__January 18__

__Speaker__- Abhishek Bhattacharya

__Title__- The Complex Projective Space - Part 1.In my talk I will identify the Complex Projective Space as a Riemannian manifold. I will identify its tangent space, compute the geodesics (great circles) and geodesic distances on the manifold, and the exponential and inverse exponential maps. The next talk on this (whenever it is) will focus on application of these ideas to the Shape space. Requirements for Audience: A little knowledge of Differential Geomtry will be good.

__January 25__

__Speaker__- John Kerl

__Title__- Computation in Finite FieldsTreatment of finite fields in undergraduate and graduate algebra courses ranges from the very brief (proofs of existence and uniqueness), to the theoretical (algebraic number theory), to the explicit (coding theory, cryptography, and designs). For the benefit of those us lacking exposure to the latter, I will talk about explicit, pencil-and-paper computations in finite fields. The goal is for us to be able to compute with finite fields as readily as we compute with, say, the rationals. Topics, depending on audience interest: * Determination of irreducible polynomials * Multiplication * Reciprocation * Automorphisms * Shanks' discrete log algorithm

__February 1__

__Speaker__- Tom LaGatta

__Title__- My Talk on Bill Faris' Review on E. T. Jaynes' Book on Bayesian StatisticsIf you are a math graduate student like me, you're well holed up in your ivory tower of pure mathematics, and rarely peek out to see how the plebes are doing down below. Every once in a while, though, your gaze slips, and you find a very tantalizing bite of applicable mathematics cross your vision. That happened to me as I was reading the January issue of the Notices and came across the eponymous review. Bill Faris wrote a really nice ten page crash course introduction to Bayesian statistics which I really enjoyed, and now I would like to share my newfound knowledge with you. So, if you are quite unfamiliar with Bayesian statistics, as I was twenty-four hours ago, then you owe it to yourself to come to this talk. And on the other hand, if you are quite familiar with Bayesian statistics, then you owe it to me to come and lambaste me for utterly misconstruing the concept.

__February 8__

__Speaker__- Abhishek Bhattacharya

__Title__- More on Bayesian TheoryLast week Tom introduced the fundamentals of Bayesian Theory, the concepts of prior and posterior distributions. In my talk I will discuss the applications of these concepts. Perhaps the most important problems in statistics are those pertaining to decision theory and Bayesian theory plays an important part in that. In my talk I will define Bayes decision rules and consider their application in estimation and classification problems.

__February 15__

__Speaker__- Cam McLeman

__Title__- Abstract NonsenseThere is probably no single construction that better epitomizes 20

^{th}-century mathematical abstraction than the notion of a category. The tools of the category theory trade have proven surpringly useful in (almost by definition) a large variety of mathematical disciplines. Further, the*language*of category theory is both potent and versatile, enabling fundamental statements at the heart of many disciplines to be formulated both concisely and precisely. Despite all such gains, however, category theory is often dismissed as being too removed from reality, and too surrounded by impenetrable notation, to merit serious attention. This talk will be a survey talk, complete with full definitions and rife with examples (including some well-known constructions you might not even realize are categorical). Finally, of course, we'll discuss where the limits of this abstraction might lead us...__February 22__

__Speaker__- Mel Koppens

__Title__- Quadratic FormsIf you enjoy algebra as much as I do, you owe it to yourself to come see my presentation at the grad talk. I am in the early stages of putting together my thesis on Quadratic Forms and will be showing you my progress. To be presented is a colorful parade of bilinear forms, isotropic & hyperbolic spaces, Witt rings and Grothendieck constructions. Basic knowledge about groups and rings should be sufficient. See you Wednesday!

__March 1__

__Speaker__- John Kerl

__Title__- A walking tour through projective spacesIn this informative and elementary talk, we will discuss glass tabletops, crystal balls, and railroad tracks. We will learn about why lines can be points, when a hole dug to China can get you back where you started, and how to compactify the real line by enlarging it. I will dip into algebraic geometry via a worked example illustrating Bezout's theorem, then venture far beyond my competence by mentioning combinatorics, topology, and number theory, then finally return to my domain of expertise with an application to computer graphics.

__March 8__

__Speaker__- Brad Weir

__Title__- Coming Home From SpaceLost in outer space? This talk will provide a general outline for how you might get back, or, if you so please, ensure you stay there forever. We'll begin with some rudiments of dynamical systems: limit cycles, stable and unstable manifolds, etc. From there, we'll heuristically develop more substantial results, like the Poincare-Bendixson theorem and KAM theory, and discuss how they relate to space travel. The prerequisites for this will be calculus, some analysis, and a substantial amount of caffeine. Time permitting, at the end of the talk we'll send audience members, picked at random, into space in order to see if they can make their way home.

