September 17
An Introduction to Galois Representations: F_p Representations in Characteristic p > 0
John Bergdall
One possible principle in studying representations of Galois groups is to restrict to a suitably defined subcategory of all representations and then hope to construct an equivalent category with an exploitable (semi)-linear algebra structure. For example, if G is the Galois group of E_s/E (char E = p > 0, E_s is separable closure of E) then the auxiliary structure to consider is the Frobenius morphism obtained by raising elements to their pth powers. I plan to give an introduction to the basic vocabulary of Galois representations while discussing an important example (the Artin representations) that. After that, we will specialize to the case of a representation G(E_s/E) -> GL_n(F_p), E as above. In this case, we can completely write down the equivalent category as a certain subccategory of finite dimensional E vector spaces and actually prove the equivalence with less pain than one might think. There should be minimal prerequisite as I plan to review all the relevant definitions.
September 24
From Matroids to Manifold Bundles and the Mapping Class Group
Keefe San Agustin
In unpublished work and private communication, Kiyoshi Igusa has defined a "category of oriented matroids". There are several strong motivations for this and the one I want to focus on here is a series of implications regarding the mapping class group of surfaces. This talk outlines the current state of affairs on this front. We define the category of regular oriented matroids and examine its homotopy type and cohomology for low rank. We then change course and give a brief outline of the Igusa-Klein higher torsion invariants for smooth manifold bundles and relate these via the Riemann zeta function to the Miller-Morita-Mumford classes for the mapping class group of punctured surfaces. Finally, we discuss the interdependence between all of these objects and end with a few facts and a few conjectures involving matroids, higher torsion classes in the Torelli group and the work of Soren Galatius on the homology of automorphism groups of free groups.
October 1
Permutations Patterns
Jonah Ostroff
A permutation (considered in one-line notation) contains a shorter permutation as a pattern if one of its subsequences has the same relative order as the pattern. The enumeration of permutations not containing a given set of patterns has been a popular area of study for some time. We will discuss a few classical results from this area, as well as some more recent developments in the enumeration of symmetric pattern-avoiding permutations, i.e. those invariant under certain subgroups of D_4. Connections to Fibonacci numbers (including a "new" identity), Tribonacci numbers, Catalan numbers, Coxeter groups, standard tableaux, and the Robinson-Schensted correspondence will be discussed. We will try not to inadvertently answer problem #6f from the current 150a problem set.
October 8
Rigidity Theory and Dynamics
Keith Merrill
Rigidity Theory is an active area of mathematical research, whose roots trace back to the astounding theorem of Mostow: Let M and N be compact hyperbolic n-manifolds for n \geq 3. Then any isomorphism f: \pi_1(M) \to \pi_1(N) is induced by a unique isometry F: M \to N. This theorem states that the isometry type of the manifold (an geometric consideration) is completely determined by the isomorphism class of its fundamental group (an algebraic consideration). In modern terms, rigidity theorems are any result in which a restriction on some structure of an object forces restrictions on other structures, e.g. isomorphisms of certain subgroups force isomorphisms of the groups themselves. In my talk I will describe some more recent advances in rigidity theory, and comment on the use of dynamics and cocycles in proving rigidity statements (the above theorem is subsumed by Zimmer's Cocycle Superrigidity Theorem, which is yet another classic example of math's ineptitude at nomenclature).
October 15
Joint Mathematics Colloquium
No GSS due to Colloquium.
October 22
What Is Morse Theory And Why It Is Sooooo Coool!
Matt Graham
I will be talking about Morse Theory. I will define and explain "classical" Morse theory (Morse functions on compact manifolds) in detail. I will talk about perfect Morse functions, Morse-Smale functions, Morse-Bott theory (a generalization which allows one to treat Lie groups), and try to give a glimpse of what infinite dimensional Morse theory is.
Morse theory connects the different fields of analysis, topology and most recently physics. Morse theory studies the properties of manifolds by studying the properties of the set of functions that can be defined on them. For a trivial example: If somebody handed you a circle and the real line, you could distinguish these spaces by noting that you can define a continuous unbounded function on one and not the other. Morse theory goes well beyond this, in many cases you can actually calculate the homology groups simply from evaluating the critical point data of a Morse function f:M-> R. It gives a handlebody structure to any compact manifold, which allows one to think of the topology of spaces (e.g. SO(n), U(n), G_k(R), G_k(C), etc.) in very concrete geometric terms (in addition to allowing one to explicitly show two manifolds are diffeomorphic geometrically). It lead to: the h-cobordism theorem of Milnor (which Smale used to prove the Poincare conjecture for dimensions greater than 4); Bott periodicity and then K-theory; was applied to the physics Yang Mills functional (and originally the Lagrangian of classical mechanics). Recently, Ed Witten received a fields medal for applying Morse theory to the space of all connections of a certain manifold.
October 29
Joint Mathematics Colloquium
No GSS due to Colloquium.
November 5
Combinatorial Symmetries of the 24-cell
Andrija Peruničić
The 24-cell is a four-dimensional regular polytope, one of the six four-dimensional solids analogous to the Platonic solids in three dimensions. As such, it makes sense to ask what the reflection symmetry group of the 24-cell is. This group is described in terms of automorphisms of R, the collection of vectors perpendicular to the planes of reflection called the root system, and is well known. One may then pose the following question: What is the group of automorphisms of R preserving linear dependence? That is, if M(R) is the linear dependence matroid associated to the root system, what is the automorphism group of the matroid? I will demonstrate how to determine this group and compare it to the geometric symmetry group. There will be non-geometric automorphisms that preserve the matroid, but not the root system.
November 12
TBD
November 19
Matt Moynihan
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