# College of Arts & SciencesDepartment of Mathematics

## Seminars & Colloquia 2012

Departmental Colloquia

## 2012

• Title: The structure of free complexes.
Speaker: Daniel Erman, University of Michigan.
Time and Place: Friday, Monday, April 16, 2012 at 2 pm in LC 312.
Abstract:Over the polynomial ring, a graded free complex consists of a sequence of matrices whose entries are polynomials and where the consecutive products of the matrices equals 0. Graded free complexes arise throughout algebra, geometry, and topology, where they are often used to compute homological invariants. I will discuss a new perspective for understanding the structure of graded free complexes. This is joint work with David Eisenbud.

• Title: Combinatorics and geometry of $E_7$
Speaker: Steven Sam (Miller Fellow, Univ. California at Berkeley)
Time and Place: Thursday, October 4 at 3:30 pm in LC 412
Abstract: Exceptional objects can be thought of as an accident in classification schemes, but often have a rich structure all to themselves. In this talk, we'll explore some of the combinatorics and geometry related to the exceptional object $E_7$ (its root system, Weyl group, Lie algebra, ...) which comes from the Cartan-Killing classification of simple Lie algebras. This object was studied by classical geometers long before this classification, and remains an object of interest today. We will discuss topics such as reflection arrangements, finite geometry, plane quartic curves, Kummer varieties, Vinberg's theta-representations, and toric geometry. The plan is to illustrate the beauty of this exceptional object in an accessible way.

• Title: Brief history of the Boltzmann-Sinai Ergodic Hypothesis
Speaker: Nandor Simanyi (Univ. of Alabama at Birmingham)
Time and place: Thursday, October 25 at 3:30 pm in LC 412.
Abstract: The Boltzmann-Sinai Hypothesis dates back to 1963 as Sinai's modern formulation of Ludwig Boltzmann's statistical hypothesis in physics, actually as a conjecture: Every hard ball system on a flat torus is (completely hyperbolic and) ergodic (i. e. "chaotic", by using a nowadays fashionable, but a bit profane language) after fixing the values of the obviously invariant kinetic quantities.
In the half century since its inception quite a few people have worked on this conjecture, made substantial steps in the proof, created useful concepts and technical tools, or proved the conjecture in some special cases, sometimes under natural assumptions. Quite recently I was able to complete this project by putting the last, missing piece of the puzzle to its place, getting the result in full generality.
In the talk I plan to present the brief history of the proof by sketching the most important concepts and technical tools that the proof required. The talk should be accessible to a wide audience, including graduate students.

• Title: Cutting and pasting in algebraic geometry
Speaker: Ravi Vakil (Stanford University).
Time and Place: Monday, November 19 at 3:30 pm in LC 412.
Abstract: Given some class of "geometric spaces", we can make a ring as follows:

• additive structure: When U is an open subset of such a space X, [X] = [U] + [X - U]

• multiplicative structure: [X x Y] = [X][Y].

In the algebraic setting, this ring contains surprising structure, connecting geometry to arithmetic and topology. I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural). A motivating example will be polynomials in one variable. (This talk is intended for a broad audience.) This is joint work with Melanie Matchett Wood.

• Title: Order and chaos
Speaker: Carl Pomerance (Dartmouth College).
Time and Place: Friday, November 30, 2012 at 3:30 pm LC412.
Abstract:Consider these curiosities: If you shuffle a deck of 52 cards 8 times, each shuffle perfectly interleaving the two half-decks, the cards will return to their starting position. But if you include the 2 jokers, so making a deck of 54 cards, it takes 52 perfect shuffles. Chaos in mathematics refers to a situation like this where a small change in the input produces a large change in the output. Here is another: The repeating period for the decimal expansion of 1/89 is 44, but for 1/91 it is 6. For 1/109 it is 108, and for 1/111 it is 3. We will see that these two chaotic functions, the number of perfect shuffles and the decimal repeating period, are connected to the order-of-an-element" function in group theory. It is perhaps ironic that chaos arises when considering order! I will discuss some progress recently made with P. Kurlberg towards a conjecture of V. I. Arnold.