Speaker: Sankalp Sundar (BSc III)
Title: The Riemann Hurwitz Formula
Abstract: The Riemann-Hurwitz formula is a formula in the theory of Riemann Surfaces, that relates the Euler Characteristic of two Riemann Surfaces given the data of a map between them. In the first half of the talk we'll introduce the Euler characteristic and prove that it is a topological invariant. We then derive the formula and look at some of its applications including a version of Fermat's last theorem for polynomials.
Pre-requisites: Knowledge of complex analysis would be helpful, but is not strictly necessary.
Venue and Time: Seminar Hall, 10th August 2023, 6:30 PM.
Write-up: Linked here.
Speaker: Madhav CS (BSc III)
Title: Context-free Games
Abstract: Walukiewicz theorem is one of the most important theorems in algorithmic game theory. It provides a way to solve context free games. The theorem holds for parity objectives but we shall see a simpler case of reachability objective. If time permits we shall see how to solve for parity objectives and the associated time complexity.
Pre-requisites: Knowledge of pushdown automata is useful, but not strictly necessary.
Venue and Time: Seminar Hall, 12th August 2023, 6:30 PM.
Speaker: Adhvik Jagannathan (BSc III)
Title: Hilbert Spaces and Partial Differential Equations
Abstract: The notion of functions as vectors of Hilbert spaces can be useful in finding solutions to certain partial differential equations. One such equation is the well known Poisson's equation, which appears multiple times in physics. We shall look at the class of problems known as abstract variational problems, look at a unique existence theorem, and its correlation with the Poisson's equation.
Pre-requisites: Familiarity with vector spaces, linear maps and the notion of a limit. Knowing the definitions of inner products, bilinear forms and Hilbert spaces is useful, but isn't necessary. All the required terminology will be introduced during the talk.
Venue and Time: Seminar Hall, 15th August 2023, 3:30 PM.
Slides: Linked here.
Speaker: Rohan Goyal (BSc III)
Title: The Sunflower Conjecture
Abstract: Sunflowers are families of sets whose Venn diagram looks like a sunflower :) More formally, sunflowers are families of sets for which pairwise intersection is the same as overall intersection. The sunflower conjecture says that for any r, there exists a constant C such that for any family of k sized sets which is larger than C^k, this family contains a sunflower with r petals. This was first proposed by Erdös and Rado.
Pre-requisites: Basic knowledge of discrete probability (high school probability). I will use the "big-O" notation somewhat freely and might use crude bounds at times which might actually be off by constant factors. So, I would like people to familiarize themselves with it somewhat. For simplicity, when I say a function f(k)=O(g(k)) then I mean that there exists a c>0 such that f(k) <= c g(k) for all large enough k.
Venue and Time: Seminar Hall, 18th August 2023, 6:30 PM.
Slides: Linked here
Speaker: Ayan Nath (BSc III)
Title: The LCM of Polynomial Sequences at Prime Arguments
Abstract: It has been conjectured that if f(x) is an irreducible polynomial with integer coefficients of degree d > 1 then, log lcm {f(n) : n < x} ~ (d-1) x log x. We investigate the analog of prime arguments, namely, lcm {f(p) : p < x}, where p denotes a prime and obtain lower bounds on it. Further, we also discuss some results regarding the greatest prime divisor of f(p). This is joint work with Abhishek Jha. Link to the paper here.
Pre-requisites: Familiarity with basic analytic number theory and Landau notations (big-O, little-o) will be helpful but not necessary.
Venue and Time: Seminar Hall, 21st August 2023, 6:30 PM.
Slides: Linked here.
Speaker: Anand Balivada (BSc III)
Title: Rotating Black Holes
Abstract: The rotating black hole solution to Einstein's field equations in (3+1) dimensions, first derived by Kerr in 1963, is the setting for some incredibly rich physics. We'll be presenting the Penrose process for extracting the rotational energy of such a black hole, analysing its physical realizability and bounds on its efficiency. A lightspeed introduction to special and general relativity, and relevant properties of Kerr black holes will be provided at the beginning of the talk.
Pre-requisites: High school physics, Linear algebra.
Venue and Time: Seminar Hall, 25th August 2023, 6:30 PM.
Material: Slides to the talk and Mathematica notebook linked here.
Speaker: Rajdeep Ghosh (BSc III)
Title: (Not) a Proof of Fermat's Last Theorem
Abstract: A sales pitch for algebraic number theory. The plan is to introduce some of the techniques and terminology that come up while trying to expand what we know about the integers and primes to larger, more exotic and more exciting places; all through the lens of a failed attempt at a proof of Fermat's Last Theorem.
Pre-requisites: None as such. Being comfortable with rings, ideals, fields and groups would be nice.
Venue and Time: Seminar Hall, 28th August 2023, 6:30 PM.
