The Department of Mathematics at Indraprastha Institute of Information Technology Delhi (IIIT-Delhi) announces its first summer program – Mathematics Summer Program at IIIT-Delhi. This programme is designed for mathematics enthusiasts who have studied mathematics for two or more years at undergraduate or postgraduate level. This program will be completely funded by IIIT-Delhi. The participants will attend the lectures on various topics in higher mathematics by experts in the respective fields. They will get the chance to talk with IIIT-Delhi faculty members about mathematics to deepen their understanding and expertise of the subject. We believe that sincere candidates will benefit from this program by having a broader perspective on higher mathematics and its applications in other scientific domains. Therefore, we strongly encourage the students – who are interested in Mathematics and hope to pursue a career in this field – to apply.
1. Crash courses on selected topics in Algebra, Analysis, Geometry and Topology.
2. Problem-solving sessions and group discussion.
The selection will solely be on the basis of merit, taking into account both academic records and a recommendation letter from a mathematics Professor who knows the individual well.
During the training at IIIT-Delhi campus, meals and accommodation will be provided to selected candidates upon request in their application.
● RAL: Real analysis lecture; DEL: Differential equations lecture; AL: Algebra lecture; GTL: Geometry and Topology lecture.
● RAT: Real analysis tutorial; DET: Differential equations tutorial; AT: Algebra tutorial; DTT: Geometry and Topology tutorial.
| Week | Speaker | Syllabus |
|---|---|---|
| 1 | Dr. Satish Kumar Pandey | In the first week, we revisit the fundamental concepts of real analysis with the objective of proving the Stone-Weierstrass theorems. In the second week, the course shifts towards functional analysis and Dr. Subhajit Ghosechowdhury shall present an analytic proof of the Stone-Weierstrass theorems towards its end. What follows is the content of the first week of the analysis course that I shall present. ● A brief primer on metric, normed linear and Banach spaces. |
| 2 | Dr. Subhajit Ghosechowdhury | Dual space of a Banach space, Dual of C(X), The Hahn-Banach Theorem, Weak Topology, Alaoglu's Theorem, The Krein-Milman Theorem, De branges proof of Stone - Weierstrass Theorem. (If time permits, Victor Lomonosov's Theorem will be mentioned as an application of the technique) |
| Week | Speaker | Syllabus |
|---|---|---|
| 1 | Dr. Ashish Kumar Pandey | Distribution theory and its applications in Poses: (1) Distribution in 1d: test functions, definition and examples of a distribution, order, derivative, product by a smooth function, support, sequences; (2) Distribution in higher dimensions; (3) Convolution of distributions: differentiation and integration inside the bracket, tensor products, convolution, application to constant coefficient PDEs; (4) Fourier transform: Schwartz space, Fourier transform in Schwartz space, space of tempered distribution and their Fourier transform; (5) Sobolev spaces; (6) Applications in Laplace and heat equations. |
| 2 | Dr. Subhashree Mohapatra | Introduction to one dimensional finite element method and its implementation. |
| Week | Speaker | Syllabus |
|---|---|---|
| 1 | Dr. Nabanita Ray | Field Theory: Example of fields, Field extensions, Algebraic and Transcendental elements of a field, Degree of a field extension, Splitting fields and Algebraic closures, Finite Fields, The fundamental theorem of Galois Theory. |
| 2 | Dr. Prahllad Deb | Towards Lie Groups through a path of Linear Groups: Concrete matrix groups - Topology of matrix groups, Groups and geometry, Quaternionic matrix groups; The matrix exponential function - Definition and elementary properties, Applications (No small subgroup theorem and One parameter subgroup theorem), The Baker-Campbell-Dykin-Hausdorff formula; Linear Lie Groups: The Lie Algebra of a Linear Lie Group, Computation of Lie Algebras of Linear Lie Groups. |
| Week | Speaker | Syllabus |
|---|---|---|
| 1 | Dr. Prahllad Deb | Basics of Differential topology: Smooth manifolds, smooth maps, submanifolds, transversality, smooth homotopy and stability theorems, Sard's theorem and application (Morse Lemma and Whitney's embedding theorem for smooth manifolds). |
| 2 | Dr. Kaushik Kalyanaraman | Elementary Applied Topology: Homology and Cohomology: Basics of Topology: Topological spaces – set theory; equivalence relations; definition of a topology; discrete, product, and quotient topology; vector bundles; sheaf theory – presheaves, sheaves, posets, direct limit, stalks, morphisms of sheaves and exactness, abstracted example: heat diffusion on a graph to understand evolution of opinions on social media. Basics of Homology: Simplicial complexes; nerve of a cover; simplicial and singular homology; snake lemma and long exact sequence; Mayer-Vietoris, Rips, and Čech complexes; computational example (via Python): holes in a sensor network. Basics of Cohomology: Simplicial, Čech, and de Rham cohomology; Hodge decomposition; abstracted example: ranking pairwise alternatives; computational example: designing vector fields. If time permits (!): CW complexes, cellular homology, Poincaré and Alexander duality. |
Affiliation: Assistant Professor, IIIT-Delhi
Area : Differential Equations
Syllabus: Distribution theory and its applications in Poses: (1) Distribution in 1d: test functions, definition and examples of a distribution, order, derivative, product by a smooth function, support, sequences; (2) Distribution in higher dimensions; (3) Convolution of distributions: differentiation and integration inside the bracket, tensor products, convolution, application to constant coefficient PDEs; (4) Fourier transform: Schwartz space, Fourier transform in Schwartz space, space of tempered distribution and their Fourier transform; (5) Sobolev spaces; (6) Applications in Laplace and heat equations.
