing a^n-1 for some n also divides b^n-1, is b necessarily a power of a? We discuss the interesting mathematics behind such questions.

Functional Data Analysis is one of the frontline areas of research in statistics. The field has grown considerably mainly due to the plethora of data types that cannot be handled and analyzed by using conventional multivariate statistical techniques. Such data are very common in areas of meteorology, chemometrics, biomedical sciences, linguistics, finance etc .The lecture series will primarily aim at introducing the field of functional data analysis. Since functional data analysis is broadly defined as the statistical analysis of data, which are in the form of curves or functions, we will start with probability distributions and random elements in infinite dimensional Hilbert spaces, concepts of mean and covariance kernel/operator, the associated Karhunen-Loeve expansion and some standard limit theorems in Hilbert spaces. We will then discuss some selected statistical inference problems involving functional data like inference for mean and covariance operators, functional principal component analysis, functional linear models, classification problem with functional data, robust inference techniques for functional data etc. We will recall some results from functional analysis as and when required during the lectures. 

Generalized Order Statistics(GOS) models and Non-homogeneous PoissonProcess (NHPP) models form a significant subclass of the many softwarereliability models proposed in the literature. First, we discuss themodels followed by some logical implications of NHPP models andestimability of the underlying parameters. We prove an importantlimitation of NHPP models for which the limit of the expected numberof failures $m(t)$ as the testing time $t\to\infty$ is finite.Specifically, the parameters of those models cannot be estimatedconsistently as the testing time approaches infinity. Then, we presenta nonparametric method for estimating $\nu$, the number of bugs present in acode, and investigate its properties. Our results show that theproposed estimator performs well in terms of bias and asymptoticnormality, while the MLE of $\nu$ derived assuming that the commonrenewal distribution is exponential may be seriously biased if thatassumption does not hold. We present a new parametric approach aswell.

Abstract:Starting with the problem of heat equation witha potential and using Lie-Trotter product formula, Feynmanhas a heuristic way of explaining Brownian Motion.We shall discuss this. If time permits we shall discussWiener and Ito integrals.

Abstract: In the first part of the talk I will talk about nef cones of divisors and pseudo-effective cones of k-cycles on products of projective bundles over curves. In the 2nd part, I will present some results about the Seshadri constants of ample line bundles on various blow-ups of projective spaces.

The concept of Operator Ideals will be explained through examples and several properties. These objects first occurred in the famous work ``The Memoirs'' of A. Grothendieck published in 1955. Since then, it has taken form of a vast theory of Operator Ideals.

Abstract: In this talk we will introduce graph colouring and chromatic number of a graph. We will discuss some application of graph colouring. Further we will present the proof of 'Perfect Graph Theorem' by Lovász.

This talk will be in elementary level and first year Integrated MSc students should attained it.

Centerless Lie tori play an important role in explicitly constructing the extended affine Lie algebras; they play similar role as derived algebras modulo center play in the realization of affine Kac-Moody algebras. In this talk we consider the universal central extension of a centerless Lie torus and classify its irreducible integrable modules when the center acts non-trivially. They turn out to be highest weight modules for the direct sum of finitely many affine Lie algebras upto an automorphism.

An $m$-order $n$-dimensional square real tensor $\mathcal{A}$ is a multidimensional array of $n^m$ elements of the form$\mathcal{A} = (A_{i_1\dots i_m})$, $A_{i_1\dots i_m} \in \mathbb{R}$, $1 \leq i_1, \dots , i_m \leq n.$ (A square matrix of order $n$ is a $2$-order $n$-dimensional square tensor.) An $m$-order $n$-dimensional square real tensor is said to be a nonnegative (positive) tensor if all its entries are nonnegative (positive). We shall discuss the Perron-Frobenius theory for nonnegative tensors. Using these results we establish a sufficient condition for the positive semidefiniteness of homogenous multivariable polynomials.

The topology of real algebraic varieties is the study of the topology of objects that can be defined real algebraically, particularly the restrictions that the real algebraic structure imposes on the topology. The main problem that motivated interest in this area is the so called Hilbert's sixteenth problem that was suggested by Hilbert in his famous address: to classify, up to isotopy, non-singular planar real algebraic curves of a given degree. Topologically, a non-singular real algebraic curve is merely the union of circles (in fact, the fixed point set a complex conjugation on a Riemann surface), and its isotopy class is simply determined by the arrangement of these circles with respect to each other. Nevertheless, a complete classification, which would mean identifying which of these arrangements can be realized as the zero set of a real polynomial in two variables of a given degree, has as yet only been successful up to degree 7. However, it has led to several tangential questions and generalizations, which we will discuss after a brief overview of the motivating problem.

I shall discuss on control problems governed by the partial differential equations-mainly compressible Navier-Stokes equations,visco-elastic flows. I shall mention some of the basic tools applicable to study the control problems. We mainly use spectral characterization of the operator associated to the linearized PDE and Fourier series techniques to prove controllability and stabilizability results. I shall also indicate how the hyperbolic and parabolic nature of equations affects their main controllability results. Then some of our recent results obtained in this direction will be discussed.  

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Recent decades have witnessed significant growth and progress in spatial statistics, with applications in agriculture, epidemiology, geology, image analysis and other areas of environmental science. In recent years, new perspectives have emerged in connecting Gaussian Markov random fields with geostatistical models, and in advancing vast statistical computations. This series of lectures will focus on basic theory and computations of spatial statistics. Topics will include conditional and intrinsic autoregressions, connections between Markov random fields and geostatistics, variogram calculations, h-likelihood methods and matrix-free computations. Applications from agricultural variety trials, environmental sciences and geographical epidemiology will be discussed.

I will discuss the distribution of some arithmetic and geometric objects and describe the basic tools used to study questions of equidistribution.

