Abstract:
Let G be a simple algebraic group over the field of complex numbers and B be a Borel subgroup of G. Let X_w be a Schubert variety in the flag variety G/B corresponding to an element w of the Weyl group of G, and let Z_w be the Bott–Samelson variety, a natural desingularization of X_w. In this talk we discuss the classification of the "reduced expressions of w" such that Z_w is Fano or weak Fano.

Abstract:
GATE is a big exam used for PG admissions by academic institutes as well as hiring by PSUs. In 2015, more than 8 lakh people appeared for GATE, all subjects combined.
GATE uses formula scoring with negative marking for multiple choice questions or MCQs, e.g., 1, 0, and -1/3 marks for correct, omitted, and wrong answer, respectively. Some questions have 2 marks, with -2/3 for wrong answers. Some have numerical answers, with no negative marking. The number of distinct scores possible is small (below 400), and the number of candidates is large (lakhs).
A modern statistical approach to evaluating MCQ exams uses item response theory (IRT; also called latent trait models). In this approach, each question has some parameters, called “difficulty” or “discrimination” etc., written abstractly as vector a, and each student has an ability or talent attribute, written abstractly as a scalar theta. The probability that a given question (with vector a) will be answered correctly is taken to be some specified function f(theta,a).
The definition of a, and choice of f, are modeling decisions. Later calculations, though complex, are routine. The aim is to estimate a for every question and theta for every candidate.
Two common IRT models are the Rasch model and the 2-parameter logistic (2PL) model, which I will describe. Since our question outcomes are not dichotomous (right/wrong) but polytomous (right/ omitted/wrong), we use the generalized partial credit model (GPCM), which I will describe. GPCM results are poor. The estimated abilities have low correlations with formula scores; these correlations vary across disciplines; and there are also clear conceptual problems in applying GPCM to GATE.
I will then present our new two-step IRT model, where the candidate first decides whether or not to attempt the question, and then (if attempting) gets it right or wrong. The corresponding mathematical model is simple, and aligned with how we believe GATE works. Results are better. The correlation with formula scores is higher, and near-constant across disciplines.
The policy implications of our model are positive. We now have a two-dimensional score of each candidate’s performance. The formula score represents an overall knowledge score, which may appeal to industry. The IRT ability estimate represents an academic potential estimate, which may appeal to academic institutes. If admission and hiring processes are no longer based on the same measure, both may benefit. A minor extra advantage is the possibility of awarding rationally derived ranks to individual candidates, with very few clashes.
As part of the talk, I will also discuss the estimation methods and numerical implementation. However, the emphasis will be on model statement, results, and policy implications. I hope that most of the talk will be accessible to all stakeholders in GATE.

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Henry-Parusinski proved that the Lipschitz right classification of function germs admits continuous moduli. This allows us to introduce the notion of Lipschitz simple germs and list all such germs. We will present the method of the classification in this talk and mention some open problems.

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In this talk, the likelihood construction is explained under different censoring schemes. Further, the techniques for estimation of the unknown parameters of the survival model are discussed under these censoring schemes.

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In this talk, the change point problem in hazard rate is considered. The Lindley hazard change point model is discussed with its application to model bone marrow transplant data. Further, a general hazard regression change point model is discussed with exponential and Weibull distribution as special cases.

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A {\em simplicial cell complex} K of dimension d is a poset isomorphic to the face poset of a d-dimensional simplicial CW-complex X. If a topological space M is homeomorphic to X, then K is said to be a {\em pseudotriangulation} of M. In 1974, Pezzana proved that every connected closed PL d-manifold admits a (d+1)-vertex pseudotriangulation. For such a pseudotriangulation of a PL d-manifold one can associate a (d+1)-regular colored graph, called a crystallization of the manifold.
Actually, crystallization is a graph-theoretical tool to study topological and combinatorial properties of PL manifolds. In this talk, I shall define crystallization and show some applications on PL d-Manifolds for d=2, 3 and 4.

