Dates: June 10-22, 2024

Venue: Indraprastha Institute of Information Technology Delhi (IIIT-Delhi)

Timings: 9 AM to 6 PM

Those selected for the summer program have been notified via email.

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.

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