×Affiliation: Assistant Professor, IIIT-Delhi
Area : Geometry and Topology
Syllabus: Elementary Applied Topology: Homology and Cohomology: Basics of Topology: Topological spaces – set theory; equivalence relations; definition of a topology; discrete, product, and quotient topology; vector bundles; sheaf theory – presheaves, sheaves, posets, direct limit, stalks, morphisms of sheaves and exactness, abstracted example: heat diffusion on a graph to understand evolution of opinions on social media. Basics of Homology: Simplicial complexes; nerve of a cover; simplicial and singular homology; snake lemma and long exact sequence; Mayer-Vietoris, Rips, and Čech complexes; computational example (via Python): holes in a sensor network. Basics of Cohomology: Simplicial, Čech, and de Rham cohomology; Hodge decomposition; abstracted example: ranking pairwise alternatives; computational example: designing vector fields. If time permits (!): CW complexes, cellular homology, Poincaré and Alexander duality.
×Affiliation: Assistant Professor, IIIT-Delhi
Area : Algebra
Syllabus: Field Theory: Example of fields, Field extensions, Algebraic and Transcendental elements of a field, Degree of a field extension, Splitting fields and Algebraic closures, Finite Fields, The fundamental theorem of Galois Theory.
×Affiliation: Assistant Professor, IIIT-Delhi
Area : Algebra, Geometry and Topology
Syllabus:
Algebra
Towards Lie Groups through a path of Linear Groups: Concrete matrix groups - Topology of matrix groups, Groups and geometry, Quaternionic matrix groups; The matrix exponential function - Definition and elementary properties, Applications (No small subgroup theorem and One parameter subgroup theorem), The Baker-Campbell-Dykin-Hausdorff formula; Linear Lie Groups: The Lie Algebra of a Linear Lie Group, Computation of Lie Algebras of Linear Lie Groups.
Geometry and Topology
Basics of Differential topology: Smooth manifolds, smooth maps, submanifolds, transversality, smooth homotopy and stability theorems, Sard's theorem and application (Morse Lemma and Whitney's embedding theorem for smooth manifolds)
Affiliation: Assistant Professor, IIIT-Delhi
Area : Real Analysis
Syllabus:
×Affiliation: Assistant Professor, IIIT-Delhi
Area : Real Analysis
Syllabus: Dual space of a Banach space, Dual of C(X), The Hahn-Banach Theorem, Weak Topology, Alaoglu's Theorem, The Krein-Milman Theorem, De branges proof of Stone - Weierstrass Theorem. (If time permits, Victor Lomonosov's Theorem will be mentioned as an application of the technique)
×Affiliation: Assistant Professor, IIIT-Delhi
Area : Differential Equations
Syllabus: Introduction to one dimensional finite element method and its implementation.
×The Department of Mathematics provides a vibrant environment for research and teaching in mathematics. The department was set up in 2017 and currently consists of 18 faculty members and over 25 PhD students. The department offers a Ph.D. program in pure and applied Mathematics. The department has a four-year undergraduate program: B.Tech. in Computer Science and Applied Mathematics (CSAM).
Indraprastha Institute of Information Technology, Delhi
Okhla Industrial Estate,Phase III
(Near Govind Puri Metro Station)
New Delhi, India - 110020
Email Id - msp@iiitd.ac.in
Contact No. - 011-26907548