A basis of the centralizer algebra for the action of the complex reflection group G(r, p, n) on the tensor product of its reflection representa- tion was given by Tanabe, and for p = 1, the corresponding partition algebra was studied by Orellana. In this talk, we first define the partition algebra for G(r, p, n) and call it Tanabe algebra. Along with the corresponding Schur–Weyl duality, using a confluence of ideas from Okounkov–Vershik approach, Clifford theory and higher Specht polynomials, we give a parametrization of the irreducible modules of Tanabe algebras and construct the Bratteli dia- gram. Furthermore, we give Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin basis of irreducible modules of Tanabe algebras. In the process, we also obtain some new results in the representation theory of complex reflection groups. The results presented in this talk form a part of a joint work with Dr. Shraddha Srivastava.

We will explain what it means to integrate differential forms on a manifold and state Stokes Theorem. We will see how it generalizes the fundamental theorem of calculus and Green's theorem. 

The Riemann-Roch Theorem is the foundation of the theory of algebraic curves. It gives a precise answer for the dimension of the space L(D) for a divisor D of an algebraic curve. The qualitative information that a Riemann surface is an algebraic curve is seen to be equivalent to more quantitative statement of Riemann-Roch.As an application of Riemann-Roch Theorem, it can be shown that any Algebraic curve can be holomorphically embedded into projective space. Any genus zero algebraic curve is isomorphic to the Riemann Sphere(C∞). Any non-degenerate smooth projective curve X in Pn of minimal degree n is a rational normal curve.

Abstract: In this talk, we will discuss the existence of weak solutions for some elliptic boundary value problems. In particular, we will discuss the weak solutions corresponding to the Laplace operator and the Biharmonic operator.

TBA

 The Channel  Assignment problem(CAP)  is a general framework focused on point-to-point communication, e.g.  in radio  or mobile telephone networks.  One of its main  threads asks for an assignment of frequencies or frequency channels to transmitters while keeping interference at an acceptable level and  making  use of the available  channels  in an  efficient  way. Thus  the main  task  of the channels  assignment  problem  is to assign channels to the station in a way that avoids interference and uses spectrum as efficient as possible. Radio  k-colorings of graphs is a variation of channels  assignment problem. For  a  simple connected graph  G with  diameter  q, and  an integer  k, 1 6 k 6 q, a radio  k-coloring of G is an assignment f of non-negative integers  to the vertices  of G such that |f (u) − f (v)| > k + 1 − d(u, v) for each pair  of distinct  vertices  u and v of G, where d(u, v) is the distance between u and  v in G.  The  span  of a radio k-coloring f , rck (f ), is the maximum  integer assigned by it to some vertex of G.  The radio  k-chromatic number,  rck (G)  of G is min{rck (f )}, where the minimum  is taken over all possible radio k-colorings f of G.  For k = q and k = q − 1 the radio k-chromatic number  of G is termed as the radio  number  (rn(G)) and antipodal  number  (ac(G)) of G respectively.  Radio  k-chromatic number  is known for very limited  families of graphs  and specific values of k (e.g.  k = q, q − 1, q − 2).For this research talk only we discuss the results which are related to the radio number  of graphs. For an n-vertex graph  G, we shall discuss about a lower bound  of rn(G) which depends  on a parameter based on a Hamiltonian path in a metric closure Gc  (an  n-vertex complete weighted graph  where weight of an edge uv is dG (u, v)).   Using this  result  we show that how we can derive  lower bounds  of rn(Cn ), rn(Pm   Pn ), rn(Cm   Cn ),  rn(Cm   Pn ) and  rn(Km   Cn ).   For  any  tree  T  an  improved  lower bound  of radio  number depending  on maximum  weight Hamiltonian path of T have been shown.  Next we discuss about an upper bound  of rn(G) by giving a coloring scheme that works for general graph  and  depends  on the partition of the vertex  set  V (G)  into  two  partite sets  satisfying  some conditions.   We investigate  the radio  number  of power of cycles (C r ), Toroidal  Grids Tm,n.  We present an algorithm  which gives a radio coloring of a graph G.  For an n-vertex graph  the running  time of this algorithm  is O(n4 ). 

Ricci flow and associated techniques have played pivotal roles in solving some long-standing open problems in Geometry and Topology in recent times.  In this talk we will discuss some properties of an ODE related to the evolution of curvature along the Ricci flow and my recent results in this context.

We shall derive some bounds on the order of the Schur multiplier of finite p-groups. We shall also classify finite p-groups having Schur multiplier of maximum order. Finally we shall discuss the Schur multiplier and covering groups of special p-groups having derived subgroup of maximum order.

This is a Short Lecture Series in Mathematics (SLSM) consisting of 4 lectures of 90 minutes each. This is the third lecture in this series.

Abstract: The first half of the mini course will be an introducing to the two classical models of random graphs (a.k.a. Erdős-Rényi random graphs) and discuss the phenomenon of phase transition. We will also discuss thresholds for monotonic properties with examples including connectivity threshold and sub-graph containment threshold.

In the second half of the course we will consider other kind of random graphs. In particular, we will discuss various models for complex networks, including Albert-Barabási preferential attachment models. We will discuss "scale-freeness", asymptotic degree distribution and "small-world phenomenon". Properties of super and sub-linear preferential attachment models and some recent developments in de-preferential attachment models will also be discussed.

If time permits we will also introduce the random geometric graphs and discuss asymptotic of the connectivity threshold.

We estimate the dimension of the representation variety of Fuchsian groups in indefinite special orthogonal group that is of nonsplit type. In the case where the Fuchsian group is a surface group, we give an exact formula (rather that just estimate) for the dimension of the representation variety whose representation `space' is any real algebraic group.

Recent decades have witnessed significant growth and progress in spatial statistics, with applications in agriculture, epidemiology, geology, image analysis and other areas of environmental science. In recent years, new perspectives have emerged in connecting Gaussian Markov random fields with geostatistical models, and in advancing vast statistical computations. This series of lectures will focus on basic theory and computations of spatial statistics. Topics will include conditional and intrinsic autoregressions, connections between Markov random fields and geostatistics, variogram calculations, h-likelihood methods and matrix-free computations. Applications from agricultural variety trials, environmental sciences and geographical epidemiology will be discussed.

