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MS COURSES

## MTS511 Advanced Real Analysis

This is a 3 credit hours course designed for a graduate degree in Mathematics. This course starts with the basic concepts of set theory and then gradually builds up preliminary concepts of real analysis such as real number system, sequence of real numbers, open sets and closed sets. Once foundation is laid down we discuss Riemann integration, measurable sets, outer measure, measurable functions, Lebesgue measure and Lebesgue integral. This course has some additional things that are very unlikely to a standard course in Advanced Real Analysis that along with Riemann Integration we also introduce Lebesgue Integration without going into formal theory of measure spaces. The course has been divided into four parts. First module is all about set theory and introductory real analysis. In the second, module we discuss Riemann theory of integration and also talk about Lebesgue integration. Third module is mainly about convergence in measure. Fourth module is actually some topics from functional analysis that will prepare students for the next course in this series, Measure and Integration Theory, in which a thorough treatment of measure spaces is given. We presuppose that student has good foundation of an undergraduate real analysis before taking this course. This course aims to be more dynamic and problems solving oriented than just proving theorems and asking students to reproduce them in the exams.

## MTS512 Measure Theory & Integration

This is a 3 credit hours course on Measure Theory designed for advanced graduate students of PhD Mathematics. Here we treat measure theory in the abstract and rigorous way. In addition, some topics from Functional Analysis have also been added in order to understand measure theory in its real spirit. Course has been divided into four modules. In the first module, mainly we define measure as a set valued function and discuss the properties of measure and Lebesgue measure in abstract setting. In the second module we define measure on sigma algebra, drive outer measure from the measure and define measurable sets. Third model defines mappings on the measure spaces. Fourth module is about defining measure on a class of locally compact Hausdorff Spaces.

## MTS513 Topics in Algebra

Algebra is the language of modern mathematics. This course introduces students to algebra through a study of group and ring theories. Group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have strongly influenced other disciplines, both inside and outside mathematics, such as geometry, number theory, cryptography, chemistry and physics. Ring theory is also an important area of abstract algebra. It** **is the study of rings which is an algebraic structure in which addition and multiplication are defined and have similar properties to those of integers. The aim of this course is to introduce the students to some of the basic ideas and results of group and ring theories through case studies.

## MTS514 Topics in Commutative Algebra

In this course, the object of study is predominantly a commutative ring, hence the title commutative algebra. We introduce theory of commutative rings along with modules on them as our main tool of representation in studying such rings. We also introduce the basic homological characterization of modules with the help of exact sequences that has many applications both in Algebra, Topology and Geometry in general, at an advanced level. Fractions and localizations are introduced with the intention of application-besides algebra- both in geometry and analysis where they arise naturally as germs of functions locally determining both the geometry and analysis, of which meromorphic functions on Riemann Surfaces is just a special case. The idea of Neothriannes is very important in obtaining strong results that has application in many branches of mathematics along with Hilbert basis theorem.

## MTS515 Advanced Numerical Analysis

This is a 3 credit hours course designed for a graduate degree in Mathematics. This course is basically Numerical Functional Analysis that deals with both theoretical and numerical issues of partial differential equations such as condition number, perturbation, spectral theory and also thorough treatment of some of the advanced methods for linear and non-linear systems.

## MTS516 Topology

This course introduces topology at graduate level covering both general and algebraic aspects. Starting from basic point set topology, one of the goal is to prepare students attending the course for applications in mathematical analysis besides topology itself at advanced level. On the other hand algebraic topology is introduced to help develop necessary tools for calculations involving invariants, and develop necessary background in dealing classification issues up to an equivalence.

## MTS521 Scientific Computing

This course is aimed at developing PDE based problem solving skills. The course takes on from a fairly basic level such as problem discretization, and carries on to a relatively advanced stage, such as developing and trying novel preconditioners for a discrete linear system. Successful students are expected to be fully capable of actually using computers to solve a wide variety of applied problems.

## MTS525 Stochastic Processes II

This course is a successor to Stochastic Processes I and requires participants to understand the basic stochastic processes, and probability space laws. From this stage it builds up the more involved concepts of Martingales and uses various examples to motivate the study. Models from Finance are also discussed to motivate continuous time Markov models. Successful students are expected to understand continuous and discrete processes and to be able to successfully apply this knowledge to solve applied problems.

