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In this talk, we present a proof of $C^{1, \alpha}$ regularity up to the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. The proof is achieved via compactness arguments combined with new boundary H\"{o}lder estimates for equations which are uniformly elliptic when the gradient is either small or large.

Biography:

Dr. Ram Baran Verma, currently working as an assistant professor in the department of Mathematics SRM University AP. He completed his Ph.D from IIT Gandhinagar in 2018. His research interest is to study the existence and regularity properties of solutions to nonlinear elliptic equations. Prior joining to SRM University AP, he was a post-doctorate at TIFR-CAM Bangalore. For more details about Dr. Verma you can visit https://srmap.edu.in/faculty/dr-ram-baran-verma

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In this talk, I will be talking about the maximum principle for the Laplace equation and its consequences, like uniqueness of solutions to the Dirichlet problem, a prior uniform estimate and gradient estimate for solution of Laplace equation etc.

Biography:

Dr. Ram Baran Verma, currently working as an assistant professor in the department of Mathematics SRM University AP. He completed his Ph. D from IIT Gandhinagar in 2018. His research interest is to study the existence and regularity properties of solutions to nonlinear elliptic equations. Prior joining to SRM University AP, he was a post-doctorate at TIFR-CAM Bangalore. For more details about Dr. Verma you can visit https://srmap.edu.in/faculty/dr-ram-baran-verma

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In this talk we will discuss my recent research works. We will consider basically three problems
1. First, I will talk about the controllability of the Landau-Lifshitz-Gilbert Equation (LLGE) in an interval in one spatial dimension with Neumann boundary conditions. The control force acting here is degenerate i.e., it acts through a fewer number of modes. We exploit Fourier series expansion of the solution. We use methods of Lie bracket generating property to establish the global controllability of finite-dimensional Galerkin approximations of LLGE. Then, we show L^2 approximate controllability of the full system.
2. Then, we discuss a trigonometric method for the stochastic Euler-Bernoulli beam equation. We consider a fully discrete approximation of the linear stochastic Euler-Bernoulli beam equation driven by additive noise. A standard finite element method is used for the spatial discretization and a stochastic trigonometric scheme for the temporal approximation.
3. Finally, we discuss my very recent work on a stochastic cross diffusion system. We have shown the existence of a global martingale solution of that system by exploiting its entropy structure. This is an ongoing work with Prof. Ansgar Jüngel at TU Wien, Austria.

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We start discussing Gaussian elimination to solve system Ax=b. Then, we discuss LU, LDU^t, LL^t (Cholesky) factorization of a matrix A and show how these factorization methods are used to solve system Ax=b. We also discuss the algorithms and operation counts for each of these methods.

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The present paper deals with the Shear wave propagation in a multi-layered magnetoelastic anisotropic monoclinic medium with finite-difference modeling to comprehend the stability criteria, phase velocity, and group velocity. Utilizing Maxwell's fundamental theory of magnetoelasticity, the problem has been constructed. Stability analysis has been conducted based on the finite-difference technique to reduce the soaring error values and control its stability. Numerical evaluation as well as a graphical representation, have been employed to enlighten the effects of different values of courant number and magnetoelasticity on the phase and group velocities.

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The teaching session will deal with the Existence and uniqueness of the solution of the initial value problem for a first-order ordinary differential equation.

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Thermoelastic vibration of micro and nano-beam resonators is described by the Euler-Bernoulli beam theory in context with two-temperature generalized thermoelasticity theory. The amplitude of deflection and thermal moment in the case of simply supported and the isothermal beam are obtained by the Laplace transform method along with the finite Fourier sine transform method and the vibration frequency is examined by the normal mode analysis when the beam-ends are clamped and isothermal. The effects of the two-temperature parameter and micro and nano-beam thickness on the amplitude of deflection, thermal moment, and thermoelastic vibration frequency of the micro and nano-beam resonator have been studied and shown with the numerical results. A comparison of the results with the corresponding results of the two-temperature generalized thermoelasticity (2TLS) and the Lord-Shulman (LS) theories is also presented. Besides, the size dependency nature of the micro and nano-beam is analyzed by analytical and numerical results.

Biography::

Bachelor of Science in Mathematics (Honors) from Gaya College (Magadh University), Gaya, Bihar, India. (2010-2013) Master of Science in Mathematics from Central University Of South Bihar, Gaya, Bihar, India. (2013-2015) Doctor of Philosophy in Mathematics from Central University Of South Bihar, Gaya, Bihar, India. (2017-2021) Thesis Title-: Effects of Phase-Lag on Thermoelastic Damping in Micro and Nano Beam Resonators under Generalized Thermoelasticity Research Interest- Mathematical Physics; Solid Mechanics; Heat Transfer; Thermoelastic Damping in Resonators, Wave Propagation Life Member of "Indian Mathematical Society (IMS)". Best Research Award, International Research Awards on New Science Inventions (NESIN 2020), ScienceFather. 13 research publication in peer reviewed reputed international journals (1 SCOPUS Index, 12 SCI Index).
Link- https://orcid.org/0000-0002-4580-6513

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The interaction between interstitial fluid flow and the deformation of the underlying porous structure gives rise to a variety of mechanisms of fluid-structure coupling. In the specific case of Biot poromechanics, this interaction occurs when the linearly elastic porous medium is saturated, and such problem is relevant to a large class of very diverse applications ranging from bone healing to, e.g., petroleum engineering, or sound isolation. We are also interested in the interface between elastic and poroelastic systems that are encountered in hydrocarbon production in deep subsurface reservoirs (a pay zone and the surrounding non-pay rock formation), or in the study of tooth and periodontal ligament interactions.
In this talk, we discuss the \textit{a posteriori} error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretisations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of degree $k$ for rotation vector, total pressure, and elastic pressure. Residual-based estimators are constructed, and upper and lower bounds (up to data oscillations) for all global estimators are rigorously derived. The methods are all robust with respect to the model parameters (in particular, the Lam\'e constants), they are valid in 2D and 3D, and also for arbitrary polynomial degree $k\geq 0$. The error behaviour predicted by the theoretical analysis is then demonstrated numerically on a set of computational examples including different geometries on which we perform adaptive mesh refinement guided by the \textit{a posteriori} error estimators.
Finally, we discuss some remarks related to other contributions.