We consider the problem of offline changepoint estimation in a network-valued time series where the entire series is observed beforehand. We analyze a CUSUM statistic for this problem, and obtain, under minimal conditions, the rate of convergence of the resulting changepoint estimator in terms of three relevant parameters: (i) the (common) network size, (ii) the (common) network sparsity, and (iii) the total number of networks in the series. We also discuss some applications. This is based on on- going joint work with Peter Bickel, Sharmodeep Bhattacharyya, and Shirshendu Chatterjee.

This is the second lecture on this topic with the following:Abstact: Radon-Nikodym theorem of measure theory was proved in 1930 and A. N. Kolmogorov laid the foundation of modern probability in 1933 where the formal played a crucial role. In this Short Lecture Series in Mathematics (SLSM), we discuss the conditioning of probability theory in the light of Radon-Nikodym theorem. Upon the interest of the audience, we may introduce the notion of martingals and their convergence theorems. We have scheduled these lectures in the F-slot of Thursday and planned to go in accordance with the pace of the audience. Basic understanding of measure theory and functional analysis will be the prerequisite.

Abstract: Stream cipher plays an important role in cryptography. The basic component of stream cipher is Linear Feedback Shift Registrar. Here we will discuss about the general construction of a LFSR based stream cipher and the cryptanalysis of stream cipher.

In this talk, we shall introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces in the context of matrix ordered spaces. We shall prove that a unital, bijective $\ast$-linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is a completely absolute value preserving. From here, we deduce that on (unital) C$^*$-algebras such maps are precisely C$^*$-algebra isomorphisms. We shall extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We shall introduce the notion of a partial isometry in an absolute matrix order unit space to describe the comparison of order projections. We shall also discuss direct limit of absolute matrix order unit spaces to show the existence of ``Grothendieck group" through order projections and prove that ``Grothendieck group" is a functor from category of ``absolute matrix order unit spaces with morphisms as unital completely absolute preserving maps" to category of ``abelian groups". Later, we define orthogonality of complete absolute preserving maps and prove that ``Grothendieck group" functor is additive on orthogonal unital completely absolute preserving maps.

Finding formulas for the number of representations of integers as sums of integer squares has a long history. In this talk we give a brief history of the problem and indicate the use of modular forms in obtaining certain formulas. We also discuss about some related questions. 

An asymptotic framework for optimal control of multi-class stochastic processing networks, using formal diffusion approximations under suitable temporal and spatial scaling, by Brownian control problems (BCP) and their equivalent workload formulations (EWF), has beendeveloped by Harrison (1988). This framework has been implemented in many works for constructing asymptotically optimal control policies for a broad range of stochastic network models.To date all asymptotic optimality results for such networks correspond to settings where the solution of the EWF is a reflected Brownian motion in the positive orthant with normal reflections.In this work we consider a well studied stochastic network which is perhaps the simplest example of a model with more than one dimensional workload process. In the regime considered here, the singular control problem corresponding to the EWF does not have a simple form explicit solution,however by considering an associated free boundary problem one can give a representation for an optimal controlled process as a two dimensional reflected Brownian motion in a Lipschitz domain whose boundary is determined by the solution of the free boundary problem. Using the form of the optimal solution we propose a sequence of control policies, given in terms of suitable thresholds, for the scaled stochastic network control problems and prove that this sequence of policies is asymptotically optimal. As suggested by the solution of the EWF, the policy we propose requires a server to idle under certain conditions which are specified in terms of the thresholds determined from the free boundary. This is a joint work with A. Budhiraja and X. Liu.

A brief history of classical Hardy inequalities and related results will be presented. Further, we talk about Generalized Hardy-Sobolev inequalities and its applications.

Functional Data Analysis is one of the frontline areas of research in statistics. The field has grown considerably mainly due to the plethora of data types that cannot be handled and analyzed by using conventional multivariate statistical techniques. Such data are very common in areas of meteorology, chemometrics, biomedical sciences, linguistics, finance etc .The lecture series will primarily aim at introducing the field of functional data analysis. Since functional data analysis is broadly defined as the statistical analysis of data, which are in the form of curves or functions, we will start with probability distributions and random elements in infinite dimensional Hilbert spaces, concepts of mean and covariance kernel/operator, the associated Karhunen-Loeve expansion and some standard limit theorems in Hilbert spaces. We will then discuss some selected statistical inference problems involving functional data like inference for mean and covariance operators, functional principal component analysis, functional linear models, classification problem with functional data, robust inference techniques for functional data etc. We will recall some results from functional analysis as and when required during the lectures.

This talk will retrace the main steps of the modern theory of prime numbers and in particular how the combinatorial sieve combined with the Dirichlet series theory to give birth to the modern representation of the primes via a linear combination of terms, some of which being "linear", while the other ones are "bilinear". This will lead us to the recent developments of Green \& Tao, Mauduit \& Rivat, Tao, Helfgott, and Bourgain, Sarnak \& Ziegler. 

Potts Model is a model of interacting spins on a crystalline lattice. In this talk we try to define the notion of Quantum Symmetry for Potts Model and try to explore the relations between Quantum Symmetry on Potts Model and Quantum Symmetry of the underlying graph structure. This is based On Joint work with Prof. Debashish Goswami.

Abstract:- See the attached file. 

Google Meet Link: meet.google.com/zxe-csox-oyb

In this talk, we will prove Caporaso-Harris formula for counting plan curves of any genus. This formula gives the answer of the following problem:How many degree d curves are there in CP2 having δ nodes and passing through d(d+3)/2 − δ generic points?

This talk will be focused on the non-classical solutions to some hyperbolic systems of conservation laws and scalar conservation laws with discontinuous flux. We use the method of vanishing pressure limit and vanishing viscosity limit to construct solutions. For scalar conservation laws with discontinuous flux, we propose a generalized weak solution based on the vanishing viscosity limit. We then introduce a numerical scheme for this scalar conservation law that effectively captures the solution and we also do its convergence analysis.  Next, we establish a Hopf-Lax type formula for the solution to the initial-boundary value problem for the 1D pressureless gas dynamics model by introducing generalized potentials.