__March 15__

__Speaker__- Spring Break, no talk

__Title__-__March 22__

__Speaker__- Tom LaGatta

__Title__- Phase Transitions for Dummies, and also for MathematiciansConsider a lattice with a particle placed at each intersection point, and to each particle associate a spin of +1 or -1. This is essentially the Ising model of statistical mechanics. If we instead allow spins of any direction (i.e., on a sphere), we have the Heisenberg model. The existence of phase transitions, like liquid-to-gas or Bose-Einstein condensation, is a central question to statistical mechanics. In this talk, I will present Peierls' classical argument that in 2 or more dimensions, the Ising model gives the existence of a phase transition at a certain critical temperature. I will also explain why this argument breaks down for the Heisenberg model, which requires at least 3 dimensions for a phase transition.

If this talks sounds abyssmally boring, I assure you it simply is a problem with my description and not the subject matter itself. I am quite excited by this material, and look forward to sharing my enthusiasm with you. If all goes well, your appetite will be whetted and you will stay for my talk in the Mathematical Physics Seminar immediately afterward at 1:00. As always, this talk will be accessible to an elementary audience—no physics or advanced mathematics background required.

__March 29__

__Speaker__- Eric Rudd

__Title__- Why Borda Is BetterArrow's Impossibility Theorem tells us there is no perfect voting mechanism—one that satisfies a seemingly benign set of highly desirable properties. Here, two classes of weighted tally election procedures are defined and a theory developed by Donald Saari is employed to find positive results within these classes. By viewing these weighted procedures for more-than-3 candidates as linear transformations over all subsets of candidates, the Borda Count voting method is identified as allowing the minimum number of subset outcome inconsistencies. Linear subspaces responsible for all paradoxes are described and a geometric description is used to expose the subspace responsible for Arrow's negative result. When this subspace influence is eliminated, only the Borda Count satisfies Arrow's original properties with no change to it's outcome or tally. This talk will be accessible to a general mathematical audience.

__April 5__

__Speaker__- Arlo Caine

__Title__- Symplectic and Poisson GeometrySymplectic geometry is a beautiful and broad field of research in mathematics. Invented by Hermann Weyl and born out of the mathematical physics of Hamiltonian mechanics its language is found in dynamical systems, differential topology, partial differential equations, and even complex algebraic geometry. Poisson geometry is a much younger subject but contains symplectic geometry as a sub-field of study. This talk will be an introduction to these two subjects woven with examples and laced with historical anecdotes. It is aimed at a fairly general mathematical audience.

__April 12__

__Speaker__- Derek Habermas

__Title__- What Next?You can wait until you are done with the quals to start prepping for the orals, and you can wait until you are done with the orals to start prepping for the dissertation. But you CANNOT wait until you are done with your dissertation to prepare to apply for a job. In fact, you will have to apply for a job during the semesters that you are working most intensely on your thesis, and it will suck up far more of your time than you expect. This is the next to last totally unfair thing about graduate school that nobody tells you. And if somebody does tell you (like Professor Ulmer told me years ago) it will probably get pushed to the "back burner" anyway, because applying for jobs does you no good if you don't graduate.

I will talk about my experience in the job application process from the realization that I was already going to miss the first deadlines to negotiating an offer with the Dean. I will talk about things I think I did right, things I should have done better, and things I wish I did years ago. And then you can push it all to your "back burner," because you have more important things to do, like graduate.

I would especially like to invite others who have recently gone through the job search to add breadth to the discussion. Specifically, since my applications were largely restricted to liberal arts colleges, anyone who could add some experience applying for post-docs would be most welcome.

__April 19__

__Speaker__- McKenzie Lamb

__Title__- Divisors, Line Bundles, and Sheaves on Riemann SurfacesIf X is a compact, connected Riemann surface, there is a correspondence between divisors, line bundles, and invertible sheaves on X. Taking suitable quotients of these spaces, this correspondence is one-to-one. I will define divisors, line bundles, and sheaves and describe the correspondence between them. This talk is intended for a general mathematical audience.

__April 26__

__Speaker__- Jeff Taft

__Title__- The Poincare Index TheoremI will be giving an outline of the proof of The Poincare Index Theorem. I will define the index of a once-differentiable vector field with a finite number of fixed points on a two dimensional surface. I will then explain why this number depends only on the topology of the surface, and not on the vector field chosen. This talk will not be rigorous, and will be completely accessible to any graduate student and mostly accessible to an undergraduate who has taken vector calculus.

__May 3__

__Speaker__- Lisa Berger

__Title__- Using Writing to Teach and Assess Mathematics in High School and Undergraduate ClassesI discuss why we should require our students to write more in mathematics, and I'll provide some examples of the work I've done in this area.