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We know how to obtain the general solution of the nth-order linear differential equation with constant coefficients. We have seen that in such cases the form of the complementary function may be readily determined. The general nth-order linear equation with variable coefficients is quite a different matter, however, and only in certain special cases can the complementary function be obtained explicity in closed form. One special case of considerable practical importance for which it is fortunate that this can be done is the so-called Cauchy-Euler equation.

Biography:

Bachelor of Science in Mathematics (Honors) from Gaya College (Magadh University), Gaya, Bihar, India. (2010-2013) Master of Science in Mathematics from Central University Of South Bihar, Gaya, Bihar, India. (2013-2015) Doctor of Philosophy in Mathematics from Central University Of South Bihar, Gaya, Bihar, India. (2017-2021) Thesis Title-: Effects of Phase-Lag on Thermoelastic Damping in Micro and Nano Beam Resonators under Generalized Thermoelasticity Research Interest- Mathematical Physics; Solid Mechanics; Heat Transfer; Thermoelastic Damping in Resonators, Wave Propagation Life Member of "Indian Mathematical Society (IMS)". Best Research Award, International Research Awards on New Science Inventions (NESIN 2020), ScienceFather. 13 research publication in peer reviewed reputed international journals (1 SCOPUS Index, 12 SCI Index).
Link- https://orcid.org/0000-0002-4580-6513

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In this lecture we discuss numerical methods for solving a system of linear equations Ax = b, where the given matrix A is real, n × n, and nonsingular and b is a given real vector in Rn . We seek a solution x that is necessarily also a vector in Rn. It is well known that such problems arise frequently in any branch of science, engineering, economics, or finance. There is no single technique that is perfect for all cases. But the available numerical methods in the literature can generally be divided into two classes: direct methods and iterative methods. In this lecture, we will discuss methods of the first type. In the absence of roundoff error, we see that such methods would yield the exact solution within a finite number of steps. The basic direct method for solving linear systems of equations is Gaussian elimination. The Gaussian elimination method proceeds by successively eliminating the elements below the diagonal of the matrix of the linear system until the matrix becomes triangular, when the solution of the system is very easy. In the end part of the lecture, we discuss the LU decomposition method.

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Before the scientific community, one of the most challenging problems is what sustains life or how living things originate from non-living things. A biological cell is assumed as the smallest unit of life. The multiple cellular processes are mainly controlled by tiny biological machines called motor proteins which move along macromolecular highways, namely microtubules, to transfer cargoes at distinct locations inside a cell. To get insight into the life-sustaining mechanism, it is necessary to understand the collective transport of motor proteins that fall into a specific category of the non-equilibrium system due to the presence of non-zero current governed by the continuous supply of energy. In recent decades, asymmetric simple exclusion process (TASEP), a Markov model which includes unidirectional particle hopping along a one-dimensional discrete lattice, has achieved the status of paradigm model to analyze stochastic non-equilibrium transport, including motor movement. The first part of the talk will be devoted to some beautiful results on the standard 1D TASEP model with fixed lattice size. Recent experimental observations suggest that the biological paths (microtubules) can polymerize or depolymerize under certain conditions, resulting in the growth or shrinkage of these paths. Modeling stochastic transport with dynamic microtubules will be explored in the second part of the talk, then validating theoretical results through experimental observations and extensively performed Monte Carlo simulations. Finally, challenges in modeling and analyzing intracellular transport will be discussed.

Biography:

I have completed BSc in Physics, Chemistry, Mathematics from CSJMU Kanpur (2012), followed by MSc in Industrial Mathematics & Informatics from IIT Roorkee (2014). I earned a Ph.D. on the topic "Mathematical Modelling of Driven Stochastic Transport Systems" from IIT Ropar (2019). After completing my Ph.D., I worked as Director's Fellow at IIT Ropar from Sep 2019 to May 2021. Since June 2021, I have been working as Assistant Professor at the Department of Mathematics, NIT Trichy, Tamil Nadu. I have published 9 research papers (all SCI) in international journals of repute like Physical Review E, Journal of Statistical Physics, Journal of Statistical Mechanics, etc.

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During my teaching seminar, I will be discussing the basic theory of the system of homogeneous first order linear ordinary differential equations with constant coefficients.

Biography:

I have completed BSc in Physics, Chemistry, Mathematics from CSJMU Kanpur (2012), followed by MSc in Industrial Mathematics & Informatics from IIT Roorkee (2014). I earned a Ph.D. on the topic "Mathematical Modelling of Driven Stochastic Transport Systems" from IIT Ropar (2019). After completing my Ph.D., I worked as Director's Fellow at IIT Ropar from Sep 2019 to May 2021. Since June 2021, I have been working as Assistant Professor at the Department of Mathematics, NIT Trichy, Tamil Nadu. I have published 9 research papers (all SCI) in international journals of repute like Physical Review E, Journal of Statistical Physics, Journal of Statistical Mechanics, etc.

