__August 30__

__Speaker__-Daniel Champion

__Title__- Infinite Ramsey TheoryRamsey Theory is a branch of combinatorial Set Theory that studies what conditions are necessary for order to appear from disorder. I will begin the talk with a classic (finite) Ramsey theorem: "In any set of 6 people, there is a subset of three people who are all either friends with each other, or there is a subset of three people who are not friends with each other." A corresponding infinite version will also be presented and proved. I will conclude the talk with several generalizations and some fascinating negative results involving larger cardinals.

__September 6__

__Speaker__- Predrag Punosevac

__Title__- Normal Forms in Local Dynamical SystemsIn this lecture I will speak on the method of normal forms in the study of diffrential equations. I will try ot address the most important aspects of the subject such as: formal problem, homological equation, resonance problem, convergence problem, Siegel's domain, Poincare-Dulac theorem, and normal forms of nilpotent operators. I will also discuss an example of an application in detail. No prior knowledge of diffrential equations will be necessary to follow the talk.

__September 13__

__Speaker__- John Kerl

__Title__- TensoramaIn our graduate careers, we encounter tensors in at least four guises: (1) as presented in abstract-algebra texts; (2) as multilinear functions (e.g. in differential geometry); (3) as multidimensional arrays; (4) as objects which transform in a certain way on change of coordinates (physicists' favorite definition). A bewildered first-year graduate student might well inquire: "Are these four completely different things, all masquerading under the same name?" Fortunately, that bewildered student has survived, more or less intact, into his second year and is able to confidently report that they are in fact ALL THE SAME THING. Wednesday at noon we'll see why, learn how to think about freedom (freeness), and enjoy a bagel or two.

__September 20__

__Speaker__-Abhishek Bhattacharya

__Title__- Statistics on ManifoldsThe mean and variance of the data form the major summary statistics in data analysis. But most of the literature is on data in Rn. What do we do if our data is non euclidean? In this talk I shall deŻne the notions of mean and variance of probability distributions on arbitrary manifolds. Then based on a random sample on the manifold, I shall talk about the sample analogues. Then I shall present important properties of these parameters. I shall deduce asymptotic properties of the sample estimates like consistency, Central Limit Theorem etc. Finally I shall apply my results to certain manifolds which include the sphere, the real projective space and the planer shape spaces.l

__September 27__

__Speaker__- Taylor Dupuy

__Title__- Angel Versus Devil ProblemThe Angel-Devil game is a game that is played on an infinite chess board where the angel and devil alternate turns. On the angel's turn he gets to move n-moves like a chess king (n is called the `power' of the angel) and on the devil's turn he is allowed to devour any one square of the chess board. The object of the game for the devil is to trap the angel on an island with a moat of thickness n around it (the angel is allowed to `fly' over moats of size n - 1) while the object of the game for the angel is to not get caught by the devil. It is an open problem whether the devil can catch an angel of power n (n > 1). John H. Conway defines a fool to be any angel who promises to always increase his y-coordinate every turn. We present the following theorems: the devil can catch a fool, the devil can catch a lax fool, the devil can catch a relaxed fool and the devil can catch an out-and-out fool. The talk will at least touch on the Blass-Conway diverting strategy and may include generalizations of the Angel-Devil game. John Conway offers a cash prize for the solution of this problem.

__October 4__

__Speaker__- Arlo Caine

__Title__- Local Representation Theorems in GeometryThis will be a leisurely paced talk about the Morse Lemma in differential topology and the Darboux Theorem in symplectic geometry. Both of these are results which guarantee the existence local coordinates in a neighborhood of a point on your manifold that represent your abstract object (a smooth function with non-degenerate critical points in the former context and the symplectic form in the latter context) in a standard way. Their proofs, an interesting blend of analysis and algebra, are simple and eloquent and we will discuss them in detail. This talk should be accessible to any student who is taking, or has taken, the geometry/topology core course.

