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# Mathematics

## OM representation of prime ideals and applications in function fields

Speaker:
Jens Bauch
Date:
Thu, Dec 10, 2015
Location:
PIMS, Simon Fraser University
Conference:
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
Let $A$ be a Dedekind domain, $K$ the fraction field of $A$, and $f\in A[x]$ a monic irreducible separable polynomial. Denote by $\theta\in K^{\mathrm{sep}}$ a root of $f$ and let $F=K(\theta)$ be the finite separable extension of $K$ generated by $\theta$. We consider $\mathcal{O}$ the integral closure of $A$ in $L$. For a given non-zero prime ideal $\mathfrak{p}$ of $A$ the Montes algorithm determines a parametrization (OM representation) for every prime ideal $\mathfrak{P}$ of $\mathcal{O}$ lying over $\mathfrak{p}$. For a field $k$ and $f\in k[t,x]$ this yields a new representation of places of the function field $F/k$ determined by $f$. In this talk we summarize some applications which improve the arithmetic in the divisor class group of $F$ using this new representation.

## An arithmetic intersection formula for denominators of Igusa class polynomials

Speaker:
Bianca Viray
Date:
Thu, Nov 12, 2015
Location:
PIMS, University of Washington
Conference:
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
Igusa class polynomials are the genus 2 analogue of Hilbert class polynomials; their roots are invariants of genus 2 curves that have complex multiplication by a fixed order. The coefficients of Igusa class polynomials are rational, but, unlike in genus 1, are not integral. An exact formula, or tight upper bound, for these denominators is needed to compute Igusa class polynomials and has applications to cryptography. In this talk, we explain how to obtain a formula for the arithmetic intersection number G1.CM(K) and how this results in a bound for denominators of Igusa class polynomials. We also explain how the formula for G1.CM(K) leads us to a generalization of Gross and Zagier's formula for differences of CM j-invariants. This is joint work with Kristin Lauter.

## Local-global principles for quadratic forms

Author:
Raman Parimala
Date:
Fri, Oct 30, 2015
Location:
PIMS, University of British Columbia
Conference:
PIMS/UBC Distinguished Colloquium
Abstract:
The classical theorem of Hasse-Minkowski asserts that a quadratic form over a number field represents zero nontrivially provided it represents zero nontrivially over its completions at all its places. We discuss analogous local global principles over function fields of p-adic curves. Such local-global principles in the general setting for homogeneous spaces have implications to the understanding of the arithmetic of these fields.
Notes:

## Landing a Faculty Position

Author:
Phillip Loewen
Date:
Fri, Oct 23, 2015
Location:
PIMS, University of British Columbia
Conference:
PIMS-Math Job Forum
Abstract:
The PIMS-Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in the Mathematics Department with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion. Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?
Notes:

## Finding Your Place at a liberal arts college

Author:
Kathryn Nyman
Date:
Fri, Oct 23, 2015
Location:
PIMS, University of British Columbia
Conference:
PIMS-Math Job Forum
Abstract:
The PIMS-Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in the Mathematics Department with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion. Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?
Notes:

## My Life in "industry"

Author:
Richard Liang
Date:
Fri, Oct 23, 2015
Location:
PIMS, University of British Columbia
Conference:
PIMS-Math Job Forum
Abstract:
The PIMS-Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in the Mathematics Department with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion. Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?
Notes:

## Signs of abelian varieties and representations

Speaker:
Matthew Greenberg
Date:
Thu, Oct 15, 2015
Location:
PIMS, University of Calgary
Conference:
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
The sign is a fundamental invariant of an abelian variety defined over a local (archimedian or p-adic) or global (number or function) field. The sign of an abelian varieties over a global field has arithmetic significance: it is the parity of Mordell-Weil group of the abelian variety. The sign also appears in the functional equation of the L-function of abelian variety, determining the parity of its order of vanishing at s=1. The modularity conjecture says that this L-function coincides with the L-function of an automorphic representation, and the sign can be expressed in terms of this representation. Although we know how to compute this sign using representation theory, this computation does not really shed any light on the representation theoretic significance of the sign. This representation theoretic significance was articulated first by Dipendra Prasad (in his thesis), where he relates the sign of a representation to branching laws — laws that govern how an irreducible group representation decomposes when restricted to a subgroup. The globalization of Prasad’s theory culminates in the conjectures of Gan, Gross and Prasad. These conjectures suggest non-torsion elements in Mordell-Weil groups of abelian varieties can be obstructions to the existence of branching laws. By exploiting p-adic variation, though, one can hope to actually produce the Mordell-Weil elements giving rise to these obstructions. Aspects of this last point are joint work with Marco Seveso.

