===Completed talks=== Talk title: Liouville's theorem and transcendence Topic area: Transcendental number theory Discipline: Mathematics Speaker: Swarnava Mukhopadhyay , B.Sc. (Hons) Mathematics 2nd Year Date: January 16, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 45-60 minutes Weblink: http://www.cmi.ac.in/~vipul/studenttalks/liouvillenumbers.pdf Prerequisites: None Status: Done Audience: Vipul, Shreevatsa, Kshitij, Nivedita, Anupam, Bodhayan, Moulik, Agnid, Preyas, Hrushikesh, Anirbit, Ravitej, Arnab Talk title: Primality testing is in P Topic area: Complexity theory Discipline: Theoretical Computer Science Speaker: Ramprasad Saptharishi , B.Sc. (Hons) Mathematics 3rd Year Date: January 17, 2007 Time: 3:30 p.m. Venue: Seminar Hall Duration: 45-60 minutes Weblink: http://www.cmi.ac.in/~ramprasad/studenttalks/primality/primality.pdf Prerequisites: None Abstract: "The result by Manindra Agrawal, Neeraj Kayal and Nitin Saxena made news when they gave the first deterministic polynomial time primality testing. Their paper 'Primes is in P' won them the Godel and the Fulkerson Prize. Primality testing has been a problem studied for a long time, and efficient determistic testing had been a long-standing open problem. In this talk, I plan to present the result as in an article by Jaikumar, Vinay and Kavitha and some open problems." Status: Done Audience: Vipul, Shreevatsa, Kshitij, Nivedita, Padmavathi, Anupam, Moulik, Preyas, Hrushikesh Talk title: Introduction to the theory of relativity Topic area: Relativity Discipline: Physics Speaker: Anirbit Mukherjee , B.Sc. (Hons) Physics 2nd Year Date: January 23, 2007 Time: 3:40 p.m. Duration: max{till the audience has patience, 100 minutes} Prerequisites: Patience Abstract: "This talk will analyze relativity from a point of view that hinges around the finiteness of the speed of light. It will be along similar lines as an earlier talk delivered by the speaker in the T. Nagar office, but the speaker plans to also discuss interconnections with quantum theory and electrodynamics. Depending on the audience response, the speaker may follow this up with another talk on the Dirac Equation" Status: Done Audience: Vipul, Shreevatsa, Padmavathi, Ravitej, Jayanth, Arpith, Arnab Talk title: Simple groups and the classification problem (Part 1) Topic area: Group theory Discipline: Mathematics Speaker: Vipul Naik , B.Sc. (Hons) Mathematics 3rd Year Date: January 24, 2007 Time: 3:30 p.m. Venue: Seminar Hall Duration: 1 hour Prerequisites: A first course in group theory Weblink [preliminary]: http://www.cmi.ac.in/~vipul/studenttalks/simplegroupspresentation.pdf Abstract: "The Classification of Finite Simple Groups is probably the biggest mathematical theorem of twentieth century, spanning over 15,000 pages. This talk is the first part of a two-talk series that addresses the questions: "What is a simple group?" and "What does the Classification Theorem state?"" Status: Done Audience: Shreevatsa, Kshitij, Nivedita, Anupam, Moulik, Padmavathi, Anirbit, Swarnava Talk title: Majority With Polylog Advantage (a sequel to the Sorting Networks talk) Topic: Complexity Theory Discipline: Theoretical Computer Science Speaker: Ramprasad Saptharishi , B.Sc. (Hons) Mathematics 3rd Year Date: January 31, 2007 Time: 5:00 p.m. Duration: 30-45 minutes Recommended: The speaker's previous talk on sorting networks Abstract: In an earlier talk on sorting networks, the speaker discussed Valiant's method for computing majority in NC1. The method was a probabilistic amplification after realising that a general majority just means that the bit occurs with probability at least 1/2 + 2/(n+1) and amplifying it in NC1. A natural question to ask is, what happens when the second quantity in the sum was much larger? How about inverse polylog (1/2 + 1/(polylog))? Do we need resources as big as NC1 to amplify it, can it be done in a smaller class? This talk will show that polylog advantage can be solved in AC0, which is a class strictly smaller than NC1. Organization of the slides: The speaker plan to have three major sections in the talks, the first one to revisit some computational complexity