Our number theory seminar slot is sometimes unoccupied (the next such date is January 23rd). We will begin filling these empty slots with "Number Theory Lab Meeting".
This is aimed at creating a research community, especially for the benefit of our graduate students, but it will be of interest to all the number theorists and aspiring or closet number theorists in the department and so I wanted to make an announcement to the whole department. Young students who are considering number theory as their future track are especially invited, of course. You'll see it listed in the seminar calendar under the number theory seminar, and more information follows below.
The basic design is modelled after CSU's "lab meeting". It's fluid and adaptable, but the basic structure, at least to start, is:
11-11:10 - tea and informal chatter
11:10 - progress reports (we'll go around the room and everyone can give an update on what they are doing, mostly research, but also relevant may be pedagogy, job applications, etc.)
after that - a totally informal 20 minute chalk talk by one member of the group, about their research interests (background, for now, later after we are all more familiar, maybe recent results).
There are enough of us that I expect we may all get to speak approximately once per semester. We'll ask for volunteers as we rotate through so you have some warning/choice about when your turn is coming. This is friendly, and not meant to add pressure.
When interest dictates, the time can be used for other things, like career discussions, practice talks, trying out a cool number theory outreach activity, learning Sage together, whatever comes up.
I'll (Kate Stange) happily take the first slot and give some background on a problem I'm interested in on January 23rd. Your job is to show up and ask a ton of questions.
Topological T-duality -- first investigated by Bouwknegt-Evslin-Mathai, Bunke-Schick, and others -- concerns a surprising duality isomorphism between twisted cohomology theories on different spaces. It has traditionally been studied for twisted K-theory and deRham cohomology, owing to a physical origin where these objects can be considered as spaces of fields in a certain field theory. In joint work with John Lind and Hisham Sati, we have established a general context in which duality theorems of this sort can be proven. This context is naturally related to Cartier duality of finite Hopf algebras, and gives a host of new examples, particularly arising from chromatic homotopy theory.
In the study of quadratic moment maps there appears to be a dichotomy between large and small representations. In the large case the moment map exhibits some features of regularity. This enables one to make qualitative and quantitative statements about the symplectic quotients. In the talk I will report on recent joint work with Gerald Schwarz and Christopher Seaton which proves that for 2-large representations the appropriately defined complex symplectic quotient has symplectic singularities. This in particular entails that the symplectic quotient is graded Gorenstein domain and is a normal variety with rational singularities. It is expected that this result generalizes to small representations as well, even though here the moment map can have all sorts of pathologies. For this conjecture it is crucial to use the definition of the symplectic quotient that involves the real radical of the ideal generated by the moment map.