In dimension 2, I shall show a proof of the classification of closed surfaces using crystallization. This concept has some important higher dimensional analogs, especially in dimensions 3 and 4. In dimensions 3 and 4, I shall give lower bounds for facets in a pseudotriangulation of a PL manifolds. Also, I shall talk on the regular genus (a higher dimensional analog of genus) of PL d-manifolds. Then I shall show the importance of the regular genus in dimension 4. Additivity of regular genus has been proved for a huge class of PL 4-manifolds. We have some observations on the regular genus, which is related to the 4-dimensional Smooth Poincare Conjecture.

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Rayleigh-Bénard convection is a classical extended dissipative system which shows a plethora of bifurcations and patterns. In this talk, I'll present the results of our investigation on bifurcations and patterns near the onset of Rayleigh-Bénard convection of low-Prandtl number fluids. Investigation is done by performing direct numerical simulations (DNS) of the governing equations. Low dimensional modeling of the system using the DNS data is also done to understand the origin of different flow patterns. Our investigation reveals a rich variety of bifurcation structures involving pitchfork, Hopf, homoclinic and Neimar-Sacker bifurcations.

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In this talk, we will discuss some recent results on the existence and uniqueness of strong solutions of certain classes of stochastic PDEs in the space of Tempered distributions. We show that these solutions can be constructed from the solutions of "related" finite dimensional stochastic differential equations driven by the same Brownian motion. We will also discuss a criterion, called the Monotonicity inequality, which implies the uniqueness of strong solutions.

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The investigation of solute dispersion is most interesting topic of research owing to its outspread applications in various fields such as biomedical engineering, physiological fluid dynamics, etc. The aim of the present study is to know the different physiological processes involved in the solute dispersion in blood flow by assuming the relevant non-Newtonian fluid models. The axial solute dispersion process in steady/unsteady non-Newtonian fluid flow in a straight tube is analyzed in the presence and absence of absorption at the tube wall. The pulsatile nature of the blood is considered for unsteady flow. Owing to non-Newtonian nature of blood at low shear rate in small vessels, non-Newtonian Casson, Herschel-Bulkley, Carreau and Carreau-Yasuda fluid models which are most relevant for blood flow analysis are considered. The three transport coefficients i.e., exchange, convection and dispersion coefficients which describe the whole dispersion process in the system are determined. The mean concentration of solute is analyzed at all time. A comparative study of the solute dispersion is made among the Newtonian and non-Newtonian fluid models. Also, the comparison of solute dispersion between single- and two-phase models is made at all time for different radius of micro blood vessels.

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Quasilinear symmetric and symmetrizable hyperbolic system has a wide range of applications in engineering and physics including unsteady Euler and potential equations of gas dynamics, inviscid magnetohydrodynamic (MHD) equations, shallow water equations, non-Newtonian fluid dynamics, and Einstein field equations of general relativity. In the past, the Cauchy problem of smooth solutions for these systems has been studied by several mathematicians using semigroup approach and fixed point arguments. In a recent work of M. T. Mohan and S. S. Sritharan, the local solvability of symmetric hyperbolic system is established using two different methods, viz. local monotonicity method and a frequency truncation method. The local existence and uniqueness of solutions of symmetrizable hyperbolic system is also proved by them using a frequency truncation method. Later they established the local solvability of the stochastic quasilinear symmetric hyperbolic system perturbed by Levy noise using a stochastic generalization of the localized Minty-Browder technique. Under a smallness assumption on the initial data, a global solvability for the multiplicative noise case is also proved. The essence of this talk is to give an overview of these new local solvability methods and their applications.

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In this talk, we present the moduli problem of rank 2 torsion free Hitchin pairs of fixed Euler characteristic χ on a reducible nodal curve. We describe the moduli space of the Hitchin pairs. We define the analogue of the classical Hitchin map and describe the geometry of general Hitchin fibre. Time permits,talk on collaborated work with Balaji and Nagaraj on degeneration of moduli space of Hitchin pairs.