we discussed at the end of the earlier talk. The second one would be to get a decent grip on AC0, NC1 and give a very vague intuitive idea as to why AC0 is strictly smaller than NC1, and then go on with the poly log amplification (which is just a bunch of inequalities actually). Status: Done Audience: Vipul, Shreevatsa, Kshitij, Nivedita, Anirbit, Hrushikesh Talk title: An introduction to Hamilton-Jacobi theory Topic area: Classical mechanics Speaker: Ravitej U , B.Sc. (Hons) Physics 2nd Year Date: February 1, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 50-60 minutes Prerequisites: None Weblink: ? Abstract: The talk will cover introductory Hamilton-Jacobi theory. It will be completely elementary and accessible to all. A sequel talk, titled "More into Hamilton-Jacobi theory and Angle-Action coordinates" will be presented soon Status: Done Audience: Vipul, Jayanth, Padmavathi, Arnab, Anirbit, Bhanukiran Talk title: Trigonometric functions and Fourier series (Part 1) Topic area: Fourier analysis Discipline: Mathematics/analysis Speaker: Vipul Naik , B.Sc. (Hons) Math 3rd Year Date: February 5, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 1 hour Prerequisites: High school calculus (differentiation and integration), trigonometry and complex numbers Recommended: Vector spaces, convergence of series, group theory Weblink: http://www.cmi.ac.in/~vipul/studenttalks/introtofourierseries.pdf Abstract: "The talk will cover the theory of periodic real-valued and complex-valued functions and the use of Fourier series to approximate these functions. It will also discuss how and why Fourier series are relevant both in pure and in applied mathematics" Status: Done Audience: Shreevatsa, Kshitij, Nivedita, Padmavathi, Arnab, Ved, Agnid Talk title: Statistical Mechanics of Complex Networks Topic area: Nonlinear dynamics Discipline: Physics Speaker: Arghya Mondal , B.Sc. (Hons) Physics 3rd Year Date: February 8, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 1 hour Prerequisites: None Weblink: ? Abstract:This talk will be based on a review paper published by Albert-Laszlo Barabasi and Reka Albert in January, 2002 carrying the same title as this talk. This paper, being one of the most famous review papers in the field of network dynamics, provides a detailed account of the short history of development of the subject, of which Barabasi took a leading part. The speaker shall take inputs from other works for developments after Jan, 2002. He hopes to conclude the talk with some information on the problem he was working on last summer. Audience: Vipul, Jayanth, Padmavathi, Arnab, Anirbit, Shiladitya (?) Talk title: Trigonometric functions and Fourier series (Part II) Topic area: Fourier analysis Discipline: Mathematics Speaker: Vipul Naik , B.Sc. (Hons) Math 3rd Year Date: Febrary 13, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 90 minutes Prerequisites: Content of the first talk (which will soon be up online) Weblink: http://www.cmi.ac.in/~vipul/studenttalks/fourierseriescontd.pdf Abstract: In this talk, the speaker will discuss the general idea of a Fourier transform, study theorems regarding convergence for Fourier series and computation of the Fourier coefficients, and also touch upon Fourier series on the reals Audience: Shreevatsa, Kshitij, Nivedita, Padmavathi Talk title: A Journey in Planet Lambda (Part 1) Topic area: Lambda calculus Discipline: Theoretical Computer Science Speaker: Ramprasad Saptharishi , B.Sc. (Hons) Mathematics 3rd Year Date: February 14, 2007 Time: 4:45 p.m. Venue: Seminar Hall Duration: 30-45 minutes Prerequisites: None Weblink: The talk would be slideless but a survey/paper is under production. That should have all the material in the talk, will provide the link once it is complete. Abstract: "This is a series of 3 (or maybe just 2) talks and 'First Contact' is an introduction to Lambda Calculus. I plan to cover as much as I can, including my recent work with Dr V.Vinay with as much intuition and motivation as possible. The first talk will start from scratch, defining lambda terms, and some interesting applications of certain terms and some basic theorems. I