BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:David Solomon (IMJ) DTSTART:20260922T120000Z DTEND:20260922T130000Z DTSTAMP:20260604T223547Z UID:SheffieldNumberTheory/60 DESCRIPTION:Title: Heisenberg SICs\, Stark Units and Weil Representations\ nby David Solomon (IMJ) as part of Sheffield Number Theory Seminar\n\nLect ure held in J-11 Hicks Building.\n\nAbstract\nSICs\, also known as equiang ular tight frames\, are configurations of $d^2$ equiangular lines in $\\ma thbb{C}^d$ whose applications in signal processing and quantum physics hav e been known and studied for more than 30 years. More recently\, numerical investigations of so-called Heisenberg SICs (which have an action of $\\m athcal{H}(\\mathbb{Z}/d\\mathbb{Z})$ via its Schr\\¨odinger representatio n) have revealed surprising\, heuristic connections with conjectural ``Sta rk units'' and hence with Hilbert’s 12th Problem over real-quadratic fie lds. Just as intriguingly\, the action of Galois on these units seems to b e connected to the action of $\\mathrm{SL}_2(\\mathbb{Z}/d\\mathbb{Z})$ on the set of Heisenberg SICs via its $d$-dimensional Weil representation as a subgroup of the automorphism group of $\\mathcal{H}(\\mathbb{Z}/d\\math bb{Z})$. In my talk\, I will first give an overview of recent SIC-related research\, as well as the Stark Conjectures (which date from the 1970s but are still largely unproven). I will explain the experimental evidence con necting SICs with Stark-Units over the field $\\mathbb{Q}(\\sqrt{(d−1)(d +3)})$. On a more specialised note and as time permits\, I will outline my recent work on the lifted Weil representation in the case $d = p^n$ and p ossible connections to a $p$-adic theory of SICs.\n LOCATION:https://researchseminars.org/talk/SheffieldNumberTheory/60/ END:VEVENT END:VCALENDAR