|
Summer 2008 Abstracts |
|
Oleg Ivrii August 29, 2008 It is easy to see that the values of {αn} are dense (and in fact, uniformly distributed) in [0,1] for irrational α. But what about {αn2}? To answer this question, I will give two approaches: first approach will be based on the idea of minimality from topological dynamics and the second will be based on the notion of ergodicity from measure-theoretic dynamics. I will present and compare the two approaches. After this, I will say a few words about recurrence properties: for instance, I will discuss the following question: how big should N(n) be so that ||αk2|| < 1/n for some 1 ≤ k ≤ N? (Here ||x|| denotes the distance from x to the closest integer). where combinatorics meets algebraic geometry Yura Burda August 22, 2008 Hurwitz numbers, the numbers of branched coverings of a two-sphere with given branching profile, can be defined in combinatorial terms alone. However, they were recently shown to be related to the geometry of moduli spaces of algebraic curves and meromorphic functions on them. In this talk, I will speak about this relation, starting with the simplest case: Cayley's count of the number of labelled trees. The talk will be mostly elementary: no knowledge of algebraic geometry will be assumed. Omar Antolín Camarena August 15, 2008 The theory of species is an elegant framework for problems in enumerative combinatorics. In many cases it makes finding generating functions for object counts (or at least equations for these functions) as easy as defining the kind of objects counted; these definitions are given as algebraic expressions in terms of species of basic combinatorial objects and a few intuitive operations. We will define the notion of species and the basic operations on them and give several examples of their use. Intersection Matrix of an Isolated Singularity II Mikhail Mazin August 13, 2008, 12:00 - 2:00, Huron 215 This is the continuation of the previous talk on 16 July; however, I will do my best to make it self contained. I will describe the relations between the intersection form of the singularity, the Seifert form and the variation operator. I will also define the Dynkin diagram of a singularity and compute it for some examples. As well as for the first talk, I'll follow the V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko book "Singularities of Differentiable Maps" v.II. Oleg Ivrii August 6, 2008, 12:00 - 2:00, Huron 215 Dr. Zeckendorf has created a remarkable way of representing numbers: express them as sums of Fibonacci numbers so that no two consecutive Fibonacci numbers appear in the representation, e.g 48 = 34 + 13 + 2 (to be written as 10100010). I thought it was just a toy, until I met the game Wyt Queens. Here, there are two piles of stones, say 15 and 22, and players alternatively take either any number of stones from one pile or an equal number of stones from both piles. The one who takes the last stone wins. I will describe how knowledge of the Zeckendorf representions makes a Wyt Queen master out of anyone. Chad Groft Aug 1, 2008 and Aug 8, 2008 The method of forcing, first invented by Paul Cohen in the 1960's to prove the independence of the Continuum Hypothesis and the Axiom of Choice, has since been turned into a very general method of approaching consistency problems. It is something of an open problem to explain why this method works or how anyone could think of it. We'll try to do so by referring to Cohen's original work, which is substantially different from what is used today. In part I, we'll examine the state of set theory before Cohen's work, including why the problems he solved were considered intractable. Part II will be devoted to how he solved them and how his methods were transformed into the modern techniques. David Li-Bland July 25, 2008 Can rigour be given to the intuition behind Differential Geometry? Can infinitesimals be handled with precision? I will try to answer these two questions, while investigating some of the foundations of differential geometry, and propose a more general category to work in. Oleg Ivrii July 23, 2008, 12:00 - 2:00, Huron 215 I will begin by solving the game Nim. If we cutoff Nim's wincode at heaps of size less or equal to 7, we get precisely the Fanocode, which corresponds to the Steiner system S(2, 3, 7). On June 18th, I explained that to pass from S(2, 3, 7) to S(3, 4, 8), we add a parity check digit to each codeword. The Mock Turtles Theorem provides a game-theoretic analogue of this extension. Similar situation arises when extending the binary Golay code. Next, I will give two more constructions of the extended binary Golay code -- the MOG (Miracle Octad Generater, created by Curtis in 1976) and another utilizing the non-adjacency matrix of the icosahedron. For the most part, the talk will be self-contained. Oleg Ivrii July 18, 2008 The (extended) binary Golay code is an error correcting code which has 12 message digits, 12 check digits and can correct up to 3 mistakes. Some know it as the (11, 