## MTS529 Stochastic Differential Equations

This course is aimed at providing students the background that they will require for stochastic analysis of financial derivatives, and developing exotic contingent claims later on. The same comprehension skills for stochastic processes and equations is expected of successful students.

## MTS533 Integral Equations

This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications.

## MTS537 Mathematical Astronomy

The purpose of this course is to provide the students with fundamental knowledge of the mathematical tools used in exploring positional astronomy. Starting with the basics of spherical trigonometry it describes the various terrestrial and celestial coordinate systems and coordinates transformations. It also addresses the issues related to time that are fundamental to astronomy and astrophysics. This course also explores the basic issues in celestial dynamics starting with Kepler’s Planetary laws and the Kepler’s equation.

## MTS539 Homological Algebra

The approach we intend to follow in this course is one that can be considered as a special case of taking Homological Algebra as a theory of function of two variables, one abelian and the other non-abelian. This is in spirit of axiomatic (Co)homology theory of Eilenberg-Steenrod, which have had its roots in (Co)homology theories of topological spaces. The other approach comes from Grothendieck which modifies it to convert into the theory of a single abelian variable, leading to algebraic geometry in its range of applications-the line which we find beyond the scope of this course. The student is assumed to have background in algebra, specially in the class of rings and modules determined by various finite and stationary conditions, and exact sequences. However, the details can be filled in whenever necessary. It is in this background we introduce very basic homological machinery that could be dealt justly in 1-semester graduate course. Thus it is expected that after completing this course, the student will be able to use it in algebraic topology and will be able to pursue his study further into (Co)homology theories of Groups, Lie Algebras and Associative Algebras.

## MTS541 Computational Algebraic Geometry

The main focus in this course is the computational aspects of algebraic geometry, hence the title. Since many major calculations in algebraic geometry, involve only calculating in the corresponding affine neighborhoods, we thus develop the main tools accordingly, hence major relevant ideas are all developed from scratch in this respect in context of affine algebraic geometry. The first step towards this goal is Hilbert’s Nullstellensatz which we introduce to establishes the theoretical dictionary needed to transfer the computations from pure algebra into geometry. In order to extend or enlarge the applications to geometry from smooth to mildly non-smooth cases, we introduce normalizations. This also helps extending the theoretical dictionary further when applied to non-singular models of affine curves. Projective setting is introduced to extend the span of the local scope of affine geometry and exemplify the local nature of affinness in the course’s computational aspects.

## MTS545 Applicable Modern Geometry I

In this course, we intend to establish the transition from vector calculus in R^n to the more general setting, that of manifold, and show that the former is just a special case of the latter via the fact that every manifold offers a calculus intrinsic to its own isomorphism class in the corresponding category. This also helps us establish a deep and rich interplay b/w topology and analysis, especially when we do integration on manifolds.

## MTS549 Algebraic Geometry I

In this course, the primary object of study is the classical algebraic variety (or a pre-variety as in EGA) with affine varieties serving as its local model. Throughout the course, everything is modeled on an algebraically closed field; however, one can extend most of the arguments to the fields of characteristic zero. We introduce the concept of a rational map giving rise to birational geometry along with resolution of singularities, very powerful aspects of algebraic geometry. It also aims at appreciation and application of the Riemann-Roch theorem, one of the most important results of algebraic geometry. For this course, the basic background in commutative algebra is assumed. However, to handle all algebraic instruments necessary for both, local and global analysis of varieties, required details can always be filled in whenever necessary. Some intersection theory is included, in both affine and projective cases, to help student develop an appreciation for the advanced topics in the context of applications.

## MTS553 Algebraic Cycles I

This course provides rigorous introduction to the most important objects and concepts of algebraic geometry and number theory. At the end of this course students will be familiar with the concept of schemes and able to define higher chow groups.

## MTS557 Arithmetic Algebraic Geometry

This is an introductory course on Diophantine geometry that deals Fermat's equations as well as Diophantine equations and inequalities. At the end of this course students will be familiar with L-functions and zeta-functions.