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Determination of modular forms is one of the fundamental and interesting problems in number theory. It is known that if the Hecke eigenvalues of two newforms agree for all but finitely many primes, then both the forms are the same. In other words, the set of Hecke eigenvalues at primes determine the newform uniquely and this result is known as the multiplicity one theorem. In the case of Siegel cuspidal eigenform of degree two, the multiplicity one theorem has been proved only recently in 2018 by Schmidt. In this talk, after discussing a refinement of the multiplicity one theorem for newforms due to Rajan, we refine the result of Schmidt by showing that if the Hecke eigenvalues of two Siegel eigenforms agree at a set of primes of positive density, then they are the same (up to a constant). We also distinguish Siegel eigenforms from the signs of their Hecke eigenvalues. The main ingredient to prove these results are Galois representations attached to Siegel eigenforms, the Chebetarov density theorem and some analytic tools.

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In this talk, I will begin with a quick introduction to the PDE perspective of the Image Processing. Then, I will introduce two new image inpainting models using Modified Cahn-Hilliard equation and multi-well potential constructed from the histogram of a given image. The construction of the multi-well potential term which completely meets the challenge of capturing all the grey scale intensities of any image will be discussed. Later the existence- uniqueness of the proposed PDE model will be dealt with. Convexity Splitting method for time and Discrete Fourier spectral method in space has been used to discretize the PDE. Consistency, Stability, and convergence of the discretized models has been shown. Finally, we will present some results of our models and some applications will be shown.

Biography:

Dr. Abdul Halim is currently working as a Postdoc in the Department of Applied Mathematics, University of Twente, Netherland. Prior to that He was working as an Assistant Professor in Munger University, Bihar. He has completed his Ph.D. from the Department of Mathematics and Statistics, IIT Kanpur. His research area includes PDE based Image Processing, PDE eigenvalue Problems, Reduced Order Modeling.

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In this talk, I will prove the existence and uniqueness theorem for first order initial value problem known as Picard’s Theorem using the Picard’s successive approximation on rectangle containing the initial point and discuss few examples.

Biography:

Dr. Abdul Halim is currently working as a Postdoc in the Department of Applied Mathematics, University of Twente, Netherland. Prior to that He was working as an Assistant Professor in Munger University, Bihar. He has completed his Ph.D. from the Department of Mathematics and Statistics, IIT Kanpur. His research area includes PDE based Image Processing, PDE eigenvalue Problems, Reduced Order Modeling.

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Fundamental group of a topological space is an important topological invariant of a space. In algebraic geometry, there are many analogues of topological fundamental groups in the literature, which are generally group-schemes. We discuss the notions of étale fundamental group, Nori's fundamental group-scheme and the S-fundamental group-schemes of an algebraic variety, and relationships among them. Understanding these fundamental group-schemes for moduli spaces is an important problem to study. The Hilbert scheme of n points on a smooth projective surface X, denoted as $Hilb^n_X$ , is an important moduli space to study. In a recent joint work with Ronnie Sebastian, we established a relationship with the S-fundamental group-scheme and the Nori's fundamental group-scheme of $Hilb^n_X$ with that of X. After discussing this result, if time permits, we indicate some future research plans along this direction.

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Flag varieties are natural generalizations of Grassmannians. There is a well-established connection between the geometry of flag varieties and representation theory of algebraic groups. Schubert varieties are subvarieties of a flag variety which parametrize families of linear subspaces of a vector space. In this talk, we consider the quotients of flag varieties and Schubert varieties for the action of a maximal torus. In the case of Grassmannian, we give a classification of smooth torus quotients of Schubert varieties. Finally, I will also give a few words about the Gromov-width of Bott-Samelson varieties, these are desingularizations of Schubert varieties.

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It is well known that the Orlicz spaces are natural generalizations of the Lebesgue spaces. We will discuss the Banach algebra structure of the Orlicz spaces associated to vector measures over compact groups. We will also discuss the Fourier transform associated to vector measures over compact groups.

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Zadeh introduced the notion of fuzzy subsets. The notions introduced by Atanassov (1983) and Atanassov&Gargov (1989) in defining intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets are interesting and useful in modelling real-life problems. An intuitionistic fuzzy set plays a vital role in decision making, data analysis, and artificial intelligence. The information received from a source is represented by an information system involving quantitative, qualitative, and incomplete information. Such incomplete information is fed into the intelligent system for enhancing accuracy using Trapezoidal Intuitionistic Fuzzy Numbers (TrIFN). The ranking of fuzzy numbers plays an important role in Decisionmaking problems and many other applications. The ranking of Fuzzy Numbers have started in the early eighties in the last century, different researchers have proposed different methods for the Ranking of Fuzzy numbers. But none of them yields a total ordering on the class of fuzzy numbers and intuitionistic fuzzy numbers. The main aim of this lecture is to introduce a ranking (ordering) principle for comparing arbitrary intuitionistic fuzzy numbers (TrIFNs) and discuss its application in decision-making.

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In the present research work, we have analysed the movement of solute in a homogeneous aquifer. Firstly, the fractional-order (1+1)-dimensional solute transport model is considered which is simply an advection-diffusion equation having source/sink term with given initial condition and first type source boundary conditions. To describe the movement of solute concerning the column length, the considered problem is solved by using shifted Legendre collocation method. Secondly, the shifted Legendre collocation method is extended to obtain the solution of the (2+1)-dimensional fractional-order nonlinear advection-reaction-diffusion model. The main feature of the present contribution is the graphical exhibitions of the effects of advection term, reaction term, and fractional-order parameters on the solution profile. To authenticate the effectiveness of the method, a drive has been taken to compare the obtained results with the existing analytical results of the integer-order form of the considered model through error analysis. The striking feature of the work is the damping effect of the field variable on the solution profile when the system approaches fractional-order from the integer-order for specified values of the parameters of the system which greatly describes the physical phenomenon where the rate of transportation is much faster than the usual one.

