Algebra & Number Theory Seminar


Fall 2021 ANT Zoominar
Fridays @ 10AM California Time 
Meeting ID: 941 6843 5061, Passcode: 821454 [Direct link]
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For further information please contact
Professor Robert Boltje

Schedule (click dates for title and abstract) 

10/15, 9am
Chan-Ho Kim Korea Institute for Advanced Studies (KIAS)
10/22 Nadia Romero University of Guanajuato
10/29 Yunqing Tang Princeton
11/5 Caroline Lassueur University of Kaiserslautern
11/12 Mandi Schaeffer Fry MSU Denver
11/19 David Holmes University of Leiden 
12/3 Baptiste Rognerud University of Paris, Jussieu
Friday, October 15, at *9* am PST

Chan-Ho Kim, Korea Institute for Advanced Studies (KIAS) 

On the p-converse to a theorem of Gross-Zagier and Kolyvagin

The celebrated Birch and Swinnerton-Dyer (BSD) conjecture predicts the equality between the rank of the rational points on elliptic curves and the vanishing order of their L-functions at s=1. One of the most striking results on the BSD conjecture is the work of Gross-Zagier and Kolyvagin: if the vanishing order is ≤1, then the predicted equality holds and the Tate-Shafarevich group is finite. Around 2013, Skinner and W. Zhang independently proved the p-converse of the aforementioned result under certain mild assumptions: If the Selmer corank is one, then the vanishing order is one. After their work, p-converse theorems are extensively studied in various forms. We discuss a simple proof of the p-converse theorem based on recent developments on Perrin-Riou's conjecture.

Friday, October 22, at 10 am PST

Nadia Romero, University of Guanajuato

Green Fields 

Green fields were discovered by Serge Bouc in 2019. To be precise, the terminology was introduced at the very end of his paper "Relative B-groups", published in 2019. A Green field is a commutative Green biset functor with no non-trivial ideals. In this talk I will present some properties of a Green field and examples of known Green biset functors which are Green fields. This is a joint work with Serge Bouc.

Friday, October 29, at 10 am PST

Yunqing Tang, Princeton

The unbounded denominators conjecture

The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, we will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem. (Joint work with Frank Calegari and Vesselin Dimitrov.)

Friday, November 5, at 10 am PST

Caroline Lassueur, University of Kaiserslautern

Classifying trivial source modules in blocks with cyclic defect groups

In the modular representation theory of finite groups the class of trivial source modules -- by definition the direct summands of the permutation modules -- is omnipresent and it is essential to understand them better in order to shed new light on main open questions concerned with the structure of blocks up to different kind of categorical equivalences. The aim of this talk is to present a concrete classification of the indecomposable trivial source modules belonging to a block B of a finite group with non-trivial cyclic defect groups. To achieve this goal, we rely on the classification of the indecomposable B-modules (by Janusz and dating back to the 1960's) and of the indecomposable liftable B-modules (by Hiß and Naehrig, 2012), both of which are determined up to Morita equivalence, hence by the Brauer tree of the block. Trivial source modules are not invariant under Morita equivalences in general. However, they are invariant under the stronger source-algebra equivalences. In this respect, besides the Brauer tree, two further parameters come into play. In particular, we will see how to recover the trivial source modules from the source algebra of the block in this case. This is joint work with Gerhard Hiß.

Friday, November 12, at 10 am PST

Mandi Schaeffer Fry, MSU Denver

The McKay--Navarro Conjecture: the Conjecture That Keeps on Giving!

The McKay conjecture is one of the main open conjectures in the realm of the local-global philosophy in character theory. It posits a bijection between the set of irreducible characters of a group with p’-degree and the corresponding set in the normalizer of a Sylow p-subgroup. In this talk, I’ll give an overview of a refinement of the McKay conjecture due to Gabriel Navarro, which brings the action of Galois automorphisms into the picture. A lot of recent work has been done on this conjecture, but possibly even more interesting is the amount of information it yields about the character table of a finite group. I’ll discuss some recent results on the McKay—Navarro conjecture, as well as some of the implications the conjecture has had for other interesting character-theoretic problems.

Friday, November 19, at 10 am PST

David Holmes, University of Leiden

How often is a divisor on a curve principal?

A divisor on an algebraic curve (or Riemann surface) is called principal if it is the divisor of a rational (or meromorphic) function. If one has a family C/S of curves, and a family of divisors D on C, it is natural to study the locus of points s in S such that D becomes principal when restricted to the fibre over S (the `double ramification' locus). We will talk about how to extend the definition of this locus to families of singular curves, and how to compute the class of the resulting locus in the (logarithmic) Chow ring of S. If time allows we will briefly discuss connections to cohomological field theories and to localisation computations in Gromov-Witten theory.

Friday, December 3, at 10 am PST

Baptiste Rognerud, University of Paris, Jussieu

Coxeter matrices of finite posets and fractionally Calabi-Yau categories

The Coxeter matrix of a finite poset is defined by an easy formula involving the incidence matrix of the poset and its inverse. If its definition is combinatorial, many of its properties are in fact shadows of representation theoretical properties of the poset. One interesting question is to understand which posets have a Coxeter matrix of finite order. We will see examples of such posets and how this phenomenon is related to the notion of fractionally Calabi-Yau categories.