## MTS561 Exploratory data Analysis

Analysis of scientific data and experiments: Design of experiments and ethical research. Data modeling management, Exploratory data analysis, Randomness and probability, Statistical analysis including linear regression, analysis of variance, logistic regression, categorical data analysis and non-parametric methods.** **The aim of this course is to provide an understanding of the nature of scientific data and the subsequent need for statistical analysis. You will develop your statistical expertise and critical judgment in scientific studies, including an awareness of ethical issues in research and analysis. You will learn about the different types of data and how each can be visualized and summarized, and how you can make conclusions and predictions from the statistical analysis. You will also see that these statistical tools are based on simple mathematical ideas and associated assumptions.

## MTS565 Mathematical Physics I

Complex Analysis: Analytic functions, Contour integration. Ordinary Differential Equations : Exact solutions, special functions Series solutions Approximation methods (WKB, perturbation theory). Linear Algebra: Vector spaces and matrices, Infinite-dimensional spaces; Fourier and other transforms. Partial Differential Equations and Boundary Value Problems: General properties, Green's functions, Boundary-value problems.

## MTS569 Statistical Data Mining & Knowledge Discovery

Development of high performance computing facilities have given the way for testing and implementation of those concepts that were assumed impossible and so were not given their proper status. This list includes complex mathematical function mapping and classification techniques, linguistic and imprecise computing and machine learning paradigms. These approaches are capable to handle complex and gigantic real world problems. Computing facilities provided the chance to make breakthrough against the conventional requirement of mathematical rigidity and formality of solutions that even become impossible due to high complexity. These techniques replaced the complexity of exactness of solution with proximity of solution. On the other hand, massive data sets pose a great challenge to many cross-disciplinary fields, including statistics. The high dimensionality and different data types and structures have now outstripped the capabilities of traditional statistical, graphical, and data visualization tools. Extracting useful information from such large data sets calls for novel approaches that meld concepts, tools, and techniques from diverse areas, such as computer science, statistics, artificial intelligence machine learning. Statistical Data Mining and Knowledge Discovery bring together a stellar panel of experts to discuss and disseminate recent developments in data analysis techniques for data mining and knowledge extraction. This carefully edited collection provides a practical, multidisciplinary perspective on using statistical techniques in areas such as marketing research, risk management, financial forecasting and classification, rule based systems for decision support systems, image and speech analysis, health informatics.

## MTS573 Statistical Machine Learning

Development of high performance computing facilities have given the way for testing and implementation of those concepts that were assumed impossible and so were not given their proper status. This list includes complex mathematical function mapping and classification techniques, linguistic and imprecise computing and machine learning paradigms. These approaches are capable to handle complex and gigantic real world problems. Computing facilities provided the chance to make breakthrough against the conventional requirement of mathematical rigidity and formality of solutions that even become impossible due to high complexity. These techniques replaced the complexity of exactness of solution with proximity of solution. On the other hand, massive data sets pose a great challenge to many cross-disciplinary fields, including statistics. The high dimensionality and different data types and structures have now outstripped the capabilities of traditional statistical, graphical, and data visualization tools. Extracting useful information from such large data sets calls for novel approaches that meld concepts, tools, and techniques from diverse areas, such as computer science, statistics, artificial intelligence machine learning. Statistical Data Mining and Knowledge Discovery bring together a stellar panel of experts to discuss and disseminate recent developments in data analysis techniques for data mining and knowledge extraction. This carefully edited collection provides a practical, multidisciplinary perspective on using statistical techniques in areas such as marketing research, risk management, financial forecasting and classification, rule based systems in decision support systems, image and speech analysis, health informatics.

## MTS577 Galois Theory

This course gives a detailed introduction to Galois theory that starts from review of group action on a set and Sylow theorem with its application. Here students will understand the concept of field extension and Galois groups. Students will also study separable and inseparable extensions.

## PhD COURSES

## MTS621 Numerical Treatment of P.D.E

This course is intended to be an introduction to numerical methods for hyperbolic partial differential equations. These equations require special treatment which do not often form part of standard numerical analysis courses for PDEs. Successful students are expected to be able to solve a wide variety of hyperbolic PDEs numerically.

