• Dr. S. Venkataraman

This 4-credit core course provides an in- depth study of key topics in abstract algebra, with a focus on group theory, rings, modules, and field theory. The course begins with Group Theory, covering group actions on sets, the class equation, direct products and sums, free groups, Sylow theorems, and finitely generated Abelian groups. Next, the course delves into Rings and Modules, exploring direct products, the Chinese Remainder Theorem, localization, polynomial rings, and the theory of modules, including homomorphisms, exact sequences, and special types of modules like free, projective, and injective modules. Linear Algebra over Rings addresses linear transformations and matrices over a ring, with an emphasis on rank, equivalence, and the determinant as a multilinear function. The course concludes with Fields and Galois Theory, where students will study algebraic and transcendental field extensions, splitting fields, and the fundamental theorem of Galois theory, along with properties of Galois extensions. This course equips students with foundational knowledge in algebra, preparing them for advanced studies and research in mathematics.

• Prof. Sujatha Varma
• Dr. Suvankar Biswas

The study of analysis at the Ph.D. level involves a rigorous exploration of the core theories and structures that underpin much of modern mathematics, particularly focusing on real and complex analysis, functional analysis, and their interconnections. Major theories include the foundational principles of measure theory, integration theory, and the study of convergence in various contexts, leading to the development of Lebesgue integration and Lp spaces. An essential area of focus is the theory of function spaces, particularly spaces of continuous, differentiable, or integrable functions, along with their dual spaces, which capture the continuous linear functionals on these spaces. Linear operators on Hilbert and Banach spaces are central topics, where students explore the properties of bounded and unbounded operators, their spectra, and the intricacies of operator theory, including the Riesz representation theorem and the Fredholm alternative. Moreover, complex analysis becomes an essential component, with a focus on complex function theory, analytic functions, and the study of singularities, residues, and contour integration.

• Prof. Deepika
• Dr. Pawan Kumar

This course aims to inculcate research techniques essential for pursuing research in mathematics. In particular the learners are taught research and publication ethics. It also teaches some standard programming languages such as R or python, and Latex for preparing manuscripts.

• Dr. Suvankar Biswas

This 4-credit elective course provides a thorough study of partial differential equations (PDEs), covering both linear and non-linear cases. It begins with First Order Linear PDEs, including transport, Laplace, heat, and wave equations, focusing on fundamental solutions, Green’s functions, and energy methods for solving non-homogeneous problems. The course then explores First Order Non-linear PDEs, emphasizing the method of characteristics, complete integrals, and boundary value problems. In the Transformation Methods section, students will learn similarity solutions, Fourier and Laplace transforms, and techniques for converting non-linear PDEs into linear ones, such as Hopf- Cole and Hodograph transforms. The course concludes with Second Order Linear PDEs, focusing on elliptic, parabolic, and hyperbolic equations, including the existence of weak solutions. This course equips students with essential analytical tools for solving complex PDEs in applied mathematics and related fields.

• Dr. S. Venkataraman

Advanced abstract algebra at the Ph.D. level offers an in-depth exploration of the abstract structures that form the foundation of modern algebraic research. Key topics include category theory, which provides a unifying framework for understanding mathematical structures and their relationships through objects and morphisms, offering powerful tools for defining and proving results in various algebraic contexts. Galois theory is another cornerstone, linking field theory and group theory to solve classical problems in algebra, such as the solvability of polynomial equations by radicals, and revealing deep insights into the symmetries of roots of polynomials. The study of representation theory explores how algebraic structures, particularly groups, rings, and Lie algebras, can be realized through linear transformations on vector spaces, providing a bridge between abstract algebra and geometric or physical applications. Additionally, commutative algebra focuses on the properties of commutative rings, modules, and ideals, offering fundamental tools for understanding algebraic geometry and number theory.

