# Seminars & Colloquia Calendar

## From Hilbert’s 12th problem to complex equiangular lines

#### Gene Kopp (Bristol)

Location: ** Hill 525**

Date & time: Tuesday, 17 September 2019 at 2:00PM - 3:00PM

Abstract: We describe a connection between two superficially disparate open problems: Hilbert’s 12th problem in number theory and Zauner’s conjecture in quantum mechanics and frame theory. Hilbert asked for a theory giving explicit generators of the abelian Galois extensions of a number field; such an "explicit class field theory" is known only for the rational numbers and imaginary quadratic fields. Zauner conjectured that a configuration of d 2 pairwise equiangular complex lines exists in d-dimensional Hilbert space (and additionally that it may be chosen to satisfy certain symmetry properties); such configurations are known only in a finite set of dimensions d. We prove a conditional result toward Zauner’s conjecture, refining insights of Appleby, Flammia, McConnell, and Yard gleaned from the numerical data on complex equiangular lines. We prove that, if there exists a set of real units in a certain abelian extension of a real quadratic field (depending on d) satisfying certain congruence conditions and algebraic properties, a set of d 2 equiangular lines may be constructed, when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s = 0 and is closely connected to the Stark conjectures over real quadratic fields. We will work through the example d = 5 in detail to help illustrate our results and conjectures.

R. Shapiro Organizer's Page

Chiara Damiolini, Ian Coley and Franco Rota -Charles Weibel Organizer's Page

Brooke Logan

Wujun Zhang Organizer's webpage

P. Gupta, X.Huang and J. Song Organizer's webpage

Swastik Kopparty, Sepehr Assadi Seminar webpage

Jeffry Kahn, Bhargav Narayanan, Jinyoung Park Organizer's webpage

Brooke Ogrodnik, Website

Robert Dougherty-Bliss and Doron Zeilberger --> homepage

Paul Feehan, Daniel Ketover, Natasa Sesum Organizer's webpage

Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward Organizer's webpage

Jason Saied Seminar webpage

Brian Pinsky, Rashmika Goswami website

Quentin Dubroff Organizer's webpage

James Holland; Organizer website

Edna Jones Organizer's webpage

Brooke Ogrodnik website

Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum Organizer's Webpage

Organizer: Luochen Zhao

Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song Organizer's Page

Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi Organizer's webpage

Simon Thomas website

Kasper Larsen, Daniel Ocone and Kim Weston Organizer's page

Joel Lebowitz, Michael Kiessling

Yanyan Li, Haim Brezis Organizer's Webpage

Stephen D. Miller, John C. Miller, Alex V. Kontorovich, Alex Walker seminar website

Stephen D. Miller

Brooke Ogrodnik, Website

Organizers: Yanyan Li, Z.C. Han, Jian Song, Natasa Sesum

Yael Davidov Seminar webpage

Kristen Hendricks, Xiaochun Rong, Hongbin Sun, Chenxi Wu Organizer's page

Fioralba Cakoni Seminar webpage

Ebru Toprak, Organizer

Organizer's webpage: Organizer's webpage

- Show events from all categories

## Special Note to All Travelers

Directions: map and driving directions. If you need information on public transportation, you may want to check the New Jersey Transit page.

*Unfortunately, cancellations do occur from time to time. Feel free to call our department: 848-445-6969 before embarking on your journey. Thank you.*