Biography:

I, Anup Singh, Assistant Professor (Guest faculty) in the Department of Mathematics, Mahatma Gandhi Central University, Motihari, Bihar. I have completed my Ph.D. in the Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, under the supervision of Prof. Subir Das on June 08, 2020. The title of the thesis is "Study and analysis of advection-reaction-diffusion equations in porous media." I did B.Sc. (Hons.) and M.Sc. at Banaras Hindu University (BHU), Varanasi. I have published 06 research articles in the reputed SCI/SCIE journals.

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Tensors (also known as multidimensional arrays or n-way arrays) have significantly impacted many areas of science and engineering in recent years. However, operations with tensors are still a major challenge and preoccupation for the scientific community. Therefore, it is appropriate to develop an infrastructure that supports reasoning about tensor computations. In this talk, we will discuss tensor singular value decomposition based on a new notion of tensor-tensor multiplication. Then I shall present tensor singular value decomposition applications to a color image compression and computation of the tensor Moore-Penrose inverse. Further, we will discuss a color image deblurring problem and the solution of the Poisson problem in a tensor-structure domain.

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I will discuss about the nonlocal operators, in particular, the fractional Laplacian, and investigate the positivity properties of nonlocal Schr\"odinger 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 Schr\"odinger operator.

Biography:

I am currently visiting Prof. Mousomi Bhakta, IISER Pune since July, 2021. Prior to this, I was a postdoctoral fellow at Montan University, Leoben, Austria and Masaryk University, Brno, Czech Republic between 2018 and 2021. I defended my doctoral thesis on 18th July, 2018 on the title" Nonlocal elliptic equations: existence and multiplicity results" from IISER Pune under the supervision of Dr. Mousomi Bhakta.

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In this talk, uniqueness and existence results for first-order ordinary differential equations will be studied. Several examples are given to illustrate the applicability of this result.

Biography:

I am currently visiting Prof. Mousomi Bhakta, IISER Pune since July, 2021. Prior to this, I was a postdoctoral fellow at Montan University, Leoben, Austria and Masaryk University, Brno, Czech Republic between 2018 and 2021. I defended my doctoral thesis on 18th July, 2018 on the title" Nonlocal elliptic equations: existence and multiplicity results" from IISER Pune under the supervision of Dr. Mousomi Bhakta.

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Systems of linear algebraic equations arise in many different problems, such as traffic flow, electrical networks, balancing chemical equations, evaluating the integrals, etc. In this talk, we will discuss the solution of systems of linear algebraic equations; the existence and uniqueness of the solution. Then we will discuss LU and Cholesky factorizations for solving special types of linear systems.

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In this talk, we begin by translating the problem of classification of Hopf algebras of a fixed dimension over an algebraically closed field into an algebro-geometric question. This naturally leads us to the classical invariant theory of the space of mixed tensors. We obtain a finite set of polynomial invariants for this space and use this to construct a set of invariants which separate closed orbits of the variety of Hopf algebras of a fixed dimension, under the action of a general linear group. We show that this gives a finite set of complete invariants for the isomorphism classes of Hopf algebras of 'small dimensions'.

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It is known that every irreducible subfactor $N\subset M$ with finite Jones index has only finitely many intermediate subfactors. We provide an explicit bound for the cardinality of this set. To achieve this, in a joint work with Das, Liu and Ren , we have introduced and investigated a notion of angle between a pair of intermediate subfactors. As a consequence, we answer a question of R. Longo published in 2003. More recently, in a joint work with Gupta we generalize the above result in the purely $C^*$-algebra setting exploiting a notion of generalized Pimsner-Popa basis.

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Various porous medium flows like enhanced oil recovery, CO2 sequestration are of significant importance to various engineers, geologists and chemists. But a mathematical insight into these is equally important to obtain certain parameters before performing the experiments and to validate the experimental findings with mathematical theories. We discuss hydrodynamic instabilities ubiquitous in such flows through mathematical modelling and performing linear stability analysis as well as non-linear simulations to understand the dynamics.

Biography:

Dr. Vandita Sharma obtained her Ph.D from department of Mathematics, Indian Institute of Technology Ropar. She has expertise in mathematical modeling and scientific computing, computational fluid dynamics, hydrodynamic stability. In particular, she has worked on modeling, computation and stability analysis of reaction-diffusion systems. She is a Gold medallist in M.Sc Mathematics and has won various competitive awards like SIAM student travel award, ICTAM travel award and ICIAM financial support, to name a few, during her PhD.

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We will present the results related to the eigenvalue statistics of random Schrodinger operators with various types of random potentials. Mostly, we will study the limiting behaviour of the sequence of point processes (or random measures) associated with the spectrum of finite volume restriction of the Schrodinger operators. We will also talk about the smoothness of the integrated density of states (IDS) and multiplicity of the singular spectrum for general Anderson type Hamiltonian.

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We will discuss a system of linear equations Ax =b and try to obtain a solution when this system is actually inconsistent. We shall discuss a couple of ways for finding the solution, building upon the disadvantages of one method and proceeding to the next method.

Biography:

Dr. Vandita Sharma obtained her Ph.D from department of Mathematics, Indian Institute of Technology Ropar. She has expertise in mathematical modeling and scientific computing, computational fluid dynamics, hydrodynamic stability. In particular, she has worked on modeling, computation and stability analysis of reaction-diffusion systems. She is a Gold medallist in M.Sc Mathematics and has won various competitive awards like SIAM student travel award, ICTAM travel award and ICIAM financial support, to name a few, during her PhD.