5, 2) biplane. Some know it as the icosahedron. It is also the Steiner system system S(5, 8, 24). Its 759 "weight 8" codewords are winning positions in the combinatorial game Mogul. The group of symmetries is the infamous Mathieu group M24. There is also a younger brother, the ternary Golay code which is just as fun (it corresponds to the Mathieu group M12 and the so-called hexad game). Intersection Matrix of an Isolated Singularity Mikhail Mazin July 16, 2008, 12:00 - 2:00, Huron 215 In this talk, I'll study the topology of the non-singular level set of an isolated singularity (Cn, 0) -> (C, 0) and the monodromy of the singularity. I'll introduce the monodromy and variation operators, vanishing cycles and distinguished bases, Picard-Lefschetz operators, Seifert form of a singularity. I'll sketch a proof of the Picard-Lefschetz formula and describe how to get the Dynkin diagram of a singularity. I'll follow the V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko book "Singularities of Differentiable Maps" v.II. Pontryagin's Construction Jordan Watts July 11, 2008 For a fixed smooth compact manifold M and a positive integer p, I will prove a wonderful (cool) theorem that states that framed cobordism classes of framed (compact) submanifolds of M of codimension p are in bijection with homotopy classes of smooth maps from M to the p-sphere. We will then use this theorem to compute some of the higher homotopy groups of spheres, following in Pontryagin's footsteps, or at least some of them. Kiumars Kaveh July 4, 2008 An algebraic variety is the solutions of a system of polynomials in several variables. The geometry of algebraic varieties is in many ways connected to the geometry of convex bodies. We'll sketch the elegant connection between the theory of convex polytopes and the so-called toric algebraic varieties. We will then discuss Hilbert theorem on degree of an algebraic variety and will present a formula for degree in terms of volume of a convex body. This is a far-reaching generalization of the toric case. I assume no background except for definition of a polynomial and projective space! Pablo Carrasco Correa June 27, 2008 In the first part of this talk I will present some results about closed geodesics in surfaces using ideas from dynamics, in particular the geodesic flow. The second part will be focused in sketching a proof of the following result of Huberd: If S is a surface of curvature equal to -1 and Ω(T) denotes the number of closed geodesics of period less than equal T then Ω(T) is roughly eT/T. Oleg Ivrii June 20, 2008 The Fano plane appears in block design, difference sets, round-robin tournaments, finite geometries, Galois theory, multiplication law of Cayley numbers (octonions), error correcting codes and many other places. I wish to (briefly) explain some of these connections and maybe talk about the symmetries of the Fano plane and similar structures. Omar Antolín Camarena June 13, 2008 As an antidote to all those serious math seminars we've had recently, I'll talk about some frivolous but I hope fun problems about partitioning spaces (mostly Rn) into geometric figures (mostly circles)! Jordan Bell June 6, 2008 I will explain what the Riemann hypothesis is and what some important implications of it are. Yura Burda May 30, 2008 I will discuss a proof of a theorem of Belyi stating that a Riemann surface can be defined over algebraic numbers if and only if it admits a meromorphic function that defines a mapping of this Riemann surface to the Riemann sphere unramified outside three points. This amazing result has lead Grothendieck to consider simple combinatorial objects - dessin d'enfants, on which the absolute Galois group acts (faithfully). After discussing the proof, I will present a potpourri of applications and examples: generalized Chebyshev polynomials, the "true" shape of planar trees and dessins describing some Hurwitz spaces. Oleg Ivrii May 23, 2008 Abstract missing. Oleg gave examples of higher homotopy groups of various CW complexes including π3(S2 v S2). Zavosh Amir-Khosravi May 16, 2008 I will first talk about the following result: if p: Rn x Rn -> Rn is a non-degenerate bi-linear map (meaning p(0,x) or p(x,0) vanishes only when x=0), then n must necessarily be 1, 2, 4, or 8. I will show, following Milnor, that one can prove the weaker result n = 2k using Stiefel-Whitney classes. Then I will define the Hopf-invariant, a map h: π2n-1(Sn) -> Z, and prove its basic properties. I will link the two previous discussions, and state the generalization, proven by J.F Adams, that h takes the value 1, only if n=1, 2, 4, 8. If possible, we will then discuss elements of the Adams' proof. Yura Burda May 9, 2008 No abstract. Yura talked about Milnor's construction of the 28 "exotic" seven dimensional spheres, homeomorphic but not diffeomorphic to S7. Oleg Ivrii May 2, 2008 No abstract. Oleg gave a proof of the Hermite-Lindemann theorem, which in particular implies that the only algebraic point on the curve y = ex is (0,1). |