Algebra & Number Theory Seminar


Fall 2020 ANT Zoominar
Fridays @ 3pm California Time 
Meeting ID: 9045829721 --- password: seminar
For further information please contact
Professor Beren Sanders

Schedule (click dates for title and abstract) 

10/9 James Cameron UCLA
10/16 John McHugh UCSC
10/23 Changho Han U of Georgia
10/30 Soumya Sankar MSRI
11/6 Andrew Kobin UCSC
11/13 Bregje Pauwels U of Sydney
11/20 Clover May UCLA
11/27 BLACK FRIDAY (holiday; no talk)

Friday, November 20, 2020

Clover May (UCLA) 

Classifying perfect complexes of Mackey functors 

Mackey functors were introduced by Dress and Green to encode operations that behave like restriction and induction in representation theory. They play a central role in equivariant homotopy theory, where homotopy groups are replaced by homotopy Mackey functors. In this talk I will discuss joint work with Dan Dugger and Christy Hazel classifying perfect chain complexes of Mackey functors for \(G=\mathbb{Z}/2\). Our classification leads to a computation of the Balmer spectrum of the derived category. It has topological consequences as well, classifying all modules over the equivariant Eilenberg--MacLane spectrum \(H\underline{\mathbb{Z}/2}\).

Friday, November 13, 2020

Bregje Pauwels (U of Sydney) 

Tannakian formalism for stacks

It is well known that the category of algebraic representations of an affine group scheme G over a field k is a k-linear abelian tensor category. Moreover, if k is algebraically closed, then the group scheme G is completely determined by its category of representations. It is thus natural to ask which abelian tensor categories are equivalent to the representation category of an affine group scheme. In case k has characteristic zero, Deligne gives an internal characterisation of such categories: this is classical tannakian duality.

Similarly, varieties, schemes, group schemes and various generalizations thereof are often studied via an associated symmetric monoidal Grothendieck category (the category of quasi-coherent sheaves or the category of representations). Lurie has shown that passage to quasi-coherent sheaves gives an embedding into the 2-category of symmetric monoidal Grothendieck categories for many algebro-geometric objects. Like before, it is natural to ask whether we can characterize the image of this embedding. In particular, one can ask which symmetric monoidal Grothendieck categories are equivalent to the category of quasi-coherent sheaves on an (Adams) stack. In this talk, I will give partial answers to these questions, both in positive characteristic and characteristic zero. I will not assume everyone knows what a stack is.

This is joint work with Kevin Coulembier.

Friday, November 6, 2020

Andrew Kobin (UCSC) 

Witt vectors, lifting problems, and moduli spaces of curves

The ring of Witt vectors is an essential tool for understanding relationships between the worlds of characteristic 0 and characteristic p algebra and geometry. In this talk, I will describe how Witt vectors provide an elegant solution to the problem of lifting field extensions from characteristic p to characteristic 0. An important application is to the lifting problem for covers of curves. This situation motivated Garuti to define a projective scheme which compactifies the ring (scheme) of Witt vectors "equivariantly" (with respect to Witt vector addition). After describing Garuti's construction and its application to covers of curves, I will introduce a new compactification in the category of stacks that I am currently using to describe moduli stacks of curves in characteristic p.

Friday, October 30, 2020

Soumya Sankar (MSRI) 

Ordinarity of curves in positive characteristic

A curve in characteristic p is called ordinary if the p-torsion of its Jacobian is as large as possible. One may ask, what is the probability that a given curve is ordinary? I will talk about various perspectives on this question as well as some heuristics about what one might expect. I will answer this question for some families of curves, namely Artin-Schreier and superelliptic curves. 

Friday, October 23, 2020

Changho Han (U of Georgia) 

Arithmetic inflection locus for pluricanonical series on hyperelliptic curves

Over the algebraic closed fields, one can find the number of inflection points of a linear series (of rank r and degree d) on a curve of genus g via the degree of the Wronskian. Over the real numbers, one can use topology to take the "oriented count" on the number of inflection points upto sign. Here, we explore the geometric meaning of a natural analogue for arbitrary fields via \(A^1\)-homotopy theory, especially finding the meaning of local degrees at an inflectionary point. No prior knowledge of \(A^1\)-homotopy theory is assumed. This is a joint work with Ethan Cotterill and Ignacio Darago.

Friday, October 16, 2020

John McHugh (UCSC) 

On the image of the trivial source ring in the ring of virtual characters of a finite group

Let \(G\) be a finite group and let \(\mathcal{O}\) be a complete discrete valuation ring whose residue field \(\mathcal{O}/J(\mathcal{O})\) is algebraically closed of positive characteristic p. The character of a finitely generated \(\mathcal{O}\)-free \(\mathcal{O}G\)-module can be considered as an element of \(R_{\mathbb{C}}(G)\), the virtual character ring of \(G\). In particular, "taking characters" gives a map from the ring \(T_{\mathcal{O}}(G)\) of trivial source \(\mathcal{O}G\)-modules to \(R_{\mathbb{C}}(G)\). By a theorem of A. Dress, the image of this map is a subring of the ring of \(p\)-rational characters. We give a "detection theorem'' that answers the question: when is a \(p\)-rational character of \(G\) contained in the image of \(T_{\mathcal{O}}(G)\)? It turns out that when \(p\) is odd every \(p\)-rational character is contained in the image. This is not the case when \(p=2\): the quaternion group \(Q_{8}\) is a counterexample. We show that, in the \(p=2\) case, any other counterexample "comes from" \(Q_{8}\). The theories of \({biset functors}\) (introduced by S. Bouc) and \({fibered  biset functors}\) (developed by R. Boltje and O. Coşkun) make this precise.

Friday, October 9, 2020

James Cameron (UCLA) 

Computing homological residue fields

Homological residue fields give a notion of residue fields in tensor triangulated geometry. They are useful, but rather abstractly defined. In many tt categories there are tt residue fields, which are easy to understand. In this talk I will relate homological residue fields to the tt residue fields that exist in examples and describe a characterization of the homological residue fields in examples from algebra, topology, and representation theory.