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Scientific

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.

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?

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?

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?

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.

Deducing Rock Properties from Spectral Seismic Data - Final Report

Speaker: 
Maria-Veronica Ciocanel
Speaker: 
Heather Hardeman
Speaker: 
Dillon Nasserden
Speaker: 
Byungjae Son
Speaker: 
Shuai Ye
Date: 
Fri, Aug 14, 2015
Location: 
Institute for Mathematics and its Applications
Conference: 
PIMS-IMA Math Modeling in Industry XIX
Abstract: 

Seismic Data in Exploration Geoscience

The recovery and production of hydrocarbon resources begins with an exploration of the earth’s subsurface, often through the use of seismic data collection and analysis. In a typical seismic data survey, a series of seismic sources (e.g. dynamite explosions) are initiated on the surface of the earth. These create vibrational waves that travel into the earth, bounce off geological structures in the subsurface, and reflect back to the surface where the vibrations are recorded as data on geophones. Computer analysis of the recorded data can produce highly accurate images of these geological structures which can indicate the presence of reservoirs that could contain hydrocarbon fluids. High quality images with an accurate analysis by a team of geoscientists can lead to the successful discovery of valuable oil and gas resources. Spectral analysis of the seismic data may reveal additional information beyond the geological image. For instance, selective attenuation of various seismic frequencies is a result of varying rock properties, such as density, elasticity, porosity, pore size, or fluid content. In principle this information is present in the raw data, and the challenge is to find effective algorithms to reveal these rock properties.

Spectral Analysis

Through the Fourier transform, frequency content of a seismic signal can be observed. The Short Time Fourier transform is an example of a time-frequency method that decomposes a signal into individual frequency bands that evolve over time. Such time-frequency methods have been successfully used to analyze complex signals with rich frequency content, including recordings of music, animal sounds, radio-telescope data, amongst others. These time-frequency methods show promise in extracting detailed information about seismic events, as shown in Figure 1, for instance.

Figure 1: Sample time-frequency analysis of a large seismic event (earthquake). From Hotovec, Prejean, Vidale, Gomberg, in J. of Volcanology and Geothermal Research, V. 259, 2013.

Problem Description

Are existing time-frequency analytical techniques effective in providing robust estimation of physical rock parameters that are important to a successful, economically viable identification of oil and gas resources? Can they accurate measure frequency-dependent energy attenuation, amplitude-versus-offset effects, or other physical phenomena which are a result of rock and fluid properties?

Using both synthetic and real seismic data, the goal is to evaluate the effectiveness of existing time-frequency methods such as Gabor and Stockwell transforms, discrete and continuous wavelet transforms, basis and matching pursuit, autoregressive methods, empirical mode decomposition, and others. Specifically, we would like to determine whether these methods can be utilized to extract rock parameters, and whether there are modifications that can make them particularly effective for seismic data.

The source data will include both land-based seismic surveys as well as subsurface microseismic event recordings, as examples of the breadth of data that is available for realistic analysis.

Figure 2: (a). Seismic data set from a sedimentary basin in Canada. The erosional surface and channels are highlighted by arrows. The same frequency attribute are extract from short time Fourier transform (b), continuous wavelet transform (c) and empirical mode decomposition (d).

Deep Learning for Image Anomaly Detection - Final Report

Speaker: 
Ross Churchley
Speaker: 
Omeiza Olumoye
Speaker: 
Emily Gunawan
Speaker: 
Morgan Schreffler
Speaker: 
Deyeob Yeo
Date: 
Fri, Aug 14, 2015
Location: 
Institute for Mathematics and its Applications
Conference: 
PIMS-IMA Math Modeling in Industry XIX
Abstract: 

The machine learning community has witnessed significant advances recently in the realm of image recognition [1,2]. Advances in computing power – primarily through the use of GPUs – has enabled a resurgence of neural networks with far more layers than was previously possible. For instance, the winning team, GoogLeNet [1,3], at the ImageNet 2014 competition triumphed with a 43.9% mean average precision, while the previous year’s winner, University of Amsterdam, won with 22.6% mean average precision.

Neural networks mimic the neurons in the brain. As in the human brain, multiple layers of computational “neurons” are designed to react to a variety of stimuli. For instance, a typical scheme to construct a neural network could involve building a layer of neurons that detects edges in an image. An additional layer could then be added which would be trained (optimized) to detect larger regions or shapes. The combination of these two layers could then identify and separate different objects present in a photograph. Adding further layers would allow the network to use the shapes to decipher the types of objects recorded in the image.

Goal of this project

An issue facing industries that deal with large numbers of digital photographs, such as magazines and retailers, is photo accuracy. Nearly all photos used in such contexts undergo some amount of editing (“Photoshopping”). Given the volume of photographs, mistakes occur [4]. Many of these images fall within a very narrow scope. An example would be the images used within a specific category of apparel on a retailer’s website. Detecting anomalies automatically in such cases would enable retailers such as Target to filter out mistakes before they enter production. By training a modern deep convolution neural network [1,5] on a collection of correct images within a narrow category, we would like to construct a network which will learn to recognize well-edited images. This amounts to learning a distribution of correct images so that poorly-edited images may be flagged as anomalies or outliers.

Keywords: neural networks, deep learning, image processing, machine learning

Prerequisites: Programming experience in Python. Experience in a Linux environment is a plus.

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