Speaker: Subhranil Deb (BSc III)
Title: Galois Representations and Iwasawa Theory
Abstract: I will be discussing Iwasawa theory on function fields and its relations to the number field case(Cyclotomic Z_p-extensions). We will talk about analogues of elliptic curves and how they can be used to construct towers of field extensions via Galois representations on their torsion points. We will define the class group in this case and will show that the p-primary part of the class module is finitely generated and torsion free. A very similar statement is also true over number fields which plays a crucial role in Iwasawa's growth formula. No prior knowledge of Iwasawa theory or number theory on function fields will be assumed.
Pre-requisites: Basic knowledge of number theory and algebra( especially group theory and Galois theory) is highly desirable. I will try to keep the talk as self contained as possible, however, understanding all the details of the proofs will need more prerequisites.
Venue and Time: Seminar Hall, 8th January 2024, 6:30 PM.
Speaker: Gautham Viswanathan (BSc III)
Title: Mario Games are Actually Pretty Hard
Abstract: This talk will be an introduction to basic ideas in complexity theory with everyone's favorite Nintendo character, Mario! In particular, we'll talk about one way to define the difficulty of problems, look at some examples of ‘difficult’ problems, and finally prove that determining whether or not a given Mario level can be beaten is ‘difficult’. In particular, this talk introduces the idea of reductions and proves that the above Mario problem is NP-hard. If time permits, we'll also show that it's PSPACE-complete and take a look at problems from other Nintendo games.
Pre-requisites: All necessary theory will be developed during the talk (including what NP and PSPACE are), so none.
Venue and Time: Seminar Hall, 11th January 2024, 6:30 PM.
Speaker: Suchir Kaustav (BSc II)
Title: The Quillen-Suslin Theorem
Abstract: In 1955, Serre posed the problem whether every finitely generated projective module over the polynomial ring k[x_1, x_2,... x_n] is free. The problem quickly became known as "Serre's conjecture" and it was quite difficult and remained open until 1976, when Quillen and Suslin independently proved the result. The main new idea introduced by Quillen is the result called "Quillen's Patching Theorem". In this talk I will present the proof of the Quillen Patching theorem, and show how to combine it with a result of Horrocks to prove the Quillen-Suslin theorem.
Pre-requisites: Some knowledge of commutative algebra, familiarity with localisations and Nakayama's lemma. It will be helpful (but not necessary) to know some sheaf theory to get the motivation behind some of the proof ideas.
Venue and Time: Seminar Hall, 18th January 2024, 6:30 PM.
Speaker: Pritish Sinha (PhD Physics)
Title: Geometric Formulation of Quantum Mechanics
Abstract: Classical mechanics is best understood in its geometrical picture: symplectic formulation of classical mechanics where we have a phase space of states endowed with a symplectic structure and a Hamiltonian function generating the time evolution. On the other hand, quantum mechanics is usually studied in algebraic or the functional analytic formulation. However, it turns out there exists an equivalent geometrical description of quantum mechanics which also admits symplectic structure, making it possible to understand quantum mechanics completely in the language of classical mechanics. Unifying this classical-quantum geometrical picture may have far reaching consequences for understanding semi-classical physics (and is already doing so) and may help us resolve many of the classical-quantum dichotomies. In the talk, we will first review the geometry of classical mechanics. Then, we will proceed to the standard formalism of finite dimensional quantum mechanics and show the existence of the geometric structures, and how it leads to a natural geometric formulation of quantum mechanics. We will mainly focus on the section 2 of this paper.
Pre-requisites: Hamiltonian formalism and some familiarity with tensors and differential forms.
Venue and Time: Seminar Hall, 22th January 2024, 6:30 PM.