Main Organisers

Gene Kopp (kopp@math.lsu.edu) 

David Solomon (david.solomon314@gmail.com)

Both organisers welcome email queries, comments and suggestions.

Dates: Biweekly on Mondays starting 8th September, 2025

Times: 9:30 CDT/10:30 EDT/15:30 BST/16:30 CEST unless otherwise indicated. Suggested duration: approx 50 minutes plus discussion.

New participants should email David to receive regular emails with updates, programme, links etc. 

Next talk

22nd Sept. 2025 

title: `Everything you always wanted to know about SICs but were afraid to ask'

speaker: Marcus Appleby

Teams link: here. Meeting ID: 350 732 000 532 7, Passcode: nw975xt2 

Abstract: The talk will begin by explaining the physical significance of SICs.  It will then go on to explain all the things that physicists and frame theorists had discovered during the two decades before the discovery of the connection with number theory: Specifically the role of the Weyl-Heisenberg and Clifford groups, the methods used for calculating them, and what is sometimes called "SIC phenomenology".  The last is sometimes compared with spectroscopy as it existed in the period before 1926:  a huge mass of empirical observations, with numerous observed regularities, but hardly any explanation.  Manin (Notices of the AMS, 56, 1268 (2009)), in response to the question "what has changed in mathematics since the appearance of computers", replied:  

"The unique possibility of doing large-scale physical experiments in mental reality arose. We can try the most improbable things. More exactly, not the most improbable things, but things that Euler could do even without a computer. Gauss could also do them. But now, what Euler and Gauss could do, any mathematician can do, sitting at his desk. So if he doesn’t have the imagination to distinguish some features of this Platonic reality, he can experiment."  

SIC phenomenology is a good illustration of this point.  The talk will conclude with a description of how number theory has played a role similar to the role quantum mechanics played with respect to spectroscopy:  it has reduced the plethora of empirical observations to some kind of order.  Nevertheless the key questions remain unanswered.  In particular we still can't prove SIC existence.

Future Talks

6th Oct. 2025

title: `Heisenberg groups over number rings: Weil representations and p-adic limits'

speaker: David Solomon

20th Oct. 2025 (TBC)

title: TBC

speaker: Steve Flammia

Subject Matter of the Seminar

The first part of the seminar title refers to the S-units of global fields predicted by the well-known rank-1, complex Stark conjecture for abelian L-functions (cf Tate's  book `Les Conjectures de Stark sur les Fonctions L d'Artin en s=0', Birkhauser, 1984). They have strong connections to Hilbert's 12th problem/explicit abelian class field theory.

The second part of the title refers to SIC-POVMs (in brief, maximal equiangular sets of lines in C^d). Research into SICs originated in Quantum Information and Design Theories and is probably less well-known to number theorists. It has, however, burgeoned over the last 25 years, driven in part by Zauner's 1999 conjecture and extensive computational work by Grassl, Scott et al. which revealed apparent links to Stark units on the one hand and representations of finite Heisenberg groups on the other.

The third part of the title refers to the fact that the seminar is also open to talks on a variety of topics related to the above two main themes. A non-exhaustive list might include: other (eg p-adic) aspects of Stark's conjectures and/or Hilberts 12th Problem, Heisenberg groups and Weil representations, Mutually Unbiased Bases in Quantum Theory etc.

This is a research seminar. Talks containing both published work and work in progress are welcome, as is participative discussion (within reason).

Past Talks

Date: 8th Sept 2025

Title: `The Shintani–Faddeev modular cocycle: Stark units from q-Pochhammer ratios'

Speaker: Gene Kopp, LSU

Slides below (now without pauses, 15/9/25):