## Juggling Mathematics & Magic

Speaker:
Ronald Graham
Date:
Thu, Sep 17, 2015
Location:
PIMS, University of Calgary
Conference:
Louise and Richard K. Guy Lecture Series
Abstract:
The popular Richard & Louise Guy lecture series celebrates the joy of discovery and wonder in mathematics for everyone. Indeed, the lecture series was a 90th birthday present from Louise Guy to Richard in recognition of his love of mathematics and his desire to share his passion with the world. Richard Guy is the author of over 100 publications including works in combinatorial game theory, number theory and graph theory. He strives to make mathematics accessible to all. Dr. Ronald Graham, Chief Scientist at the California Institute for Telecommunications and Information Technology and the Irwin and Joan Jacobs Professor in Computer Science at UC San Diego. Dr. Ronald Graham, Chief Scientist at the California Institute for Telecommunications and Information Technology and the Irwin and Joan Jacobs Professor in Computer Science at UC San Diego, will the present the lecture, Juggling Mathematics & Magic. Dr. Graham’s talk will demonstrate some of the surprising connections between the mystery of magic, the art of juggling, and the some interesting ideas from mathematics. Ronald Graham, the Irwin and Joan Jacobs Professor in Computer Science and Engineering at UC San Diego (and an accomplished trampolinist and juggler), demonstrates some of the surprising connections between the mystery of magic, the art of juggling, and some interesting ideas from mathematics. The lecture is intended for a general audience.

## Sparse Recovery Using Quantum Annealing - Final Report

Speaker:
Aritra Dutta
Speaker:
Geonwoo Kim
Speaker:
Meiqin Li
Speaker:
Carlos Ortiz Marrero
Speaker:
Mohit Sinha
Speaker:
Cole Stiegler
Date:
Fri, Aug 14, 2015
Location:
Institute for Mathematics and its Applications
Conference:
PIMS-IMA Math Modeling in Industry XIX
Abstract:
The main optimization problem in many applications in signal processing (e.g. in image reconstruction, MRI, seismic images, etc.) and statistics (e.g. model selection in regression methods), is the following sparse optimization problem. The goal is finding a sparse solution to the underdetermined linear system Ax = b, where A is an m x n matrix and b is an m-vector and m ≤ n [2]. The problem can be written as min (over x) ||x||₁ subject to Ax = b. There are several approaches to this problem that generally aim at approximate solutions, and often solve a simplified version of the original problem. For example passing from ℓ-norm to ℓ₁-norm yields an interesting convexification of the problem [1]. Moreover the equality Ax = b does not cover noisy cases in which Ax + r = b for some noise vector r min (over x) ||x||₁ subject to ||Ax - b||₂ ≤ σ. Extensive theoretical [6, 7] and practical studies [5, 8] have been carried on solving this problem and various succesfull methods adopting interior-point algorithms, gradient projections, etc. have been tested. The discrete nature of the original problem also suggests possibility of viewing the problem as a mixed-integer optimization problem [3]. However common methods for solving such mixed-integer optimization problems (e.g. Benders’ decomposition) iteratively generate hard binary optimization subproblems [4]. The exciting possibility that quantum computers may be able to perform certain computations faster than digital computers has recently spiked with the quantum hardware of D-Wave systems. The current implementations of quantum systems based on the principles of quantum adiabatic evolution, provide experimental resources for studying algorithms that reduce computationally hard problems to those that are native to the specific evolution carried by the system. In this project we will explore possibilities of designing optimization algorithms that use the power of quantum annealing in solving sparse recovery problems. References [1] Emmanuel J. Cand´es, Justin K. Romberg, and Terence Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics, 59(8):1207–1223, 2006. [2] Simon Foucart and Holger Rauhut. A Mathematical Introduction to Compressive Sensing. Birkh¨user Basel, 2013. [3] N.B. Karahanoglu, H. Erdogan, and S.I. Birbil. A mixed integer linear programming formulation for the sparse recovery problem in compressed sensing. In Acoustics, Speech and Signal Processing (ICASSP), 2013 IEEE International Conference on, pages 5870–5874, May 2013. [4] Duan Li and Xiaoling Sun. Nonlinear Integer Programming. International Series in Operations Research & Management Science. Springer, 2006. [5] Ewout van den Berg and Michael P. Friedlander. SPGL1: A solver for large-scale sparse reconstruction, June 2007. http://www.cs.ubc.ca/labs/scl/spgl1. [6] Ewout van den Berg and Michael P. Friedlander. Probing the pareto frontier for basis pursuit solutions. SIAM Journal on Scientific Computing, 31(2):890–912, 2008. [7] Ewout van den Berg and Michael P. Friedlander. Sparse optimization with least-squares constraints. SIAM J. Optimization, 21(4):1201–1229, 2011. [8] Ewout van den Berg, Michael P. Friedlander, Gilles Hennenfent, Felix J. Herrmann, Rayan Saab, and ¨Ozg¨ur Yilmaz. Algorithm 890: Sparco: A testing framework for sparse reconstruction. ACM Trans. Math. Softw., 35(4):29:1–29:16, February 2009.