will try to cover as much ground as possible, hoping to keep the talk completely understandable as an invariant. Notes/Requests from the Speaker: I attempted to make it as clear as possible for the Primality talk as well, but I didn't do a good job there. One problem I noticed was that slides forced me to explain only what was on them and I just didn't give enough insight and motivation that I wished to give. Hence this time (also motivated by Anirbit's talk) want to make it slideless, plan to give the talk on the board. I hope to give as much of insight and motivation as possible to make it enjoyable. The board should prevent me from handwaving and help me do a better job this time. My only intention to give the talks is to make the audience understand and appreciate. The purpose is lost if I do something on the board but doesn't reach the audience. Please stop me if I'm fast, wrong, not clear, not enough motivation, whatever. Lambda calculus is a truly remarkable area and the beauty is lost if First Contact is a disaster." Audience: Vipul, Shreevatsa, Nivedita, Kshitij, Anirbit, Preyas Talk title: Chronicles of Planet Lambda -- Return of the Birds (Part 2 of 3) Topic area: Lambda calculus Discipline: Computer Science Speaker: Ramprasad Saptharishi , B.Sc. (Hons) Mathematics 3rd Year Date: March 7, 2007 Time: 4:30 p.m. Venue: Seminar Hall Duration: 1 hour Recommended prerequisites: The first part (held on February 14th) Abstract: "The last talk introduced lambda terms, and motivated their power in computability theory (simulated arithmetic, boolean algebra, recursion) through the combinator birds defined. This talk we shall dwell a little more on the Y combinator and recursion. I will then go on to prove some interesting theorems here, analogous to some popular theorems in the turing machine setting. I hope to motivate the power of the birds, in this talk, especially the starling and the kestral. The talk should be reachable to all who are a little familiar with lambda terms (which hopefully should be the case if you had attended the last talk)." Audience: Vipul, Shreevatsa, Nivedita, Kshitij, Anirbit, Preyas Talk title: Structure of S^3 Topic area: ? Discipline: Mathematics Speaker: Anirbit Mukherjee , B.Sc. (Hons) Physics 2nd Year Date: March 8, 2007 Time: 6:30 p.m. Venue: Seminar Hall Duration: 60-75 minutes Prerequisites: Elementary group theory, elementary topology, high-school coordinate geometry, high school knowledge of complex numbers Recommended prerequisites: Electrodynamics, acquaintance with manifolds Abstract: "This talk shall not be as intense in contents as my previous physics talks but it shall be intended with a view to motivate the audience into certain beautiful regions of mathematics and physics and a platform to share the excitement of the starting points of some of the roads which I have been recently following which lead to some of the beautiful insights into physics through mathematics , which I hope to understand someday. The talk shall be started with a small study of teh different avatars of S^1 and S^2 and hence build up the techniques and the intuition to elucidate the structure of S^3 which shall be done in detail.( the crux of the talk ). I shall NOT probe into the study of the homotopy groups of these spheres If time permits then from the above analysis I shall try to pull-out as a by-product a motivation of the concept of "principle G ( a lie group)-bundles over a manifold X" and introduce a road which shows how it forms the general framework for electrodynamics to reside in ( time permitting ) { I might take up the topics mentioned in the last paragraph in a next talk since I anticiapate that time ( and energy :-) and depending on how much of write-up I can prepare ) may not permit it to be covered in this talk }" Audience: Vipul, Shreevatsa, Padmavathi, Arpith, Hrushikesh Talk title: Classification of finite simple groups (Part II): use of Sylow theory Topic area: Group theory Discipline: Mathematics Speaker: Vipul Naik , B.Sc. (Hons) Math 3rd Year Date: March 13, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 90 minutes Weblink: http://www.cmi.ac.in/~vipul/studenttalks/sylowconstraints.pdf Prerequisites: Basic knowledge of group theory; attending the first talk will suffice Audience: Shreevatsa, Nivedita, Kshitij, Anupam, Anirbit, Padmavathi Talk title: Structure of S^3 (Part II) Topic area: Differential geometry/topology Discipline: Mathematics Speaker: Anirbit Mukherjee , B.Sc. (Hons) Physics 2nd Year Date: March 15, 2007 Time: 7:00 p.m. Venue: Seminar Hall Duration: 60 minutes Prerequisites: Roughly, the material covered in the first talk Abstract: "This talk today will be of a very short duration and will probably be of less than 60 minutes. It will be a completion of some of the strings that I had started during my last talk and could not complete it. I will mostly deal with elucidation of a map or a construction which I had motivated during my last talk and which attempts to tie together all the n-dimensioanl spheres on a common platform and will motivate their further study. I shall *not* deal with the approach to the spheres from the topological groups point of view in this talk , but I shall focus on it only in my next talk." Audience: Vipul, Padmavathi, Arnab Talk title: The story of the symmetric group (Part I) Topic area: Group theory/combinatorics/representation theory Discipline: Mathematics Speaker: Vipul Naik , B.Sc. (Hons) Math 3rd Year Date: March 20, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 90 minutes Weblink: http://www.cmi.ac.in/~vipul/studenttalks/thesymmetricgroup.pdf Prerequisites: High school combinatorics (permutations and combinatorics), elementary group theory Abstract: "The set of all permutations from a set to itself has been an object of interest to combinatorialists, group theorists and representation theorists for a long time. Many interesting properties and tools have been developed to study this set. In the first talk of the series, I shall explore some basic combinatorial and algebraic properties of the symmetric groups, including Young subgroups, binomial coefficients, conjugacy classes, leader and guard representations, and Stirling numbers. This will pave the way for the discussing of the representation theory of symmetric groups, which will be the subject matter of later talks in the series". Audience: Shreevatsa, Kshitij, Nivedita, Anupam, Moulik, Anup, Arpith Talk title: Finding solutions by approximation Topic area: Calculus/analysis Discipline: Mathematics Speaker: Vipul Naik , B.Sc. (Hons) Mathematics 3rd Year Date: March 26, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 75 minutes Weblink: http://www.cmi.ac.in/~vipul/studenttalks/approximatingsolutions.pdf Prerequisites: High school mathematics Abstract: "This talk will deal with various techniques to find roots of equations, as well as solutions to differential equations, using techniques of successive approximation. It will also explore how such techniques can be used to prove existence and uniqueness of solutions and determine the qualitative characteristics of these solutions." Audience: Shreevatsa, Kshitij, Nivedita, Belliappa, Anupam, Anirbit Talk title: Chronicles of Planet Lambda: Reflection -- The Final Frontier Topic area: Lambda calculus Discipline: Theoretical Computer Science Speaker: Ramprasad Saptharishi , B.Sc. (Hons) Mathematics 3rd Year Date: April 4, 2007 Time: 5:00 p.m. Venue: Seminar Hall Duration: 45 minutes Abstract: "The final part of the trilogy would be to mimic universal turing machines in the lambda calculus setting. Once we see the importance of universal turing machines, the bird I acting as a trivial UTM would motivate us to find 'encodings' of lambda terms. Using the combinator bases {S,K} and {X}, we shall defined encodings and corresponding self-interpretters and some small improvements on recent results in this area." Audience: Vipul, Shreevatsa, Kshitij, Nivedita Talk title: Conjugacy class-representation duality and the Extensible Automorphisms Problem (Part 1) Topic area: Representation theory of finite groups Discipline: Mathematics Speaker: Vipul Naik , B.Sc. (Hons) Mathematics 3rd Year Date: April 5, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 90 minutes Weblink: http://www.cmi.ac.in/~vipul/studenttalks/ccrdpresentation.pdf Prerequisites: Group theory, linear algebra (a first course) Abstract: "This talk will introduce linear representation theory, and describe the relation between conjugacy classes in a group and irreducible representations of the group. In this talk, I will also introduce the Extensible Automorphisms Problem and indicate how this approach is helpful for the problem. The second part of the talk shall focus on the partial solution to the Extensible Automorphisms Problem using this approach." Audience: Shreevatsa, Kshitij, Nivedita, Anupam Talk title: The Heat Equation Topic area: Differential equations Discipline: Mathematics Speaker: Vipul Naik , B.Sc. (Hons) Mathematics 3rd Year Date: April 9, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 90 minutes Weblink: http://www.cmi.ac.in/~vipul/studenttalks/theheatequation.pdf Prerequisites: High school calculus and differential equations, elementary linear algebra Abstract: "This talk will focus on defining the heat equation, and study both the general properties of solutions and the ways of solving it, subject to boundary conditions. Some generalizations of the principle to arbitrary reaction-diffusion equations will also be considered" Audience: Shreevatsa, Kshitij, Nivedita, Moulik, Anirbit, Padmavathi, Swarnava Talk title: Semidirect products and the Extensible Automorphisms Problem (Part 2) Topic area: Group theory Discipline: Mathematics Speaker: Vipul Naik , B.Sc. (Hons) Mathematics 3rd Year Date: April 12, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 90 minutes Weblink: http://www.cmi.ac.in/~vipul/studenttalks/semidirectproductsandextautos.pdf Prerequisites: The first talk and similar stuff Abstract:"This talk will review the Extensible Automorphisms Problem, which was mentioned in a previous talk. It will then proceed to recall the result that had been proved in the previous talk (on class automorphisms being equal to linearly pushforwardable automorphisms). After that, the notion of semidirect product will be explored and we will see how this, along with the representation theory result seen earlier, tells us that every extensible automorphisms of a finite group must preserve conjugacy classes. We shall also look at a few select illustrative examples." Audience: Shreevatsa, Kshitij, Nivedita, Anupam Talk title: The story of the symmetric group (Part II) Topic area: Group theory/combinatorics/representation theory Discipline: Mathematics Speaker: Vipul Naik , B.Sc. (Hons) Math 3rd Year Date: April 19, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 90 minutes Weblink: http://www.cmi.ac.in/~vipul/studenttalks/thesymmetricgrouppart2.pdf Prerequisites: High school combinatorics (permutations and combinatorics), elementary group theory, material covered in the first talk Abstract: "In this talk, I shall review some of the combinatorial structures introduced in my previous talk on the symmetric group. I shall then proceed to look at the following: uniform descriptions of all symmetric groups, generating sets and sorting algorithms, and Young tableaux and their relation with the representation theory of symmetric groups" Audience: Shreevatsa, Kshitij, Nivedita Talk title: Polynomial rings and their automorphisms Topic area: Commutative algebra and invariant theory Discipline: Mathematics Speaker: Vipul Naik , B.Sc. Hons) Math 3rd Year Date: April 23, 2007 Time: 6:15 p.m. Venue: Seminar Hall Duration: 90 minutes Weblink: http://www.cmi.ac.in/~vipul/studenttalks/polynomialringsandtheirautomorphisms.pdf Prerequisites: High school knowledge of polynomials, basic idea of linear algebra and group theory Abstract: "In this talk, I shall look at the polynomial ring in finitely many variables over a field, and at the structure of the automorphism group of this polynomial ring. I will also look at the relation between subgroups of the automorphism group and subrings of the polynomial ring (the so-called Galois correspondence) and shall consider some interesting results on when these subrings are free and when they are finitely generated." Audience: Shreevatsa, Kshitij, Nivedita, Anupam