__October 11__

__Speaker__- Tom LaGatta

__Title__-Bridging the Communication Gap, or Physicists Are Sloppy Jerks and Mathematicians Are Anal-Retentive ElitistsMathematicians and physicists have different vocabularies, different ways of looking at the world, and even different techniques of proof. Yet, so often the two are studying the same thing. It seems a shame to have so many smart people working independently for a common goal, but not cooperating or even communicating. In this talk, I will outline a number of historical and modern cases where the two camps overcame their differences to accomplish great things together. Most importantly, by coming to this talk you will have an opportunity to see and meet students on the other side of the spectrum

__October 18__

__Speaker__- Brad Weir

__Title__- Funding for Dummies (or, People too Lazy to Teach)Tired of teaching College Algebra 8 times like Bob? Want to go to burning man, but feel bad about missing class? Get funding! You'll be able to spend all your time doing "research" (read "watching baseball"), or using "matlab" (read "checking myspace and watching youtube movies"). This talk will serve as a forum on funding. I will start by discussing outside and internal sources of funding and summer internship opportunities. Throughout, a number of students (including myself) will share their experiences. The goal of the talk is that the non-funded audience member leave understanding both the abstract theory and physical applications of becoming funded. Following the talk, we'll have a raffle for lifetime VIGRE funding.

__October 25__

__Speaker__-Daniel Champion

__Title__- Scissor CongruenceCan a circle be cut into pieces that can be re-positioned to form a square? Two planar bodies can be defined to be equivalent provided one can be "cut" into finitely many pieces which can be re-assembled to form the other. This equivalence is called scissor congruence. The famous Wallace-Bolyai-Gerwien theorem states that any two polygons of equal area are scissor congruent. However, the circle is not scissor congruent to any polygon. I will begin with a proof of the Wallace-Bolyai-Gerwien theorem, then I will prove that smooth bounded planar bodies with non-zero genus are not scissor congruent to any polygon. I will conclude the talk with examples of invariants that can preclude scissor congruence.

__November 1__

__Speaker__- Casey Warmbrand

__Title__- Partitions, Young tableaux, and the Plancherel measure.We will be discussing partitions of n. A useful representation of such a partition, is as a collection of left-justified dots, called a Ferrers diagram. From this we construct Young tableaux. We will make use of the Robinson-Schensted algorithm, which will yield us a bijection between pairs of standard Young tableaux of the same shape and permutations in S_n. This bijection will allow us to push forward the uniform measure on S_n to obtain the Plancherel measure on partitions of n. We will conclude by showing some of the techniques involved in the large n asymptotic analysis of this measure.

__November 8__

__Speaker__- Ben Dyhr

__Title__-The Bessel Stochastic Differential Equation; Implications for Drunk Humming Birds.A Brownian process in d dimensions is a random, continuous path, Bt , from [0,„) to Rd such that, for fixed t, Bt is a d-dimensional normal distribution with variance t. We will derive and consider the equation that |Bt| = |(Bt(1), Bt(2), ... , Bt(d) | = [(Bt(1))2 + (Bt(2))2 + ... + (Bt(n) )2]1/2 satisfies, called the Bessel stochastic differential equation (SDE). The dimension, d, is an important parameter in this SDE that changes the qualitative nature of the solution as it varies. This observation will lead to interesting characterizations of a d-dimensional Brownian motion and the perils of humming bird alcoholism.

__November 15__

__Speaker__-Grant Peterson

__Title__- A Brief History of Probability Theory__November 22__

__Speaker__- Donna Harris

__Title__-The Role of General Education Mathematics Courses in Fulfilling the Goals of the UniversityThe appropriate role of a given course is determined by the institution's goals. The three topics of focus will be: 1) Some historical reasons for studying mathematics, 2) How university goals have been determined and, arguably, ought to be determined, and 3) the implication of the university's purpose to the structuring of general education mathematics courses. As part of the latter two topics, we will consider what constitutes a community of learning and the consequences of losing a common aim.

__November 29__

__Speaker__- Victor Piercey

__Title__- Hamiltonian Vector Fields and the Darboux Theorem.A symplectic manifold is a pair $(M,\omega)$ where $M$ is an even dimensional smooth manifold and $\omega$ is a closed, nondegenerate 2-form on $M$. Symplectic manifolds form the natural setting for classical dynamics because the symplectic structure contributes the existence of Hamiltonian vector fields and Darboux coordinates. Hamiltonian vector fields and Darboux coordinates in turn give rise to Hamilton's equations on a coordinate neighborhood. In this talk, we will define symplectic manifolds and show how the symplectic structure gives rise to Hamiltonian vector fields. We will then show how these vector fields can be used to prove the Darboux theorem, which states that every point in a symplectic manifold has a neighborhood with Darboux coordinates. Finally, we will see how Hamilton's equations arise on a neighborhood with such coordinates. This talk will be accessible to any graduate student who has studied the material covered in MATH 534A. In particular, current students in MATH 534A should be able to follow the discussion. A handout with most of the details will be available on my department web page by the day of the talk, if not sooner.