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In 1970, Davenport and Schmidt studied a Diophantine property of pairs of real numbers; it concerned those pairs for which the classical Dirichlet theorem can be improved. They showed that the set of Dirichlet-improvable pairs, while small in the sense of having zero Lebesgue measure, has full Hausdorff dimension. We study a similar Dirichlet-improvable property, where the approximations are made using an arbitrary norm rather than the supremum norm, and show the same result. To this end this, we recast the Dirichlet-improvable property into a dynamical property of certain orbits in the space of unimodular lattices, and prove a Hajos-Minkowski type result in the geometry of numbers.

Biography:

I graduated from Brandeis University in 2020 with a thesis in the area of metric Diophantine approximation. This area of research comes under number theory and studies the approximation properties of real numbers by rational ones. I am also interested in the related areas of ergodic theory, Lie groups and geometry of numbers.

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Bordification of spaces obtained by adjoining a boundary at infinity has created a lot of interest in recent times. One such well known object is the Gromov boundary of hyperbolic metric spaces. In this talk, we will give some recent developments made in bordification of spaces.

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Let A and B be two elliptic curves defined over the field of rational numbers with equivalent residual Galois representations at an odd prime p. We compare the parity of p-Selmer rank and the root number of A and B over a number field K. As a consequence, in many cases, we show that the p parity conjecture over K holds for A if and only if it holds for B.

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Recent developments in the p-adic Langlands program allow us to revisit classical conjectures in Iwasawa Theory which help in understanding the arithmetic of Galois representations arising from elliptic curves, modular forms and automorphic forms. This talk will cover various aspects of my research domain. We will present results in Iwasawa Theory and p-adic Hodge Theory and combine them to frame the signed Iwasawa main conjectures for non-ordinary automorphic representations. Alongside we will also discuss Greenberg’s p-rationality conjecture and use it to construct Galois representations with big open image in reductive groups. Furthermore, we will discuss results in p-adic functional analysis and show the existence of rigid analytic vectors in the crystalline representations arising from the p-adic Langlands program. These works are a culmination of several projects with various collaborators.

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Moser-Trudinger inequality (1971) is connected to the embedding results of Sobolev spaces (W^{1,p}) for certain critical exponent (p). Classical Moser-Trudinger inequality was generalised to a singular form by Adimurthi-Sandeep in 2006. In this talk, I will discuss the issue of the existence of extremal functions for such inequality. This talk is based on a joint work with G.Csato and V. H. Nguyen.

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The approximate analytical solutions for the system of fractional order Van der Pol equations with independent initial profiles are investigated. The influence of two main physical parameters such as angular frequency (a) and the amplitude (ω) are included for the study. The effect of the physical parameters on phase portrait and the time-history curves for various values of fractional orders are plotted and discussed. It is found that the variations of in-phase and out-of phase periodic solutions and convergence rate strongly depend on the initial conditions. Based on the HAM method, the convergence rate, accuracy, and efficiency of the governing equations are demonstrated, which exhibit meaningful structures and advantages in science and engineering.

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The second order Partial differential equations (PDEs) are extremely important in both mathematics and physics. This talk provides a review on the nomenclature of the geometrical figures, if B2 − 4AC > 0 the partial differential equation is said to be hyperbolic; if B2 − 4AC = 0 the equation is parabolic, and if B2 − 4AC < 0 the equation is elliptic. Also, illustrate these on real life situational examples and its solution behaviour lucidly.

Biography:

I have studied all my education starting from 1st class to Ph. D in Government Schools/ Social Welfare Residential Hostels and Government Universities. I have completed my Ph.D in Mathematics from the Department of Mathematics, Central University of Pondicherry, Pondicherry India under the guidance of Prof. Rajeswari Seshadri.(Ph.D. from IISc Bangalore) in the areas of Solutions for nonlinear differential equations. Also, I have completed my one year Post Doctoral Fellow (PDF) research work in Applied Mathematics/Mechanics from School of Naval Architecture, Ocean and Civil Engineering(1st Rank in World by GLOBAL RANKING OF WORLD OF ACADEMIC SUBJECTS-2018), Shanghai Jiao Tong University, Shanghai in China under the guidance of senior Scientist and Distinguished Professor. Shi Jun Liao (HAM method Founder and 2016 Most Highly Cited Researchers in Global Mathematics).
I have "5 years" of teaching experience and “1 year” Post Doctoral fellow experience and Published 28(SCI/SCOPUS/UGC) international journal articles/conference proceedings and have communicated 4 more papers. I also presented 5 papers and gave 8 lectures at various conferences and seminars. I have participated in more than 74 Conferences / Seminars / Workshops and presented papers in some of them. I have received prestigious fellowship “Rajiv Gandhi National Fellowship(JRF & SRF) from UGC, New Delhi, Govt of India during my M.Phil and Ph.D. I have received renowned the SJTU Post Doctoral Fellowship from Shanghai Jiao Tong University, Shanghai, China in 2016 and Prof. Meenakshi Sundaram Memorial Best paper award in Applied Mathematics by APSMS in 2017. Also awarded QITCS Post Doctoral Scientist fellow from Shandong University, Govt of China-2020(Not availing due to Covid). Recently, I have received CONFIRMATION LETTER FOR HIGH LEVEL FOREIGN TALENTS from CHINA (i.e. prestigious “China Talent Visa(R)” i.e, Foreign High-Level Talents Fellow Researcher among all 100 International fellows from Government of China) for 2 years free VISA services for work as a Postdoctoral Scientist from 6th April, 2020 to 6th April, 2022 at Qingdao Institute for Theoretical and Computational Sciences, Shandong University, Qingdao, China under the supervision of Director, Prof. Wenjian Liu.