Abstract: Normal distribution has been used quite extensively in different areas of science and technology both in theory and practice. Although, it has several desirable properties, it has its own limitations also. Recently, various non-normal distributions (both univariate as well as multivariate) have been proposed in the literature for data analysis purposes. In this talk, we will consider different non-normal distributions and discuss their properties and provide applications in different areas.

This talk consists of two parts. In the first part, we discuss the convergence of finite volume method for solving non-linear aggregation-breakage equation. The pro of re lie s on showing the consistency of the scheme and Lipschitz continuity of numerical fluxes. It is investigated that the technique is second order convergent independently of the meshes for pure breakage problem while for aggregation and coupled problems, it depends on the type of grids chose n for the computations. Next, we show the efficient representation of d-point correlation functions for a Gaussian random field. To avoid the curse of dimensionality for d > 2, a truncated KarhunenLo´eve expansion of the random field is used together with the low rank Tensor Train decomposition. The target application of this work is the computation of statistics ofthe solution of linear PDEs with random Gaussian forcing terms.

This is a topic in classical algebraic K-Theory. I will recall definitions of elementary linear group, elementary symplectic group, linear transvection group, symplectic transvection group and symplectic group w.r.t. any alternating form. These groups have natural action on the set of unimodular elements. I will briefly discuss how bijections between orbit spaces of unimodular elements under different group actions are established. Finally, I will talk about an application of these results, namely improving injective stability bound for K1 group.

This will be the last lecture of a series of five on Finite Fields.

We will consider the basic material on the geometric approach to partial differential equations and symmetries, including an introductory part on the geometry of jet spaces.

C*-tensor categories are important descriptors of generalized symmetries appearing in non-commutative analysis and mathematical physics. An important algebra associated to a rigid semisimple C*-tensor category $ \mathcal{C} $ is the tube algebra $ \mathcal{A}\mathcal{C} $. The tube algebra admits a universal C*-algebra, hence has a well behaved representation category. Further, this representation category provides a useful way to describe the analytic properties of initial C*-tensor categories, such as amenability, the Haagerup property, and property (T).With a brief motivation from different directions, in this talk, I will move on to describing the annular algebra $\mathcal{A}\Lambda$ associated to a rigid C*-tensor category $ \mathcal{C} $. The annular representation category of $ \mathcal{C} $ is the category of $*$-representations of the annular algebra $\mathcal{A}\Lambda$. I will then present a description of the annular representation category of free product of two categories with an application to the Fuss-Catalan subfactor planar algebra.We then move onto oriented extensions of subfactor planar algebras (or equivalently singly generated C*-2-categories), which are a class of singly generated C*-tensor categories (or equivalently oriented factor planar algebras). I will end the talk with few problems which could extend this work. 

In 1980s Goldman introduced various Lie algebra structures on the free vector space generated by the free homotopy classes of closed curves in any orientable surface F. Naturally the universal enveloping algebra and the symmetric algebra of these Lie algebras admit a Poisson algebra structure. In this talk I will define and discuss some properties of these Poisson algebras. I will briefly explain their connections with symplectic structure of moduli space and the skein algebras of $F\times [0,1]$. This is an ongoing work with Prof. Moira Chas.

Abstract: Horrocks' Theorem is an indispensable tool to prove Quillen-Suslin Theorem, which gives a sufficient condition for finitely generated projective modules to be free. In this talk, we will talk about a version of Horrocks' Theorem which uses concepts of completable unimodular rows over a ring. First, we will prove the theorem for local rings, then generalize it using the Local-Global Principle. 

Abstract: I will introduce to the basics of the machinery of motivic homotopy theory and explain some of the foundational aspects of the theory in the study of algebraic varieties.

Let F be a non-Archimedean local field with ring of integers O and finite residue field k of odd characteristic. In contrast to the well-understood representation theory of the finite groups of Lie type GL_n(k) or of the locally compact groups GL_n(F), representations of groups GL_n(O) are considerably less understood. In this talk we construct a family of continuous complex-valued irreducible representations of the groups GL_n(O) called regular representations. This talk is based on a joint work with Roi Krakowski and Uri Onn.

An automorphism T of a locally compact group is said to be distal if the closure of the T-orbit of any nontrivial element stays away from the identity. We discuss some properties of distal actions on groups. We will also relate any distal group with the behaviour of convolution powers of probability measures on it. (This would be a survey talk and most of it would be accessible to most people.)

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.

TBA

The critical group of a graph is an interesting isomorphic invariant. It is a finite abelian group whose order is equal to the number of spanning forests in the graph. The Smith normal form of the graph's Laplacian determines the structure of its critical group. In this presentation, we will consider a family of strongly regular graphs. We will apply representation theory of groups of automorphisms to determine the critical groups of graphs in this family.

Abstract:  Hardy - Littlewood maximal function is one of the most fundamental operators in real analysis with many applications in different areas. In the seminar we will discuss about the Hardy- Littlewood function on sphere and its properties. Also the maximal function on sphere for weighted integral in case of Dunkl operator will be discussed.

TBA

Abstract: Employing solution of heat equation, we prove Taylor's theorem with Peano form of the remainder. In addition, we derive the Taylor series of an infinitely differentiable function under the additional assumption that the n'th derivative does not grow faster than the n'th power of some fixed positive constant.

Googlemeets link: https://meet.google.com/pdg-ymra-wkv

All are cordially invited. 

