Speaker: Ryan Hota
Title: The Equivalence of the Axiom of Choice with its Counterparts
Abstract: The Axiom of Choice appears to be simple and obvious at first, and yet some of its equivalent forms (Well Ordering Principle, Zorn’s Lemma, Vector Space Basis’) and consequences (Non-measurable sets, Banach-Tarski Paradox, Comparable Cardinalities) are anything but. To bridge the gap between this apparent simplicity and controversial consequences, one needs to provide proof of these equivalences and implications. Thus we will aim to prove the Axiom of Choice equivalent two of its other (apparently stronger) forms, using only basic principles.
Pre-requisites: Only Basic understanding of Set theory
Venue and Time: 16th August, 6:30 PM, Seminar hall
Speaker: Madhav CS
Title: The Path Contractibility Problem
Abstract: The Path Contractibility Problem is natural and seemingly simple in computational topology. This problem is however undecidable. We will try to prove this result and develop tools in the talk.
Pre-requisites: TOC ( Turing machine, undecidability, reductions), some basic idea of fundamental groups would be nice but unnecessary. Some results from combinatorial group theory will be assumed/not proven due to lack of time.
Venue and Time: 23rd August, 6:30 PM, Seminar hall
Speaker: Deepro Sarkar
Title: Fair Division and Some aspects of Social Choice Theory
Abstract: Ever been locked in a cold war over who cleans the bathroom or does the dishes? Or felt the tension rise as you and your siblings argue over Grandma’s antique old clock? Welcome to the quirky world of algorithmic fair allocation! In this talk, we’ll uncover how to divide indivisible things—like household chores, course assignments for lecturers, and inheritance—without sparking a riot. You’ll learn how algorithms can keep everyone happy (or at least not plotting revenge) by ensuring fair splits of everything from the least-favorite chores to the most coveted courses. We’ll cover a wide range of scenarios, skipping the deep dives in favor of breadth, and highlight some clever solutions from CMI that help you solve life’s everyday decisions—without casualties!
Pre-requisites: Discrete math, especially graph theoretic notions. A bit of complexity theory might help, but absolutely not necessary
Venue and Time: 30th August, 6:30 PM, Seminar hall
Speaker: Rooptak Mazumder
Title: Which traces are spectral?
Abstract: We look at ideals of compact operators on a separable Hilbert space. We look at a characterisation of a class of ideals, such that any trace on such an ideal satisfies Lidskii’s formula. It turns out that this class of ideals is characterized by logarithmic sub-majorisation. This result is due to F.Sukochev , D.Zanin.
Pre-requisites: Linear Algebra (eigenvalues and stuff), Hilbert Spaces, Compact operators
Venue and Time: 6th September, 6:30 PM, Seminar hall
Speaker: Shashank Ingalagavi
Title: The Cauchy surface area formula with Minkowski sums
Abstract: The Cauchy surface area formula in R^n states that the surface area of a convex shape is proportional to the average surface area of its projection onto a plane, where the average is taken over all orientations of the shape. We shall see see a proof of this result using Minkowski's Theorem, a result relating the mixed volume function of convex shapes to the volume of any linear combination of the shapes.
Pre-requisites: None
Venue and Time: 13th September, 6:30 PM, NKN hall
Speaker: Arjun Maneesh Agarwal
Title: Bubbles that Burst: Why markets rise, fall and who comes out on top?
Abstract: What do GameStop, FTX, Paytm, and Nykaa have in common? Each experienced massive crashes in recent times. But how do these crashes unfold, and why do they continue to occur? In this talk, we’ll explore the fundamentals of market microstructures and how stocks are theoretically priced to maintain stability.
Pre-requisites: Some familiarity with game theory. Meaning of equilibrium in sequential and non-sequential games, backward induction, signaling.
Venue and Time: 11th October, 6:30 PM, seminar hall
Speaker: Ojas Kumar
Title: What is the Riemann hypothesis?
Abstract: The Riemann hypothesis, first stated in Riemann's 1859 paper on the prime counting function, is a central problem in mathematics that has so far remained unresolved. This talk will give a quick introduction to the complex analysis needed, followed by the definition of the zeta function and a proof of its functional equation. Finally, we will describe the Riemann hypothesis and consider some of its consequences. A secondary goal of the talk is to give an introduction to several important ideas in analytic number theory (including a proof of Dirichlet's theorem on arithmetic progressions). Much of the material for the talk comes from the first chapter Edwards' great book on the zeta function, so you can take a look at it and see if you're interested.
Pre-requisites: Calculus; It would help to know Complex Analysis, but we will rapidly cover the ideas needed
Venue and Time: October 18, NKN Hall, 6:30 PM
Speaker: Soumyajoy Kundu
Title: Conformal Prediction: A Framework for Reliable Uncertainty
Abstract: Conformal Prediction is a powerful, model-agnostic framework for generating valid prediction intervals or sets that adapt to the uncertainty in any given prediction. It offers a versatile approach for both regression and classification tasks, ensuring that the uncertainty quantification remains reliable, regardless of the underlying model. In this talk, we will explore the core principles of Conformal Prediction, its theoretical guarantees, and practical applications.
Pre-requisites: Predictive Modelling, Statistical Inference
Venue and Time: 25th october, 6:30 PM, Seminar hall
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, November 15, 6:30 PM.