• Secured 1st rank in All India Entrance (PU CET) for M.Phill. in Mathematics at Pondichetty University ( A Central University), Pondicherry in 2009.
• Secured 1st rank, All India Selections for Ph.D. in Mathematics at University of Hyderabad in 2011.
• Secured 1st rank in All Andhra Pradesh state Entrance (KU CET) for M. Sc in Mathematics, Kakatiya University (State University), Warangal in 2005.

My research area is on application of approximate

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Dynamical systems is the branch of mathematics which investigates how a system evolves with time. In my talk I will discuss two broad sects of this subject One-Dimensional Dynamical Systems and Complex Dynamical Systems in the spirit of "Developing efficient machinery to make conclusions about the dynamics of a given map f from reduced information". In the former part, we will accomplish the above stated goal by studying the forcing relation among over-rotation numbers of points under a map f and concocting algorithms to compute out its over-rotation interval I(f) = [r(f), 1/2]. It is known that this single number r(f) epitomizes the limiting dynamical behaviour of points of the interval which in turn portrays the dynamical complexity of the map f. In the later discipline we will accomplish the stated goal by developing efficient topological models for the parameter space of complex polynomials. This will allow us to partition the whole parameter space into parts depending upon dynamics. Then if we just knew the location of a polynomial in the parameter space we can immediately unveil its dynamics.

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The Theory of Ordinary Differential Equations is playing an ever more important role in provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classicaltechniques of mathematics. Examining stability of solutions is one of the central problems in the Theory of Ordinary Differential Equations. In my talk I will present an introduction to this topic focussing on Systems of Linear Differential Equations. Important components of this disquisition entails : determining stable , unstable and centre subspaces of a given linear system , classification of equilibrium points of a linear system and use of Liapunov function in ascertaining stability of a linear system. We will elaborate upon the particular case of Linear systems in the plane.

Abstract:

Individuals in an ecosystem may exhibit different behavioural strategies for their long term survival. Mathematical modelling of different ecological phenomenon by systems of non-linear ordinary differential equations with non-negative initial conditions can give us important insights of ecological dynamics. The systems may be autonomous or non-autonomous and depending upon different scenarios particular forms can be chosen. A linear stability analysis of an autonomous system tells us about the stability of the equilibrium points of the systems and provide understanding about the bifurcation scenarios. For a non-autonomous system that is when the species are seasonally forced (parameters of the systems are explicitly dependent on time), coincidence degree theorem is useful to check the existence of periodic solutions of the systems. Apart from deterministic system, the effects of environmental fluctuations on a system can be analysed by considering the stochastic set up. It is observed that in the study of erythropoiesis mathematically, the influence of stem cell dynamics on the subsequent process of erythrocyte production has been less explored in the recent studies. Combine study of these processes under the framework of multi-scale modelling may provide important insights of the collective cell dynamics.

Biography:

Jyotirmoy Roy had completed his doctoral degree form IIEST, Shibpur in the year 2021. His research interests include mathematical modelling and numerical analysis of different ecological, eco-epidemiological and biological phenomenon. He had published ten articles in different journals and conferences.

Abstract:

Continued and remarkable empirical successes of increasingly complicated machine learning models such as neural networks without a sound theoretical understanding of success and failure conditions can leave a practitioner blind-sided and vulnerable, especially in critical applications such as self-driving cars and medical diagnosis. As such, there has been an enhanced interest in recent times in research on building interpretable models as well as interpreting model predictions. In this talk, I will discuss various theoretical and practical aspects of interpretability in machine learning along both these directions through the lenses of feature attribution and example-based learning. In the first part of the talk, I will present novel theoretical results to bridge the gap in theory and practice for interpretable dimensionality reduction aka feature selection. Specifically, I will show that feature selection satisfies a weaker form of submodularity. Because of this connection, for any function, one can provide constant factor approximation guarantees that are solely dependent on the condition number of the function. Moreover, I will discuss that the cost of interpretability accrued because of selecting features as opposed to principal components is not as high as was previously thought to be.
In the second part of the talk, I will discuss the development of a probabilistic framework for example-based machine learning to address ``which training data points are responsible for making given test predictions?“. This framework generalizes the classical influence functions. I will also present an application of this framework to understanding the transfer of adversarially trained neural network models.

Short Biography:

Dr. Rajiv Khanna is currently a Visiting Faculty Researcher at Google Research, and an incoming Assistant Professor at the Department of Computer Science at Purdue University. Previously, he was a postdoc at the Department of Statistics at UC Berkeley and was also associated with the Foundations of Data Analytics Institute (FODA) at UC Berkeley. Before that, he was a Research Fellow in the program of Foundations of Data Science at the Simons Institute for the Theory of Computing, also at UC Berkeley.
His research is focussed on elucidating mechanisms of success/failure conditions of machine learning through optimization, learning theory and interpretability. His work on beyond worst-case analysis on the Column Subset Selection won the best paper award at NeurIPS 2020. He earned his PhD in Electrical and Computer Engineering at UT Austin.

Abstract:

In this talk, after briefly describing my research areas, I focus on the topic of importance sampling (IS). The standard IS method uses samples from a single proposal distribution and assigns weights to them, according to the ratio of the target and proposal pdfs. This naive IS estimator, generally does not work well in multiple target examples as the weights can take arbitrarily large values making the estimator highly unstable. In such situations, alternative generalized IS estimators involving samples from multiple proposal distributions are preferred. Just like the standard IS, the success of these multiple IS estimators crucially depends on the choice of the proposal distributions. For selecting these proposal distributions, we propose three methods based on a geometric space filling coverage criterion, a minimax variance approach, and a maximum entropy approach, respectively. The proposed methods for selecting proposal densities are illustrated using some examples.