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The formalism of an ``abelian category'' is meant to axiomatize the operations of linear algebra. From there, the notion of ``derived category'' as the category of complexes ``upto quasi-isomorphisms'' is natural, motivated in part by topology. The formalism of t-structures allows one to construct new abelian categories which are quite useful in practice (giving rise to new cohomology theories like intersection cohomology, for example).
In this talk we want to discuss a notion of punctual (=``point-wise'') gluing of t-structures which we formulated in the context of algebraic geometry. The essence of the construction is classical and well known, but the new language leads to several applications in the motivic world.

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Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, product of elementary Abelian groups and dihedral groups.
Restricting them to finite skeleta constructs equivariant maps between spaces which are related to the topological Tverberg conjecture. This answers negatively a question of \"Ozaydin posed in relation to weaker versions of the same conjecture. Further, it also has consequences for Borsuk-Ulam properties of representations of cyclic and dihedral groups.
This is joint work with Samik Basu.

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Quantile regression provides a more comprehensive relationship between a response and covariates of interest compared to mean regression. When the response is subject to censoring, estimating conditional mean requires strong distributional assumptions on the error whereas (most) conditional quantiles can be estimated distribution-free. Although conceptually appealing, quantile regression for censored data is challenging due to computational and theoretical difficulties arising from non-convexity of the objective function involved. We consider a working likelihood based on Powell's objective function and place appropriate priors on the regression parameters in a Bayesian framework. In spite of the non-convexity and misspecification issues, we show that the posterior distribution is strong selection consistent. We provide a “Skinny” Gibbs algorithm that can be used to sample the posterior distribution with complexity linear in the number of variables and provide empirical evidence demonstrating the fine performance of our approach.

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Given a closed smooth Riemannian manifold M, the Laplace operator is known to possess a discrete spectrum of eigenvalues going to infinity. We are interested in the properties of the nodal sets and nodal domains of corresponding eigen functions in the high energy (semiclassical) limit. We focus on some recent results on the size of nodal domains and tubular neighbourhoods of nodal sets of such high energy eigenfunctions (joint work with Bogdan Georgiev).

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I will define affine Kac-Moody algebras, toroidal Lie algebras and full toroidal Lie algebras twisted by several finite order automorphisms and classify integrable representations of twisted full toroidal Lie algebras.

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In recent years, one major focus of modeling spatial data has been to connect two contrasting approaches, namely, the Markov random field approach and the geostatistical approach. While the geostatistical approach allows flexible modeling of the spatial processes and can accommodate continuum spatial variation, it faces formidable computational burden for large spatial data. On the other hand, spatial Markov random fields facilitate fast statistical computations but they lack in flexibly accommodating continuum spatial variations. In this talk, I will discuss novel statistical models and methods which allow us to accommodate continuum spatial variation as well as fast matrix-free statistical computations for large spatial data. I will discuss an h-likelihood method for REML estimation and I will show that the standard errors of these estimates attain their Rao-Cramer lower bound and thus are statistically efficient. I will discuss applications on ground-water Arsenic contamination and chlorophyll concentration in ocean. This is a joint work with Debashis Mondal at Oregon State University

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It is a well-known result from Hermann Weyl that if alpha is an irrational number in [0,1) then the number of visits of successive multiples of alpha modulo one in an interval contained in [0,1) is proportional to the size of the interval. In this talk we will revisit this problem, now looking at finer joint asymptotics of visits to several intervals with rational end points. We observe that the visit distribution can be modelled using random affine transformations; in the case when the irrational is quadratic we obtain a central limit theorem as well. Not much background in probability will be assumed. This is in joint work with Jon Aaronson and Michael Bromberg.