## MTS625 Financial Mathematics I

This course develops concepts of financial mathematics, mainly for pricing financial derivatives. Another aim of the course is to develop and re-visit stochastic calculus concepts applied to options and different exotic contingent claims.

## MTS629 Financial Mathematics II

This course aims to develop numerical methods for solving different PDEs related to mathematical finance. Tool-development in Matlab and C also forms a part of this course. Successful students are expected to be comfortable solving different finance problems related to pricing of derivatives.

## MTS637 Computational Astronomy

In this course advanced techniques of computations of the major celestial phenomenon based on positional astronomy.

## MTS645 Applicable Modern Geometry II

In this course, we intend to go beyond the study of the fundamental instruments of differential geometry of manifolds and investigate some of the algebraic and topological invariants associated to a manifold, along with some algebraic techniques which are useful in handling modern research tools. We introduce theory of Lie groups and Lie algebras which have applications in theoretical physics where they naturally arise in solving problems, for instance, by translating inherent symmetries via transformation groups. After completing this course the student will be expected to use the techniques learned in theoretical physics, or pursue his studies further in differential geometry.

## MTS649 Algebraic Geometry II

This course is the core of modern algebraic geometry as pioneered by Grothendieck and his "French-School". In this course we have just touched the basic ideas that form the basic language of modern algebraic geometry, the language of sheaves and schemes. Even though, it is considerably difficult to adopt a main line towards a significant goal from just a scratch, but we still hope that with the very basics of cohomological machinery and basic ingredients of intersection theory along with big results of Hirzebruch-Riemann-Roch and Hodge-Index-Theorem, that we have introduced by the end of semester, will help and prepare student with some advanced topics in intersection theory, specially in case of low dimensional algebraic geometry, where he should find himself prepared, to some extent, for handling intersection form determining intersection theory on complex algebraic surfaces. Also, equipped with the basic technical language of schemes, a student is expected to be prepared to venture into the geometric invariant theory pioneered by Mumford and Deligne, very important tool in moduli theory. On the other hand, a comprehensive introduction to sheave theory is developed to help student learn the basic tools required, with a view towards application in classification accounts in Algebraic Geometry, in context of Moduli Theory of sheaves on schemes.

## MTS657 Polylogarithms

This course can also be studies after algebraic cycles I as well. Students will be familiar with polylogs, infinitesimal and tangential versions of polylogs and their relations with the groups generated by geometric configurations.

## MTS661 Multivariate Statistical Analysis

Multivariate analysis arises with observations of more than one variable when there is some probabilistic linkage between the variables. In practice, most data collected by researchers in virtually all disciplines are multivariate in nature. In some cases, it might make sense to isolate each variable and study it separately. In most cases, however, the variables are interrelated in such a way that analyzing the variables in isolation may result in failure to uncover critical patterns in the data. Multivariate data analysis consists of methods that can be used to study several variables at the same time so that the full structure of the data can be observed and key properties can be identified. This course covers estimation, hypothesis tests, and distributions for multivariate mean vectors and covariance matrices. We also cover popular multivariate data analysis methods including multivariate data visualization, maximum likelihood, principal components analysis, multiple comparisons tests, multidimensional scaling, cluster analysis, discriminant analysis and multivariate analysis of variance, multiple regression and canonical correlation, and analysis of repeated measures data. Coursework will include computer assignments.

## MTS665 Mathematical Physics II

This is an advanced level course that builds the basic mathematical techniques to be used in exploring deeper issues in the theoretical and particle physics.** **

## MTS671 Monomial Algebra

Monomial ideals are ideals in polynomial rings that can be described in combinatorial and geometric terms. These descriptions make monomial ideals quite accessible by allowing us to employ intuition and tools from discrete mathematics and geometry to study them. In spite of their simplicity, monomial ideals are powerful tools. For example, in algebraic combinatorics they are used to attach algebraic invariants to finite simple graphs and, more generally, simplicial complexes. These invariants have led to the solutions of several important problems in combinatorics.

## MTS691 Topics of Special Interest I

To be described by the supervisor offering the course.

## MTS692 Topics of Special Interest II

To be described by the supervisor offering the course.