Abstract:

Maximum likelihood estimation is probably the most common method of estimation in use in frequentist statistics. After discussing the basic idea of the maximum likelihood estimator (MLE), we will define MLE and go over several examples. Finally, we will discuss some nice properties of the MLE.

Short Biography:

Dr. Roy is an Associate Professor in the Department of Statistics at Iowa State University. Currently, he is serving two statistics journals - Journal of Computational and Graphical Statistics and Sankhya, Series B as an Associate Editor. His research interests are Markov chain Monte Carlo, importance sampling, high dimensional data analysis, model selection and Bayes and empirical Bayes methods.

Abstract:

D & C paradigm is commonly used in designing efficient algorithms. Here, the problem is divided into subproblems, each subproblem is solved recursively, and the solutions to the subproblems are combined to get the overall solution. In this talk, I will discuss this design strategy with the example of efficiently finding the closest pair of points. Here, given n points in a plane, the goal is to find a pair of points that have the smallest Euclidean distance between them. I will show that the brute force approach here leads to a time complexity of O(n^2), while a D & C strategy leads to time complexity of O(n log^2 n). This complexity can be reduced further. Depending upon the availability of time, I would also summarize the use of D & C strategy for the problem of efficiently multiplying two matrices.

Abstract:

Algorithmic scalability is a challenge in many complex systems and machine learning applications. Inexact computing is one solution to this scalability challenge. With the focus on the problem of efficiently reducing large-scale dynamical systems, we propose the use of approximate (and hence, faster) linear solves in a set of model reducing algorithms. Theoretically, we prove that under mild conditions (easily satisfiable), these algorithms are backward stable with respect to the error introduced by inexact linear solves. Experimentally, we demonstrate that while reducing an industrial disk-brake model of size 1.2 million, one of our approximate algorithms leads to relative savings of up to 64% in the total computation time (when compared with the corresponding exact algorithm). In absolute terms, this leads to a saving of 5 days.
Another solution to this scalability challenge is sampling, which leads to a reduction in the problem size. With the focus on the problem of efficiently grouping plant species, we propose a class of probabilistically-sampled spectrally-clustered algorithms. Using experiments on multiple Soybean datasets comprising of thousands of species, we demonstrate the superiority of our approach over the current standard in-terms of both the cluster quality (up to 45% better) as well as the time complexity (an order-of-magnitude lesser).

Short Biography:

With Master’s and Ph.D. degrees in Mathematics and Computer Science (from Virginia Tech), Dr. Kapil Ahuja has a strong interdisciplinary focus. After graduating from VT, he received his postdoctoral training from the Max Planck Institute in Magdeburg. Since then, he has established his independent research program in Mathematics of Data Science and Simulation (MODSS) at IIT Indore, where he is currently working as an Associate Professor. In the recent past, he has also held visiting professor positions at TU Braunschweig, TU Dresden, and Sandia National Labs.
Dr. Ahuja’s core research interests are in Machine Learning, Optimization, Game Theory, and Numerical Methods. Dr. Ahuja's research output includes external funding worth more than half-a-million USD and over forty high-quality publications. While achieving this, he has graduated 4 PhD students with 1 more to graduate soon. On the teaching end, Dr. Ahuja has received the best teacher award four times at IIT Indore, and administratively, recently, he headed International Affairs at IIT Indore as its founding Dean.

Abstract:

Finitely many agents have preferences on a finite set of alternatives, singlepeaked with respect to a connected graph with these alternatives as vertices. A probabilistic rule assigns to each preference profile a probability distribution over the alternatives. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles at which all peaks are on leaves of the tree and, thus, extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leaves. Finally, the two results are combined to obtain a general characterization for every connected graph by using its block tree representation.

Abstract:

In this talk, I will define real-valued random variables on a probability space and talk about the probability distribution and distribution function of such a random variable. Then I will go through the details of some well-known random variables such as Binomial, Poisson, Normal, etc., and their different moments. At the end of this talk, I will touch on some inequalities and various modes of convergence.

Short Biography:

Dr. Soumyarup Sahdukhan has obtained his Ph.D. in Economics (28th August 2020) from the Economic Research Unit of ISI Kolkata. During his Ph.D., he has worked in the areas of mechanism design, random social choice theory, and division problem, and has published in top journals such as Mathematics of Operations Research, Journal of Economic Theory, Economic Theory, and Journal of Mathematical Economics. He did his B.Sc. in Statistics from Narendrapur Ramakrishna Mission (with 85% marks) and M.Stat from ISI (with distinction). Currently, he is working as a CV Raman post-doctoral fellow in the Department of Computer Science and Automation of IISc Bangalore. Previously, he has worked as a visiting scientist in the Applied Statistics Unit of ISI Kolkata under the supervision of Professor Mridul Nandi.

Abstract:

In this talk we examine the relationship between maxi–min, Bayes and Nash designs for some hypothesis testing problems. In particular we consider the problem of sample allocation in the standard analysis of variance framework and show that the maxi–min design is also a Bayes solution with respect to the least favourable prior, as well as a solution to a game theoretic problem, which we refer to as a Nash design. In addition, an extension to tests for order is provided.

Abstract:

In this talk a simple linear regression model is considered. An ordinary least square technique is carried out for the estimation of the model parameters. Elementary properties of the estimators are discussed. To check the significance of the parameters, relevant hypotheses and associated statistical tests are formulated. A data example is given to illustrate the methodology.

