The graduate student seminar meets from 3:30 to 4:30 pm Thursdays in Goldsmith 317. Want to give a talk? Email Steve Hermes at srhermes@etc... for details.

In topology, the singular cochain complex of a space comes with some extra structure which allows us to define an associative product called the cup product. This structure of a chain complex with an associative product appears throughout various branches of mathematics, leading us to the notion of a differential graded algebra (dga). Unfortunately, these associative products are not well-behaved with respect to homotopy: a homotopy equivalent chain complex may not inherit an associative product from one that does.

In this talk we will discuss how we can alleviate this problem by introducing the notion of A_{∞}-algebras. Along the way we will see how to cure all sorts of non-ideal behaviour of dgas.

By the pigeonhole principle of Dirchlet, if we partition 'many' objects into a 'small' number of classes, there will be 'many' objects contained in a single class. Suppose that these objects are now interrelated in a 'small' number of ways. Given a set of 'many' such objects, will there be 'many' objects all of which are related in a single way? Ramsey Theory studies this question. I will begin by surveying the Ramsey theory of the edge colourings of finite graphs and hypergraphs, then moving on to partition calculus (the Ramsey theory of infinite sets), an important branch of set theory.

Let X be a hyperelliptic curve of genus g defined over a finite field. Kedlaya developed an algorithm for (quickly) counting the number of points on this curve. It can also be used for computing its zeta function. The goal of my talk will be to describe this algorithm.

First, I will describe how to lift objects associated to the curve to a p-adic setting. The advantage of doing so is that there is a well-behaved cohomology theory, called Monsky-Washnitzer cohomology, which allows us to use the Weil conjectures. With their help, we will be able to explicitly compute the zeta function of the curve under consideration.

You will be able to follow the algorithm even if you've never dealt with the p-adic numbers before, and are only now learning about cohomology!

Department policy states that graduate students must see the definitions of the CAT(0) condition and of Right-Angled Artin Groups at least once per semester. In addition to fulfilling our quota, we will describe the relationship between a cube complex and its hyperplanes. Finally, we'll outline an application: a "geometric" proof of a theorem from combinatorial group theory.

No Seminar due to Brandeis-Harvard-MIT-Northeastern Colloquium

No Seminar due to Brandeis-Harvard-MIT-Northeastern Colloquium

Computing the genus of an arbitrary knot can be difficult. Adding a contact structure to R^{3}, allows us to compute a lower bound for the genus of an arbitrary Seifert surface, of any Legendrian knot. Since any knot can be nicely approximated by a Legendrian knot, this is also a lower bound for the genus of the knot. I will show that the trefoil is not the unknot.

A Geometric Method to Solve the Word Problem in Right-Angled Artin Groups

Given a finitely generated group, the word problem asks if there is an algorithm to determine if a word in the generators of the group is the identity. The word problem is not solvable in general, but it is in the case of right-angled Artin groups. I will give a geometrically flavored proof of this fact.

Previous Semesters