Speaker: Krishna Menon (PhD Math)
Title: Combinatorics of Hyperplane Arrangements
Abstract: In this talk, we discuss some problems in enumerative combinatorics that arise in the context of hyperplane arrangements. A hyperplane arrangement is a finite collection of affine hyperplanes in some Euclidean space. For example, in the plane, a hyperplane arrangement is just a finite collection of lines. One natural question to ask is how many 'pieces' a collection of lines breaks the plane into. In general, the regions of an arrangement are the connected components of the space obtained by deleting its hyperplanes from the Euclidean space. Counting regions of arrangements is an active area of research in enumerative combinatorics. There are several classes of arrangements where regions correspond to interesting combinatorial objects. For example, permutations, trees, and Dyck paths all appear as regions of certain arrangements. We will also discuss other combinatorial questions related to arrangements and solve these questions for some concrete classes of arrangements. As for most topics in enumerative combinatorics, the prerequisites for this talk are minimal. Although it might initially seem like we'll be using geometry/topology, we quickly reduce our problems to those such as counting integer tuples that satisfy certain properties.
Venue and Time: Seminar Hall, 31st August 2023, 6:30 PM
Slides: Linked here.
Speaker: Ritabrata Bhattacharyya (BSc III)
Title: Burnside's p-q Theorem
Abstract: The main goal of this talk is to introduce some basic notions of Representation Theory and give a proof of Burnside's theorem, which states that any finite group of size p^a q^b is Solvable.At first, we will see Representations, morphisms between them, Characters, Schrur's Lemma, Orthogonality Relations, Frobenius Divisibility, etc. Then we give a proof of Burnside's Theorem.
Pre-requisites: Being comfortable with Linear Algebra and Group Theory is fine.
Venue and Time: Seminar Hall, 4th September 2023, 6:00 PM.
Slides: Linked here.
Speaker: Ayan Nath (BSc III)
Title: Alterations
Abstract: The resolution of singularities is a fundamental question in algebraic geometry: can we transform any algebraic variety V into a nonsingular variety W using a birational map? While Hironaka solved this for varieties over characteristic 0 fields, it remains open in positive characteristic for dimensions greater than 3. In 1995, de Jong introduced "alterations", a variation of birational maps allowing finite extensions of the function field, and proved that any algebraic variety can be "altered" to a nonsingular variety, even in positive characteristic! In this talk, we provide an overview of de Jong's proof and, time permitting, discuss applications to cohomology.
Pre-requisites: Scheme theory; blow-ups (roughly equivalent to chapters 1-17, 21, 22, 24 of Vakil's FOAG July 31, 2023 draft); for the application part, familiarity with a cohomology theory will be helpful (e.g., étale, singular, etc).
Venue and Time: Seminar Hall, 7th September 2023, 6:30 PM.
Slides: Linked here.
Speaker: Gautham Viswanathan (BSc III)
Title: Decipher Title Via Googling Cantor, Halting, Computability, Cardinality
Abstract: This talk aims to talk about diagonalization in various avatars throughout math and CS. We begin with Cantor's classic proof that the real numbers are 'bigger' than the rationals, then segue into theoretical CS by using this fact to give a pretty one line proof that there exist problems that computers cannot solve. We then talk about a specific instance of such a problem, namely, the halting problem, and if time permits, see that diagonalization and the related philosophical idea of self reference lead to horrible horrible consequences for the foundations of mathematics.
Pre-requisites: None.
Venue and Time: Seminar Hall, 11th September 2023, 6:30 PM.
Speaker: Shubham Kumar (BSc III)
Title: Introduction to Morse Theory
Abstract: Morse theory studies the structure of a manifold by studying functions on it. The key idea is that certain differentiable functions on a manifold, called Morse functions hold a lot of information about the structure of the manifold. In this talk, I will introduce and prove some basic results like Morse lemma, Morse inequalities and homotopy type at critical values. Using these results we will see that a manifold with two nondegenerate critical points is always a sphere. Other applications of these techniques include the proof of the classification of surfaces and even the Poincare conjecture for n>= 5.
Pre-requisites: Knowledge of multivariable differential calculus is essential.
Venue and Time: Seminar Hall, 18th September 2023, 6:30 PM.
Speaker: Swaminathan Rajagopalan (BSc III)
Title: The Uniformization Theorem
Abstract: The Uniformization theorem is a landmark theorem that classifies all simply connected Riemann surfaces. The theorem is fundamental in hyperbolic geometry. In this talk we outline a topology and complex analysis flavoured proof of the same. We also classify surfaces according to their universal cover and see that almost every surface has the unit disc as its universal cover.
Pre-requisites: Topological notions like connectedness and compactness, some complex analysis. It will be nice to know what a covering space, as we will appeal to them in a small portion of the talk.