Short Biography:

Dr. Satya Prakash Singh is currently working as an Assistant Professor at the Department of Mathematics, IIT Hyderabad. He did M.Sc. (Mathematics) from IIT Delhi and M.Sc. (Statistics) from IIT Kanpur. He obtained his Ph.D. from IIT Bombay. He did his Postdoc at the University of Haifa, Israel. His research interests are optimal experiments for cluster and crossover studies and designing optimal experiments under order restrictions.

Abstract:

Short Biography:

Dr. Neeraj Bhauryal is currently a postdoctoral fellow at TIFR-CAM, Bangalore. He obtained his Masters and PhD in Mathematics from TIFR-CAM Bangalore. His current research interests include Nonlinear Partial Differential Equations, Conservation Laws and Stochastic Analysis.

Abstract:

Short Biography:

Dr. Paramita Pramanick is currently working as a Research Associate in the Department of Mathematics, Indian Institute of Science, Bangalore. Dr. Pramanick completed her B.Sc. and M.Sc. from Jadavpur University, and her Ph.D. from Indian Institute of Science, Bangalore in 2021. Her research focuses on Functional Analysis, Operator Theory: more speci fically, operator theory on Hilbert space of analytic functions, reproducing kernel Hilbert modules, homogeneous operators, subnormal operators, hyponormal operators and trace inequalities.

Abstract:

Hierarchical Bayesian modeling of high-dimensional datasets is a very active area of research in geostatistics and spatial extremes. In this work, we estimate extreme sea surface temperature (SST) hotspots, i.e., high threshold exceedance regions, for the Red Sea, a vital region of high biodiversity. We analyze high-resolution satellite-derived SST data comprising daily measurements at 16703 grid cells across the Red Sea over the period 1985–2015. We propose a semiparametric Bayesian spatial mixed-effects linear model with a flexible mean structure to capture spatially-varying trend and seasonality, while the residual spatial variability is modeled through a Dirichlet process mixture (DPM) of low-rank spatial Student’s t-processes (LTPs). By specifying cluster-specific parameters for each LTP mixture component, the bulk of the SST residuals influence tail inference and hotspot estimation only moderately. Our proposed model has a nonstationary mean, covariance, and tail dependence, and posterior inference can be drawn efficiently through Gibbs sampling. In our application, we show that the proposed method outperforms some natural parametric and semiparametric alternatives. Moreover, we show how hotspots can be identified and we estimate extreme SST hotspots for the whole Red Sea, projected until the year 2100, based on the Representative Concentration Pathway 4.5 and 8.5. The estimated 95% credible region for joint high threshold exceedances includes large areas covering major coral reefs in the southern Red Sea. Some future research directions include parametric and nonparametric Bayesian approaches for spatial geostatistics and extremes, Bayesian hotspot estimation, and extreme event attribution.

Abstract:

The last three decades have seen an explosion of interest in spatial and spatiotemporal problems. The increased availability of inexpensive and high-speed computing facilities worked as an indispensable tool for fitting realistic spatial statistical models and drawing inferences, and currently, the applications of spatial statistics span a large number of scientific disciplines. Spatial data are generally divided into three categories: point-referenced data, areal unit data, and spatial point patterns. We discuss examples of different types of spatial data and possible scientific questions. Gaussian processes are at the heart of this area of statistics, and hence we review some necessary multivariate statistical inference techniques. We discuss a brief overview of linear models, which provides some necessary insights about spatial mean modeling. The variogram is a common exploratory technique for understanding spatial dependence, and we cover the main concepts. Some spatial covariance models and some simple examples of kriging are discussed. Finally, we also cover some areal data modeling approaches. All the concepts are illustrated with R programming and real data examples.

Short Biography:

Arnab Hazra is currently a postdoctoral fellow of statistics at the CEMSE Division at King Abdullah University of Science and Technology, Saudi Arabia. He obtained his Bachelor of Statistics and Master of Statistics degrees from the Indian Statistical Institute, Kolkata, and his Ph.D. in Statistics from North Carolina State University, Raleigh, US, in 2018. His current research interests include spatial statistics, extreme value analysis, heavy-tailed processes, high-dimensional data analysis, hierarchical Bayesian modeling, and environmental statistics.

Abstract:

Short Biography:

Dr. Kapil Kant completed his B.Sc. in Mathematics from University of Allahabad in 2013. After that, he obtained his M.Sc. and Ph.D. in Mathematics from Indian Institute of Technology, Kharagpur in 2015 and 2020, respectively. His broad area of research is numerical analysis and scientific computing.

Abstract:

Short Biography:

Subhabrata (Subho) Majumdar is a Senior Inventive Scientist in the Data Science and AI Research group of AT&T Labs. His research interests have two focus areas - (1) statistical machine learning - specifically predictive modelling and complex high-dimensional inference, and (2) trustworthy machine learning methods, with emphasis on human-centric qualities such as social good, robustness, fairness, privacy protection, and causality.
Subho has a PhD in Statistics from the School of Statistics, University of Minnesota under the guidance of Prof. Anshu Chatterjee. His thesis was on developing inferential methods based on statistical depth functions, focusing on robust dimension reduction and variable selection. Before joining AT&T, Subho was a postdoctoral researcher at the University of Florida Informatics Institute under Prof. George Michailidis. Subho has extensive experience in applied statistical research, with past and present collaborations spanning diverse areas like statistical chemistry, public health, behavioral genetics, and climate science. He recently co-founded the Trustworthy ML Initiative (TrustML), to bring together the community of researchers and practitioners working in that field, and lower barriers to entry for newcomers. Link to his webpage: https://shubhobm.github.io/.