Let S ⊂ CP 3 be ’almost any’ projective surface of degree ≥ 4. ClassicalNoether-Lefschetz theorem states that any curve C ⊂ S can be written as an intersectionC = S∩ S 0 where S 0 is some surface in CP 3. Grothendieck-Lefschetz theorem generalizesthis result for higher dimension.In this survey talk we will discuss these theorems and the corresponding results forvector bundles. We will study related aspects of vector bundles over hypersurfaces andcomplex projective spaces. Our emphasis will be on extendibility theorems and varioussplitting criterion for vector bundles.We will also mention some recent results and open problems. The talk should beaccessible to a graduate student.

In this lectures, we would discuss probabilistic model of primes leading to heuristics about their distribution. We would see many surprising irregularities popping up alongside expected results. A survey of several recent and important results would be presented in a way accessible to non-experts.

Functional Data Analysis is one of the frontline areas of research in statistics. The field has grown considerably mainly due to the plethora of data types that cannot be handled and analyzed by using conventional multivariate statistical techniques. Such data are very common in areas of meteorology, chemometrics, biomedical sciences, linguistics, finance etc .The lecture series will primarily aim at introducing the field of functional data analysis. Since functional data analysis is broadly defined as the statistical analysis of data, which are in the form of curves or functions, we will start with probability distributions and random elements in infinite dimensional Hilbert spaces, concepts of mean and covariance kernel/operator, the associated Karhunen-Loeve expansion and some standard limit theorems in Hilbert spaces. We will then discuss some selected statistical inference problems involving functional data like inference for mean and covariance operators, functional principal component analysis, functional linear models, classification problem with functional data, robust inference techniques for functional data etc. We will recall some results from functional analysis as and when required during the lectures. 

Abstract: This is an informal talk on Random graphs and its relation to complex analysis. NB: This talk is organized as a part of ongoing "Advanced Instructional School on Stochastic Processes". Please note the unusual timing.

Abstrat: We prove an h-principle for poisson structures on closed manifolds. Equivalently
we prove h-principle for symplectic foliation (singular) on closed manifolds. On open
manifolds however the singularities could be avoided and it is a known result by Fernandes
and Frejlich [1].

References
[1] Fernandes, Rui Loja; Frejlich, Pedro An h-principle for symplectic foliations. Int. Math. Res. Not. IMRN
2012, no. 7, 1505–1518. (Reviewer: David Iglesias Ponte)
Presidency University, Kolkata, India., e-mail:mukherjeesauvik@yahoo.com,

In this talk, I will present Lax-Oleinik type explicit formulafor the solution of some balance laws and discuss its Structure Theory.

 I will discuss about the nonlocal operators, in particular, the fractional Laplacian, and investigate the positivity properties of nonlocal Schrödinger type operators, driven by the fractional Laplacian by developing a criterion that links the positivity of the spectrum of such operators with the existence of certain positive supersolutions, thereby establishing necessary and sufficient conditions for the existence of a configuration of poles that ensures the positivity of the corresponding Schrödinger operator.

For a given nonempty subset L of the line set of the projective plane PG(2,q), a blocking set with respect to L (or simply, an L-blocking set) is a subset B of the point set of PG(2,q) such that every line of L contains at least one point of B. Let E (respectively; T, S) denote the set of all lines which are external (respectively; tangent, secant) to an irreducible conic in PG(2,q). We shall discuss minimum size L-blocking sets of PG(2,q) for L = E, S, T, SUT, EUT, SUE.

Abstract. Bozejko and Speicher in 1994 defined a von-Neumann algebra corresponding to the    qij -deformed commutation relation. In this talk we will define the von-Neumann algebra and study its factoriality under some restriction. 

Abstract: I shall talk about the Gelfand-Naimark-Segal construction and prove that any general C*-algebra is isometrically ∗-isomorphic to a norm closed subalgebra of B(H) for some Hilbert space H.

Abstract: River basin geomorphology is a very old subject of study initiated by Horton (1945). Various statistical models of drainage networks have been proposed. Each such model is a random directed graph with its own nuances. In recent years physicists have been interested in these models because of the commonality of such branching networks in many areas of statistical physics (see Rodríguez-Iturbe  and Rinaldo (1997) for a detailed survey). We discuss the geometric features of one such model and also its scaling limit. The scaling limit of this model is the Brownian web, which has lately been the focus of extensive study among probabilists. Most of this talk will be accessible to a general audience.

For a nonnegative potential function q and a given locally finite graph G, we study thecombinatorial Schr¨odinger operator Lq(G) = ∆G + q with Dirichlet boundary condition on aproper finite subset S of the vertex set of G such that the induced subgraph on S is connected.Let Υp = {q ∈ Lp(S) : q(x) ≥ 0, Px ∈Sqp(x) ≤ 1}, for 1 ≤ p < ∞. We prove the existenceand uniqueness of the maximizer of the smallest Dirichlet eigenvalue of Lq(G), whenever thepotential function q ∈ Υp. Furthermore, we also establish the analogue of the Euler-Lagrangeequation on graphs.

Abstract  In the 18th century, while dealing with astronomical and geodesic measurements, the scientists were confronted with a statistical problem, which in those days was described as "the problem of combining inconsistent equations". People who worked on this problem and contributed towards its solutions include Euler,  Laplace, Gauss and Legendre among many others. I shall discuss the history of the problem and how it eventually led to the invention of the method of least squares.

Let M be a compact manifold without boundary. One can define a smooth real valued function of the space of Riemannian metrics of M by taking Lp-norm of Riemannian curvature for p ≥ 2. Compact irreducible locally symmetric spaces are critical metrics for this functional. I will show that rank 1 symmetric spaces are local minima for this functional by studying stability of the same at those metrics. I will also exhibit examples of symmetric metrics which are not local minima for it. In the 2nd part of my talk I will talk about Wilking’s criterion for Ricci Flow. B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators C(S), which are nonnegative in a suitable sense, to every AdSO(n,C) invariant subset S ⊂ so(n,C). We show that if S is an AdSO(n,C) invariant subset of so(n, C) such that S ∪ {0} is closed and C+(S) ⊂ C(S) denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in C+(S) also admits a metric with curvature operator in C+(S) (b) The normalized Ricci flow on any compact Riemannian manifold M with curvature operator in C+(S) converges to a metric of constant positive sectional curvature.