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The action of Gl_n(F_q) on the polynomial ring over n variables has been studied extensively by Dickson and the invariant ring can be explicitly described. However, the action of the same group on the ring when we go modulo Frobenius powers is not completely solved. I'll talk about some interesting aspects of this modified version of the problem. More specifically, I'll discuss a conjecture by Lewis, Reiner and Stanton regarding the Hilbert series corresponding to this action and try to prove some special cases of this conjecture.

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One way to understand representations of a group is to ‘restrict’ the representation to its various subgroups, especially to those subgroups which give multiplicity one or finite multiplicity. We shall discuss a few examples of restriction for the representations of p-adic groups. Our main examples will be the pairs $(GL_2(F), E^*) and (GL_2(E), GL_2(F))$, where $E/F$ is a quadratic extension of $p$-adic fields. These examples can be considered as low-rank cases of the well known Gross-Prasad conjectures, where one considers various ‘restrictions’ simultaneously. Further, we consider a similar ‘restriction problem’ when the groups under consideration are certain central extensions of $F$-point of a linear algebraic groups by a finite cyclic group. These are topological central extensions and called ‘covering groups’ or ‘metaplectic groups’. These covering groups are not $F$-point of any linear algebraic group. We restrict ourselves to only a two fold covers of these groups and their ‘genuine’ representations. Covering groups naturally arise in the study of modular form of half-integral weight. Some results that we will discuss are outcome of a joint work with D. Prasad.

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It is a well-accepted practice in experimental situations to use auxiliary information to enhance the accuracy of the experiment i.e., to reduce the experimental error. In its simplest form of use of auxiliary information, data generated through an experiment are statistically modeled in terms of some assignable source(s) of variation, besides a chance cause of variation. The assignable causes comprise ‘treatment’ parameters and the ‘covariate’ parameter(s). This generates a family of ‘covariate models’ - serving as a ‘combination’ of ‘varietal design models’ and ‘regression models’. These are the well-known Analysis of Covariance
(ANCOVA) Models. Generally, for such models, emphasis is given on analysis of the data [in terms of inference on treatment effects contrasts] and not so much on the choice of the covariate(s) values. In this presentation, we consider the situation where there is some flexibility in the choice of the experimental units with specified values of the covariates. The notion of 'optimal' choice of values of the covariates for a given design set-up so as to minimize variance for parameter estimates has attracted attention of researchers in recent times. Hadamard matrices and Mixed orthogonal array have been conveniently used to construct optimum covariate designs with as many covariates as possible.

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Recent increase in the use of 3-D magnetic resonance images (MRI) and analysis of functional magnetic resonance images (fMRI) in medical diagnostics makes imaging, especially 3-D imaging very important. Observed images often contain noise which should be removed in such a way that important image features, e.g., edges, edge structures, and other image details should be preserved, so that subsequent image analyses are reliable. Direct generalizations of existing 2-D image denoising techniques to 3-D images cannot preserve complicated edge structures well, because, the edge structures in a 3-D edge surface can be much more complicated than the edge structures in a 2-D edge curve. Moreover, the amount of smoothing should be determined locally, depending on local image features and local signal to noise ratio, which is much more challenging in 3-D images due to large number of voxels. In this talk, I will talk about a multi-resolution and locally adaptive 3-D image denoising procedure based on local clustering of the voxels. I will provide a few numerical studies which show that the denoising method can work well in many real world applications. Finally, I will talk about a few future research directions along with some introductory research problems for interested students. Most parts of my talk should be accessible to the audience of diverse academic background.

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We shall discuss a new method of computing (integral) homotopy groups of certain manifolds in terms of the homotopy groups of spheres. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we verify a conjecture of J. C.
Moore. This is joint work with Somnath Basu.

