Bay Area Algebraic Number Theory and Arithmetic Geometry Day 14

Saturday, April 29, 2017
University of California, Santa Cruz
Room: McHenry Library, room 1240


Speakers:

Kristin Lauter, Microsoft Research
Aaron Pollack, Stanford University
Karl Rubin, University of California Irvine
Eugenia Rosu, University of California Berkeley
Katherine Stange, University of Colorado Boulder


Schedule:

9:30-10:00 Coffee/Bagels
10:00-11:00 Kristin Lauter
11:00-11:15 Coffee Break
11:15-12:15
Aaron Pollack
12:15-1:45
Lunch
1:45-2:45 Karl Rubin
2:45-3:10
Coffee Break
3:10-4:10
Eugenia Rosu
4:10-4:30
Break
4:30-5:30
Katherine Stange
6:00
Dinner, Avanti
Please RSVP to sdasgup2 (at) ucsc (dot) edu





Titles and Abstracts:

Kristin Lauter, "Supersingular Isogeny Graphs in Cryptography"
Supersingular Isogeny Graphs were proposed for use in Cryptography in 2006 by Charles-Goren-Lauter, and are currently being considered for submission to several tracks of the 2017 NIST Post-Quantum Cryptography International Competition. These are Ramanujan graphs whose nodes are supersingular elliptic curves and edges are isogenies between them. This talk will introduce the hard problems and cryptographic applications in this space, and discuss a surprising connection to quantum arithmetic.

Aaron Pollack, "Twisted orbit parametrizations and lifting laws"
I will explain how one can obtain twisted versions of some of Bhargava's higher composition laws. The key technical idea is a way of relating elements in the open orbit of one prehomogeneous vector space to elements in the minimal nonzero orbit of another.

Karl Rubin, "Heuristics for the growth of Mordell-Weil ranks in big abelian extensions of number fields"
I will discuss some questions about the distribution of modular symbols attached to elliptic curves. By expressing L-values in terms of modular symbols, we obtain heuristics for Mordell-Weil ranks. For example, these heuristics predict that every elliptic curve over Q has finite Mordell-Weil rank over the compositum of all Z_p-extensions of Q. This is joint work with Barry Mazur.

Eugenia Rosu, "Integers that can be written as the sum of two rational cubes"
The Birch and Swinnerton-Dyer conjecture predicts that we have non-torsion rational points on an elliptic curve iff the L-function corresponding to the elliptic curve vanishes at 1. Thus BSD predicts that a positive integer N is the sum of two cubes if L(E_N, 1)=0, where L(E_N, s) is the L-function corresponding to the elliptic curve E_N: x^3+y^3=N. We have computed several formulas that relate L(E_N, 1) to the trace of a modular function at a CM point. This offers a criterion for when the integer N is the sum of two cubes. Furthermore, when L(E_N, 1) is nonzero we get a formula for the number of elements in the Tate-Shafarevich group.

Katherine Stange, "Circle packings, thin orbits and the arithmetic of imaginary quadratic fields"
Integral Apollonian circle packings are certain fractal packings of the plane with circles of disjoint interior, and integer curvatures. The set of curvatures which appears has been of recent interest as a challenging problem in the study of orbits of thin groups. Work of Bourgain, Fuchs and Kontorovich culminated in the demonstration that density one of the integers appear as curvatures, up to a congruence restriction. In this talk, we'll rediscover Apollonian circle packings as part of the essential nature of the Gaussian integers and their Diophantine approximation, generalize to other quadratic fields to discover new circle packings, and discuss the extension of results on curvatures to these and other Kleinian packings.


Parking (Updated 4/27; I will be updating this on the morning of the event to guide people that have not RSVPd where to park if our reserved spaces get full):

This is Alumni Weekend, so parking will be at a premium on campus. Therefore we have reserved 13 specific parking spots in the Hahn parking lot, which is the closest large lot next to the math building. Click here to see the location of the reserved spots within Hahn Parking Lot, and look at paragraph (1) below for an explanation about how to find Hahn Parking Lot. About 10 of the reserved spots have been claimed by people that have emailed me. If you have not reserved a spot by emailing me, but would like to park, then please send me an email now at sdasgup2 (at) ucsc (dot) edu. (I will be checking my email, even on the morning of the conference.) We will have graduate students in the parking lot directing you to the spots to park and giving you the permits that must be placed on the dashboard in order to park there.

Here is the general parking information for campus and some recommendations on where to park if the reserved spots get full. Throughout campus, parking is free and no permit is required on Alumni Weekend, as long as you park in a legal parking space and do not block emergency access, or park in spaces that are posted and reserved for special use such as university vehicles. You can park all day at a metered spot with no fee.

(1) Hahn Parking lot (there are spots other than the ones we will reserve): There is a foot bridge that connects the Hahn parking lot to McHenry Library. If you click on this map and zoom in two levels, the bridge is indicated by a gray line above the blue pointer and below the "Hahn Student Services" building. Halfway through the bridge, there is a path that leads to the left, but you want to ignore this path and continue straight until the end of the bridge. After crossing the bridge, make a sharp left slightly downhill to enter the building at the first level. (The main entrance is actually on the second level, so if you enter at the main entrance you will have to take the elevator down one level.)

Here are printable maps: McHenry Library, and more specifically how to get to Hahn parking lot and walk to McHenry.

(2) If Hahn parking lot is full, there are some "secret spots" in the back of McHenry Library. This is called
Lot 120. There are 11 metered parking spots where you can park for free on Saturday (for as long as you want) without a permit. Here is a printable map.

(3) If those spots are full, you can try Lot 126, the Performing Arts lot. Here is a printable map.

(4) Core West. If all else fails, the largest parking garage on campus is
Core West. Here are some directions.

Long story, short: I think you really want to avoid searching for parking on this hectic weekend on campus, so please email me to get a reserved spot!


Registration:

There is no formal registration, but if you plan to attend, we would appreciate an email to sdasgup2 at ucsc dot edu to help plan the event, especially if you plan to attend the dinner afterwards.


Dinner:

There will be a dinner following the conference at 6:00pm, at Ristorante Avanti in Santa Cruz. Please send an email to sdasgup2 at ucsc dot edu if you plan to attend. Graduate students will be partially subsidized at the dinner. We thank the UC Santa Cruz Mathematics Department, the Math Research Center at Stanford University, and the UC Berkeley Mathematics Department for partial financial support.