Lecture series abstract: We will survey the von Neumann algebraic approach to locally compact quantum groups in the sense of Kustermans and Vaes, roughly following the treatment by Van Daele. We will begin with an intro to the theory of weights of von Neumann algebras. We will then proceed to describe locally compact quantum groups. If there is any time left, we will present some recent progress in the field. The prerequisite for attending the talks is some basic knowledge of C*-algebras (and perhaps also von Neumann algebras, depending on the audience).

The Schur multiplier of a finite group is the second cohomology group with complex coefficients. It was introduced in the beginning of 20th century by Issai Schur in his work on projective representations. Since then they have been proved to be a powerful tool in group theory. To determine the Schur multiplier of a given finite group is often a difficult task. Therefore it is of interest to provide bounds for the numerical qualities such as the order, exponent and the rank of the Schur multiplier. In this talk we shall provide sharp lower and upper bounds for the order of some special p-groups and classify their covering groups.

The first talk will be a leisurely introduction to dynamics of group actions on homogeneous spaces of Lie groups.

The E_2-term of the Adams spectral sequence may be identified with certain derived functors, and this also holds for other Bousfield-Kan types spectral sequence. In this talk, I'll explain how the higher terms of such spectral sequences are determined by truncations of functors, defined in terms of certain (spectrally) enriched functor called mapping algebras. This is joint work with David Blanc.

We will continue with the discussion on (co)tangent bundle of a manifold. We will define the differential of a smooth map and state the implicit function theorem. We will also work out some concrete examples. 

Abstract: Simply put, a graph is nothing but a set of points ("vertices"), along with edges which connect them. A planar graph is a special type of graph- one where the edges do not cross each other ( i.e., meet only at the vertices). In this talk, we will discuss an interesting problem in planar graph theory- that of trying to colour the vertices with as few colours as possible, subject to the condition that no two adjacent vertices have the same colour. We will go on to prove the Five Colour Theorem, which says that for any planar graph, there exists such a colouring utilising only five colours. We will further discuss the Four Colour "Conjecture".

The only prerequisites are curiosity and an eagerness to learn!

Regression analysis is a branch of statistics that provides us with tools to model a relationship between variables. In this presentation, we will discuss two types of regression models: the classical linear regression and non-parametric regression. Linear regression analysis aims to fit a linear model to data with the help of Least Square Estimates. For this model, we will discuss the techniques for checking the appropriateness of the model, the tests for determining the significance of regressor variables, the methods for dealing with multi-collinearity, and the procedure for incorporating categorical (or qualitative) variables into our regression model. We will also talk about a few topics from non-parametric regression such as density estimation and smoothing. Most of the concepts will be illustrated in statistical software R using real-life data-sets.

Abstract : I shall discuss how Monte Carlo methods can be used to solve complex optimization problems. In particular, I shall discuss the simulated annealing algorithm.

The aim of this talk is to describe some conjectures which are part of the Langlands program.The talk is aimed at non-experts and should be accessible to a broad audience. 

This is a Short Lecture Series in Mathematics (SLSM) consisting of 4 lectures of 90 minutes each. This lecture schedule in this series is the following:

1st Lecture: Monday, October 09, 2017 - 11:30 to 13:00

2nd Lecture: Tuesday, October 10, 2017 - 11:30 to 13:00

3rd Lecture: Wednesday, October 11, 2017 - 16:30 to 18:00

4th Lecture: Thursday, October 12, 2017 - 15:30 to 17:00

Abstract: The first half of the mini course will be an introducing to the two classical models of random graphs (a.k.a. Erdős-Rényi random graphs) and discuss the phenomenon of phase transition. We will also discuss thresholds for monotonic properties with examples including connectivity threshold and sub-graph containment threshold.

In the second half of the course we will consider other kind of random graphs. In particular, we will discuss various models for complex networks, including Albert-Barabási preferential attachment models. We will discuss "scale-freeness", asymptotic degree distribution and "small-world phenomenon". Properties of super and sub-linear preferential attachment models and some recent developments in de-preferential attachment models will also be discussed.

If time permits we will also introduce the random geometric graphs and discuss asymptotic of the connectivity threshold.

References:

   1. Random Graphs by Svante Janson, Tomasz Łuczak and Andrzej Rucinski;
   2. Random Graphs by Béla Bollobás; 
   3. Random Graphs and Complex Networks by Remco van den Hofstad;
   4. Random Geometric Graphs by Mathew Penrose. 

Recent decades have witnessed significant growth and progress in spatial statistics, with applications in agriculture, epidemiology, geology, image analysis and other areas of environmental science. In recent years, new perspectives have emerged in connecting Gaussian Markov random fields with geostatistical models, and in advancing vast statistical computations. This series of lectures will focus on basic theory and computations of spatial statistics. Topics will include conditional and intrinsic autoregressions, connections between Markov random fields and geostatistics, variogram calculations, h-likelihood methods and matrix-free computations. Applications from agricultural variety trials, environmental sciences and geographical epidemiology will be discussed.

Abstract: In this talk, we will discuss some arithmetic and distribution type results on various partition functions, namely, Andrews’ singular overpartitions, cubic and overcubic partition pairs, and Andrews’ integer partitions with even parts below odd parts. We use some dissection formulas of Ramanujan’s theta functions, and arithmetic properties of modular forms and eta-quotients to study the distributions and to find some infinite family congruences of these partition functions. Also, using Radu’s algorithm on modular forms, we prove a conjecture on cubic partition pairs.

In the first part of the talk, we will discuss briefly the theory of Equidistribution. In the second part of the talk we will see the distribution of gaps between eigenvalues of Hecke operators in both horizontal and vertical settings. Moreover, we will deduce a stronger version of multiplicity one theorem for the space of cusp forms of weight k and level N from the joint Sato-Tate conjecture. This is a joint work with M. Ram Murty. Finally, using recent developments in the theory of l-adic Galois representations we will study the normal number of prime factors of sums of Fourier coefficients of eigenforms. The final part is a joint work with M. Ram Murty and V. Kumar Murty.

