Mathematics Colloquium Fall 2010

September 29, 2010 ***Will take place at 10:00 a.m.***

C_2-cofiniteness of permutation orbifolds

Toshiyuki Abe, Ehime University, Japan

October 5, 2010

The Three-Body Problem, Shape Space, and Brake Orbits

Richard Montgomery, UCSC

"The" problem goes back to Newton. I sketch some of its history, including contributions of Euler, Lagrange, and Poincare. I describe my two strongest results. Both use the shape sphere, a two-sphere whose points are oriented congruence classes of triangles and which inherits -- via invariant theory -- a natural (mass independent) conformal structure. I end with work-in-progress with Rick Moeckel (Minnesota) and Andrea Venturelli (Avignon) on brake orbits: solutions for which all velocities vanish at some instant. The topological study of brake orbits (and the Seifert and Weinstein conjectures) began with a paper of Seifert written early during World War II. I've posted two open problems regarding brake orbits on the addictive math site "Mathoverflow" in a question titled "Dropping three bodies." I will discuss these questions at the end.

October 12, 2010

Representations of the Lie Algebra of Finitary Infinite Matrices

Ivan Penkov, Bremen

In this talk I will try to answer the following question. What are "finite-dimensional representations" of the Lie algebra of finitary infinite matrices? As this Lie algebra has no non-trivial finite-dimensional representations, a more appropriate question is: what is a good category of "small" representations of the Lie algebra of finitary infinite matrices? The proposed answer is: "absolute" intergrable weight modules. A recent result of Vera Serganova, Elizabeth Dan-Cohen and myself claims that this category is Koszul.

October 18, 2010

Adding Numbers and Shuffling Cards

Persi Diaconis, Stanford

When a few large numbers are added in the usual way, "carries" accrue along the top. These carries form a Markov chain with an "amazing transition matrix" (Holte).It turns out that this same matrix occurs in the analysis of riffle shuffling cards, in the Hilbert series associated to the Veronese embedding, and in the character theory of the symmetric group. Properties derived in one mathematical area allow a full analysis of questions in others. All of this is joint work with Jason Fulman.

October 26, 2010

Two-dimensional mod p representations of the Galois group of Q

Dr. Ken Ribet, Berkeley

A famous 1987 conjecture of J-P. Serre linked irreducible two-dimensional mod p representations of the Galois group of the rational field with mod p reductions of classical modular forms. In an important breakthrough, this conjecture was proved in the last few years by C. Khare and J.-P. Wintenberger. Serre's conjecture may potentially be generalized in a number of ways: one could consider base fields other than Q; one could consider reductive groups other than GL(2); one could allow reducible representations. My talk will focus on the case of reducible representations, where the theory is not very well developed at all. Although I have proved some relevant theorems, there are puzzling examples that make it difficult to formulate a clean conjectural statement.

November 16, 2010

Arctic circles, domino tilings and square Young tableaux

Dan Romik, UC Davis

Domino tilings are ways of covering a region in the plane with "dominos", i.e., 2x1 or 1x2 rectangles. Domino tilings of a particular family of regions known as Aztec Diamonds behave in especially interesting ways. In particular, when such a tiling is chosen (uniformly) at random, one observes the "arctic circle phenomenon", where the domino configuration is frozen into a deterministic pattern outside of a curve called the "arctic circle". Another arctic circle phenomenon was observed in random objects sampled from a very different class of objects called "square Young tableaux". In this talk, I will discuss these results and their history and describe recent work in which I tie the two different arctic circle theorems together in an interesting way, by applying techniques borrowed from the analysis of square Young tableaux to prove things about random domino tilings. Another class of well-known combinatorial objects, namely alternating sign matrices, will also make a cameo appearance.

November 23, 2010

Dimensions of Fixed Point Spaces

Robert Guralnick, USC

Let G be an irreducible subgroup of GL(n,F) with F a field. There are classical problems related to problems about dimensions of fixed spaces of elements in G (including psuedoreflection groups and Frobenius kernels). We will discuss two recent results (answering conjectures from Peter Neumann's 1966 thesis) on this topic. The first is about the the average dimension of a fixed space of an element (when G is finite or compact) and the second is about the minimum dimension of a fixed space of an element. (Robert Guralnick is a leading group theorist and managing editor of Transactions of the American Mathematical Society.)

November 30, 2010

Unknotted Curves, Embedded Minimal Surfaces and Total Curvature

Jaigyoung Choe, Korea Institute for Advanced Study, Stanford

There is a well known theorem that a Jordan curve is unknotted if its total curvature is at most 4π (by Fáry-Milnor). There is also a theorem that a minimal surface has no self-intersection if its boundary has total curvature ≤ 4π (by Ekholm-White-Wienholtz). 

In this talk I will introduce some recent developments related with these two theorems.