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The theory of pseudo-differential operators provides a flexible tool for treating certain problems in linear partial differential equations. The Gohberg lemma on unit circle estimates the distance (in norm) from a given zero-order operator to the set of the compact operators from below in terms of the symbol. In this talk, I will introduce a version of the Gohberg lemma on compact Lie groups using the global calculus of pseudo-differential operators. Applying this, I will obtain the bounds for the essential spectrum and a criterion for an operator to be compact. The conditions used will be given in terms of the matrix-valued symbols of operators

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Let $F$ be a $p$-adic field. The restriction of an irreducible admissible representation of $GL_{2}(F)$ to its maximal tori was studied by Tunnell and Saito; and they provide a very precise answer. In particular, one gets multiplicity one. This can be considered as the first case of the Gross-Prasad conjectures.

We will discuss a metaplectic variation of this question. More precisely, we will talk about the restriction of an irreducible admissible genuine representation of the two fold metaplectic cover $\widetilde{GL}_2(F)$ of $GL_2(F)$ to the inverse image in $\widetilde{GL}_2(F)$ of a maximal torus in $GL_2(F)$. We utilize a correspondence between irreducible admissible genuine supercuspidal representations of the metaplectic group widetilde{SL}_2(F)$ to irreducible admissible supercuspidal representations of linear group $SL_2(F)$. This is a joint work with D. Prasad.

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We will describe the problem of mod p reduction of p-adic Galois representations. For two dimensional crystalline representations of the local Galois group Gal(Q¯ p |Qp ), the reduction can be computed using the compatibility of p-adic and mod p Local Langlands Correspondences; this method was first introduced by Christophe Breuil in 2003. After giving a brief sketch of the history of the problem, we will discuss how the reductions behave for representations with slopes in the half- open interval [1, 2). In the relevant cases of reducible reduction, one may also ask if the reduction is peu or tr`es ramifi´ee. We will try to sketch an answer to this question, if time permits. (Joint works with Eknath Ghate, and also with Sandra Rozensztajn for slope 1.)

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Let M be a compact manifold without boundary. 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 prove that rank 1 symmetric spaces are local minima for this functional by studying stability of the functional at those metrics. I will also show examples of irreducible symmetric metrics which are not local minima for it.

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Unfolding operators have been introduced and used to study homogenization problems. Initially, it was introduced for problems with rapidly oscillating coefficients and porous domains. Later, this has been developed for domains with oscillating boundaries, typically with rectangular or pillar type boundaries which are classified as non-smooth.
In this talk, we will demonstrate the development of generalized unfolding operators, where the oscillations of the domain can be smooth and hence it has wider applications. We will also see the further adaptation of this new unfolding operators for circular domains with rapid oscillations with high amplitude of O(1). This has been applied to homogenization problems in circular domains as well.

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We describe three approaches to the classical p-completion(or localization) of a topological space: as spaces, through cosimplicial space resolutions, and through mapping algebras – and show how they are related through appropriate "universal" systems of higher cohomology operations. All terms involved will be explained in the talk.

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We deal with the following eigenvalue optimization problem: Given a bounded open disk $B$ in a plane, how to place an obstacle $P$ of fixed shape and size within $B$ so as to maximize or minimize the fundamental eigenvalue $lambda_1$ of the Dirichlet Laplacian on $B setmunus P$. This means that we want to extremize the function $rho \rightarrow lambda_1(B setminus rho(P))$, where $rho$ runs over the set of rigid motions such that $rho(P) subset B$. We answer this problem in the case where $P$ is invariant under the action of a dihedral group $D_{2n}$, and where the distance from the center of the obstacle $P$ to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of $P$ coincide with a diameter of $B$. The maximizing and the minimizing configurations are identified.

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We deal with the following eigenvalue optimization problem: Given a bounded open disk B in a plane, how to place an obstacle P of fixed shape and size within B so as to maximize or minimize the fundamental eigenvalue
λ1 of the Dirichlet Laplacian on B \ P . This means that we want to extremize the function ρ → λ1 (B \ ρ(P)), where ρ runs over the set of rigid motions such that ρ(P) ⊂ B. We answer this problem in the case where P is invariant under the action of a dihedral group D 2n, and where the distance from the center of the obstacle P to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of P coincide with a diameter of B. The maximizing and the minimizing configurations are identified.