We will define what is a Laplacian on a smooth Riemannian Manifold. We will then define what are harmonic forms and state the Hodge Decomposition Theorem. In particular, we will explain how it implies Poincare Duality. We will then go on to define the notion of a weak solution and explain how elliptic regularity of the Laplacian, implies the Hodge Decomposition Theorem. We will review the basic notions of smooth manifolds and keep prerequisites to a minimum. Those who are interested in Analysis and PDE are encouraged to attend.  

In this talk we will be discussing about vanishing pressure limit for the equation

u_t + (u^2/2)_x = 0

ρ_t + (ρu)_x = 0,

with initial data

u(x, 0) = u_0(x),

ρ(x, 0) = ρ_0(x),

where u is the velocity component and ρ is the density component. This equation is considered as one of the model for the large scale structure formation of universe. In this direction we find some partial results for Reimann type initial data.

Abstract: In this talk, we will prove a fundamental theorem on differential geometry, namely the embedding theorem due to Hassler Whitney, which shows that every smooth manifold can be embedded into a Euclidean space.

Abstract: Tensor is a multidimensional array (for example matrix is a tensor of order 2). Tensors often arises from the discretizations of multidimensional functions that are involved in the numerical treatment of complex problems in many different areas of natural, financial or social sciences. The direct numerical treatment of these arrays leads to serious problems like memory requirements and the complexity of basic operations (they grow exponentially in d). In the last decade the approximation of multidimensional arrays has become a central issue in approximation theory and numerial analysis. The main idea of the approximation of a tensor is decomposing the given tensor as sums of outer products of vectors. In the language of functions, it is an approximation of multivariable functions by sums of products of univariate functions. Tensor decompositions has lot of applications in image processing, quantum chemistry, data mining, machine learning stochastic partial differential equations etc.In the matrix case (i.e tensor of order 2), the singular value decomposition (SVD) represents a matrix as sum of outer product of vectors. SVD algorithm requires O(n3) arithmeticoperations (if the matrix is of size n × n). So it is very expensive when the matrix dimensions are large. Various inexpensive techniques of low rank approximation based on skeleton/cross approximation are available in the literature. SVD and its applications, other low rank approximation techniques like RRQR, Interpolative decomposition, randomized algorithms, skeleton/cross approximation techniques will be discussed in the talk. Canonical,Tucker, Tensor Chain and Tensor Train formats for higher order tensors will be introduced.

Local Index formula is at the heart of the so called hard Riemannian aspects of Noncommutative Geometry. We will try to see why thisis so important. However so far we have only one computation of the local index formula. We will discuss two more computations. 

A major theme of interest in mathematics is the study of Geometric structures on manifolds. Geometric structures are modelled on nice topological spaces with a group of automorphisms acting transitively on them. The spaces of negative curvature are widely studied from this viewpoint. A special class of these being real and complex hyperbolic spaces. In this talk, we will first focus on the geometry of complex hyperbolic spaces. In this pursuit, we will provide algebraic classification of the isometries of these spaces, and surface group representations into their isometry groups. This is achieved by means of algebraic data given by the traces of these isometries, and some other conjugacy invariants.In the second half of this talk, we will focus on hyperbolic structures on closed oriented surfaces of negative Euler characteristic. To this end, we enter into the realm of mapping class group which serves as a significant tool in connecting the geometry and topology of surfaces. In particular, we will give a method to decompose a periodic mapping class into irreducible components, and provide some interesting algebraic and geometric applications. We will conclude this talk by connecting the study of these mapping classes to the theory of “ribbon” graphs.

I will present interactions among 2-dimensional conformal field theory, which is a kind of quantum field theory in physics, theory of operator algebras and the Moonshine conjecture which predicted mysterious relations between the finite simple group Monster and the elliptic modular function. I will emphasize representation theoretic aspects and do not assume anyknowledge of these theories. 

It is well known that the Orlicz space is a natural generalization of the Lebesgue spaces. A vector measure is a countably additive Banach space valued measure. We discuss the Banach algebra structure on the Orlicz spaces associated to a vector measure over compact groups. We also discuss the Fourier transform of functions that are integrable with respect to vector measures over compact groups. We also introduce and study the convolution of functions from Lp-spaces associated to a vector measure.

In this talk we explore the problem of determining the dimension of the L^2 [0, ∞)span of n linearly independent L^2_ loc [0, ∞) functions and the fascinating geometry associatedwith it. This problem appears in the context of Weyl’s limit classification problems of formallysymmetric differential expressions on [0, ∞).

For simple versus simple hypothesis testing, the famous Neyman-Pearson lemma (1933) provides the best fixed sample size test procedure that has minimum type II error among all the tests that control type I error at some prescribed level. However, it is well-known that even such a best test cannot guarantee control of type II error while retaining the type I error control as long as the sample size is fixed. Abraham Wald (1943) extended the Neyman-Pearson lemma for sequentially observed data by introducing the Sequential Probability Ratio Test (SPRT) that simultaneously controls type I and type II error probabilities at some prefixed levels. In this talk, I will be discussing this test procedure along with boundary conditions, average sample size, the optimality property of SPRT and truncated SPRT. If time permits, I will discuss how to extend this idea for testing k (>2) simple-vs-simple hypotheses with the goal of selecting the correct hypothesis. This problem is known as Multi-hypothesis sequential probability ratio test (MSPRT).

Abstract : I shall begin with the description of the lattice points counting problem in Euclidean spaces. Namely, establishing an (asymptotic) error estimate for the number of points that the lattice of integral points has in a Euclidean ball of large radius, as the radius goes to infinity. 
This problem has a very long history and vast literature is available in obtaining error estimates for Euclidean dilates of the unit ball as well as various other convex bodies. 
In the first half of the talk, I shall sketch the Fourier spectral method which originated with Minkowski for Euclidean balls. This classical method does not give the best possible error estimates for the balls, but it gives optimal error in the class of Euclidean dilates of convex bodies with surfaces having non-vanishing Gaussian curvature at all points.