• Prof. Deepika

Probability theory is a fundamental branch of mathematics that provides a rigorous framework for modeling and analyzing uncertainty. It plays a crucial role in various disciplines, including statistics, machine learning, finance, physics, and engineering. Probability theory extends beyond basic concepts such as probability spaces, random variables, and distributions to more advanced topics like measure-theoretic probability, stochastic processes, and asymptotic theorems. Key results, such as the Law of Large Numbers and the Central Limit Theorem, form the foundation for statistical inference and decision-making under uncertainty. Additionally, concepts like conditional probability, Bayesian inference, and Markov processes are essential for developing probabilistic models in research.

• Prof. Deepika

Optimization theory is a fundamental area of mathematical research that focuses on finding the best possible solutions to complex problems under given constraints. It plays a critical role in fields such as machine learning, operations research, economics, engineering, and data science. Optimization theory extends beyond basic techniques like linear and nonlinear programming to advanced topics such as convex analysis, duality theory, stochastic optimization, and variational methods. Key results, including the Karush-Kuhn-Tucker (KKT) conditions, Lagrange multipliers, and interior-point methods, provide the theoretical foundation for solving constrained and unconstrained optimization problems. Additionally, modern research explores high- dimensional optimization, non-convex optimization, and applications in deep learning and reinforcement learning.

• Dr. Pawan Kumar

This course aims to foster skills among the research scholars for learning and using graph theory concepts in real world scenarios. It also gives enough theoretical background to the learners so that they can pursue research in this area. The broad topics covered are fundamentals of graphs with some standard algorithms, connectivity, matching, colouring and eigenvalues of graphs.

• Dr. S. Venkataraman

The course on Algebraic Number Theory is a 4-credit elective that spans a total of 60 hours, divided into 30 lecture hours, 10 hours of tutorials, and 20 hours of library work. The syllabus covers a wide range of topics, starting with integral domains, including unique factorization domains (UFD), principal ideal domains (PID), fields, and Galois theory in the separable case, along with norm and trace. In elementary number theory, the course delves into congruences, residues, non- residues, the Legendre symbol, the Jacobi symbol, and the quadratic reciprocity law. The section on algebraic integers covers integral elements, integrally closed domains, the ring of algebraic integers, and integers in both quadratic and cyclotomic number fields. The course also explores the concept of an integral basis, including finitely generated modules, the discriminant, and the discriminants of quadratic and cyclotomic number fields. The decomposition and extension of ideals are discussed through Dedekind’s theorem, Dedekind domains, the extension of ideals, and the decomposition of prime numbers in quadratic and cyclotomic fields. Further topics include ring extensions, focusing on the ring of fractions, traces and norms in ring extensions, and the discriminant of a ring extension. The different and discriminant section covers the relative trace and norm of an ideal, as well as the relative discriminant and different of algebraic number fields. The syllabus also includes the decomposition of prime ideals in Galois extensions, covering decomposition, inertia, and ramification. The class group section discusses ideal classes, the class group, Minkowski’s theorem, and estimates of the discriminant. Finally, the course covers units, including roots of unity, units of quadratic fields, units of cyclotomic fields, and Dirichlet’s theorem.

• Dr. S. Venkataraman

The course is worth 4 credits and spans a total of 60 hours, which includes 30 lecture hours, 10 hours of tutorials, and 20 hours of library work. The syllabus is divided into several key topics:

• Linear Representations: This section covers the definition and basic examples of linear representations, subrepresentations, irreducible representations, tensor product of representations, symmetric square, and alternating square.
• Character Theory: This part delves into the character of a representation, Schur’s lemma and its basic applications, orthogonality relations for characters, decomposition of the regular representation, number of irreducible representations, canonical decomposition of a representation, and explicit decomposition of a representation.
• Subgroups, Products, Induced Representations: Topics include Abelian subgroups, products of two groups, and induced representations.
• The Group Algebra: This section discusses representation and modules, decomposition of the group algebra, the center of the group algebra, basic properties of integers, integrality properties of characters, and their applications.
• Induced Representations: This part covers the character of an induced representation, Frobenius reciprocity formula, restriction to subgroups, Mackey’s irreducibility criterion, normal subgroups, applications to the degree of the irreducible representations, and the semi-direct product of Abelian groups.