Venue and Time: Seminar Hall, 21st September 2023, 6:30 PM.
Speaker: Léo Gratien (ENS)
Title: Function field analogue of Zarközy theorem
Abstract: Take a set of integers A, and consider the differences b-a in A-A. If A is big enough, one can ask if A-A avoid all squares, or cubes, for example. Following from a theorem of Zarközy, this is impossible if A has a positive density. However many questions about the estimate of |a in A up to n| are still open. I will present the function fields analogue of this theorem, based on a work of Ben Green. It turns out much better estimates can be obtained. The talk will contain the entire proof, which is rich and creative ; and still elementary (— no hidden algebraic geometry !).
Pre-requisites: None.
Venue and Time: Seminar Hall, 12th October 2023, 6:30 PM.
Speaker: Milan Paul (BSc III)
Title: Proofs from THE BOOK
Abstract: I will present the proofs of some interesting theorems mostly taken from the book "Proofs from THE BOOK" by Aigner and Ziegler. The proofs are quite ingenious and hard to come up with but at the same time, are accessible to anyone familiar with high-school math. I will be presenting some theorems related to irrational numbers (number theory) and Borromean rings (Euclidean geometry).
Pre-requisites: None.
Venue and Time: Seminar Hall, 19th October 2023, 6:30 PM.
Speaker: Vedant Neema (BSc II)
Title: Language of the Machine
Abstract: High level languages like Haskell, Python and C are already well known. I'll give a brief introduction to what machine language is. A flavour of assembly for the x86 architecture called fasm will be discussed. I'll conclude by discussing optimizations that can happen at the lower level of instruction to the CPU and resources to learn more.
Pre-requisites: Basic programming constructs
Venue and Time: Seminar Hall, 23rd October 2023, 6:30 PM.
Speaker: Sankalp Sundar (BSc III)
Title: The Ax-Grothendieck Theorem
Abstract: The Ax-Grothendieck theorem states that all injective polynomial mappings from Cⁿ to itself are also surjective. Ax's proof of this theorem was one of the first applications of model theory to Algebraic Geometry. I will develop the basic model theory and present Ax's proof of the theorem. Along the way we will see how model theory can be thought of as "algebraic geometry without fields" and how compactness in logic is really a topological statement.
Pre-requisites: Familiarity with finite fields will help.
Venue and Time: Seminar Hall, 26th October 2023, 6:30 PM.
Speaker: Rajdeep Ghosh (BSc III)
Title: Why the quintic is insolvable: symmetry galo(is)re!
Abstract: The fundamental question we want to answer is why there is no general formula that produces the roots of a quintic polynomial using nice radicals (there is one for quadratics, cubics and quartics). There is in fact a more fundamental obstruction to this. We answer this using Galois theory, which will be (not so rigorously) developed in the talk.
Pre-requisites: Knowing how to quotient groups and a tolerance for hand-waving.
Venue and Time: Seminar Hall, 30th October 2023, 6:30 PM.
Speaker: Soumyadeep Paul (BSc III)
Title: Most efficient binary encoding of a message
Abstract: We will first define the problem of Source Coding and what we mean by optimality of a code in terms of average word length. We will then construct Huffman code "the most efficient code" and prove that it is optimal. We will also show a lower bound of average word length in terms of entropy and prove Shannon's noiseless channel coding theorem.
Pre-requisites: Basic knowledge of discrete probability.
Venue and Time: Seminar Hall, 2nd November 2023, 6:30 PM.
Speaker: Bijayan Ray (BSc III)
Title: Circuit Complexity
Abstract: We start with the perfect matching problem to motivate polynomial identity testing. We introduce arithmetic circuits for polynomial representation, including determinants and permanents. We formally present PIT, solvable in randomized polynomial time. We explore resolved cases like sparse polynomials and depth-3 whitebox diagonal circuits. Then we explore cases: general depth-3 diagonal circuits and orbit of sparse polynomials where PIT still remains unsolved. In conclusion, we mention a few other open problems in arithmetic circuits and progress made towards their partial solutions so far.
Pre-requisites: None.
Venue and Time: Seminar Hall, 6th November 2023, 6:30 PM.
Speaker: Ritabrata Bhattacharyya (BSc III)
Title: Hilbert's Seventh Problem
Abstract: This is one of the famous of Hilbert's Problems and a very important result in Transcendental Number Theory. It states that for an algebraic number x and an irrational number y, x^y is always transcendental. This was proved by Gelfond and Schneider in 1934. In this talk we will see some examples and facts about transcendental numbers and give a proof of Gelfond-Schneider Theorem and Hermite–Lindemann Theorem.
Pre-requisites: Familiarity with Algebraic Numbers and Complex Analysis is desirable.
Venue and Time: Seminar Hall, 9th November 2023, 6:30 PM.