In the last half of the talk, I shall discuss the lattice points counting problem in the context of the Heisenberg groups, for families of balls corresponding to certain radial and homogeneous norms, including the canonical Cygan-Koranyi norm. 

In the end, I shall briefly mention the scope of this method discussing more general nilpotent Lie groups. 

This talk is based on my joint work with Amos Nevo and Krystal Taylor which is available on arXiv. Most of this talk should be accessible to those who are familiar with basic functional analysis. 

• Title: Gaussian Elimination in classical groups
• Abstract: The classical row-column operations provide an efficient algorithm to solve word problem in the matrix group GL(n,k). We will see an analogues result for orthogonal and symplectic groups too.

In this lectures, we would discuss probabilistic model of primes leading to heuristics about their distribution. We would see many surprising irregularities popping up alongside expected results. A survey of several recent and important results would be presented in a way accessible to non-experts.

This will be the third lecture of a series of five on Finite Fields.

In this talk, we will report on recent work (with J. Hilgert and S. Hansen, Paderborn) relating resonances and scattering poles on Riemannian symmetric spaces of rank one. We use boundary values in the sense of Kashiwara and Oshima to show that resonances and scattering poles coincide, along with their residues. Our methods also enable us to give a new and simple proof of the Helgason's conjecture in the rank one case. Time permitting, we'll mention progress made for symmetric spaces of higher rank. 

Abstract: This will be an introductory talk towards the theory of finite simple groups and their corresponding geometries. We shall discuss some examples which will be accessible to master level students.

First I shall introduce the moduli space of curves over complex numbers.This has been a central object of study in Algebraic Geometry with connections in mathematical physics, number theory and complex dynamics. This is also the primary object of study in my research. I shall then talk about a stratification of the moduli space by affine varieties and mention some partial results in this direction. Changing topics I shall then talk about more recent work on some moduli spaces of elliptic curves with applications to number theory.

Digital signature is a fundamental cryptographic primitive. It enables an entity to validate the authenticity and integrity of a digital document. In this talk, I will be discussing Group signature, Nominative Signature and Multi-signature. Group signature scheme enables any group member to produce a signature anonymously on behalf of the group and in case of misbehavior, the signer can be traced out. In a nominative signature scheme, a nominator and a nominee jointly produce a signature such that only the nominee can verify the signature. Multisignature is a powerful cryptographic primitive that helps to reduce the bandwidth taken by N signatures from O(N ) to O(1). It provides a group of signers the ability to sign collaboratively a common message in such a way that the size of the multisignature remains the same as that of a single signature and the verifier gets convinced that the message has been signed by all the signers.

Abstract:  In this talk, we aim to introduce the process of mathematical modeling using differential equations. Basic compartmental models for spread of infectious diseases (mathematical epidemiology) will be described. In particular, models on the impact of awareness on epidemic outbreak will be studied. Also, methods to analyze the local stability of the equilibrium states of the system will be discussed.

Prerequisite: High school level knowledge of Ordinary Differential Equations

 We study the limiting behavior of the solutions of Euler equations of one-dimensional compressible fluid flow as the pressure like term vanishes. This system can be thought of as an approximation for the one dimensional model for large scale structure formation of universe. We show that the solutions of former equation converges to the solution of later in the sense of distribution and agrees with the vanishing viscosity limit when the initial data is of Riemann type. 

Abstract: In the first half of the talk we define unimodular rows over a ring R and see some of its basic properties. The completability of such unimodular rows can be viewed in terms of some special multiplicative subgroups of the matrix ring over R. In the second half, we define projective modules. Free modules are projective. In 1976 Quillen-Suslin gave a sufficient condition under which the converse is true. The completability of unimodular rows over certain polynomial rings plays an important role to show the fact that finitely generated projective modules over such ring are free. 

Abstract:

The Lie groups were discovered by Sophus Lie around 1880 while searching for a framework to analyze the continuous symmetries of differential equations in much the same way as permutation groups are used in Galois theory for analylyzing the discrete symmetries of algebraic equations. One of the important idea in the theory of Lie groups is to replace the gobal object, the group, with its local or linearized version which Lie called its “infinitesimal group”, now known as its Lie algebra. Around 1940, after Elie Cartan’s beautiful classification of finite dimensional semisimple Lie algebras, Lie algebras emerged as an independent branch of Algebra. An important class of infinite dimensional Lie algebras generalizing the finite dimensional semisimple Lie algebras were discovered independently by Victor Kac and Robert Moody in 1968.  An important family of these infinite Lie algebras is known as affine Lie algebras. These Lie algebras have proved to be very important with interactions in many areas of mathematics and physics. One such interaction is with number theory, particularly combinatorial identities.  In this talk I will give an overview of some applications in this direction.

In this talk we will see the well posedness results for the nonlinear Schr\"{o}dinger equation for the magnetic Laplacian on $\R^{2n}$, corresponding to constant magnetic field, namely the twisted Laplacian on $\C^n$ with power type nonlinearity $\lambda |u|^{\alpha} u$. We establish the well posedness in certain first order Sobolev spaces associated to the twisted Laplacian. The approach is via the spectral theory of the Schr\"{o}dinger propagator for the twisted Laplacian, and local existence is proved using Strichartz estimates established for the same.Using blowup analysis and conservation laws, we conclude global well posedness in the defocussing case (\lambda>0) with $0\leq \alpha< 2/(n-1)$ and also in the focussing case (\lambda<0) with $0\leq \alpha< 2/n$.We also prove finite time blow up in the focussing case (\lambda<0) with $2/n\leq \alpha< 2/(n-1)$. In this talk we also see a Hardy-Sobolev inequality for the twisted Laplacian on $\C^n$. We also show that the inequality is optimal in the sense that weight can not be improved.