__Summer 2018 Project Descriptions __

__Summer 2018 Project Descriptions__

Below are brief descriptions of research projects in mathematics from faculty members who wish to supervise qualified undergraduate students.

Unless otherwise specified, each project is available to one student only.

#### Please be sure to refer to the appropriate page for full details on how to apply - NSERC USRA Instructions or VPR USRA Instructions

**Supervisor: Drs Nathan Ilten & Mattia Talpo
**

__PROJECT: Is the toric boundary Cohen-Macaulay?__

A toric variety is an algebraic variety XΣ obtained from a combinatorial object Σ, a collection of compatible polyhedral cones in a vector space R n . These objects are widely studied in algebraic geometry, especially because several geometric aspects about these spaces can be “explicitly computed” via the combinatorial properties of Σ. Every such variety contains (C ∗ ) n as an open subset, and the complement ∆Σ = XΣ \ (C ∗ ) n is called the toric boundary. Cohen-Macaulayness is a property of algebraic varieties that relaxes smoothness. The intuitive idea is that every point of the variety can be described by using (dim X)-many “nice” equations. The Cohen-Macaulay property is important for a number of reasons: for example, it has nice implications for computations in local cohomology and duality theory, and several classes of useful singular varieties are Cohen-Macaulay. Even though very few of them are smooth, every toric variety is Cohen-Macaulay. The goal of this project is to understand whether the toric boundary is also always CohenMacaulay. This is important for the study of ideally smooth logarithmic schemes, that are some kind of spaces locally modelled on the boundary of a toric variety. The student working on this project will become familiar with several ideas from commutative algebra and algebraic geometry. The computer program Macaulay2 (or a similar computer algebra program) will be useful to experiment with the problem, before trying to approach it from a theoretical point of view.

**Requirements**: Applicants should have background knowledge in abstract algebra (e.g. at least MATH 340, MATH 440/441 preferable). If you are interested in applying, please contact Dr. Ilten at nilten@sfu.ca.

**Supervisor: Dr Nathan Ilten**

**Email: ****nilten@sfu.ca**

__PROJECT: Tangent cones in truncated power series rings__

Given an ideal I ⊂ C[x1, . . . , xn], its tangent cone is the ideal generated by inf for all f ∈ I, where inf is the lowest degree part of f. Seen geometrically, the tangent cone cuts out limits of secant lines between 0 and p, as p approaches 0 in the variety V (I) defined by the ideal I — hence the name. There are essentially two algorithms (due to Mora and Lazard) which can be used to calculated the tangent cone of an ideal. They can be seen as relatives of Buchberger’s algorithm, which calculates a Gr¨obner basis for an ideal. The key difference is that the “term order” underlying the tangent cone is not well-ordered. The goal of this project is to adapt the known algorithms to quickly calculate the tangent cone up to a fixed degree d for ideals of the form I + hx1, . . . , xni d . This has important applications in deformation theory and the lifting of syzygies. In addition to theoretical work, the student will be expected to implement their algorithm in the computer algebra system Macaulay2. In carrying out this project, the student will be exposed to a variety of ideas from algebraic geometry and commutative algebra..

**Supervisor: Dr Nathan Ilten**

**Email: ****nilten@sfu.ca**

__PROJECT: Algorithms for Cayley structures__

A Cayley structure on a finite set A ⊂ Z n is a linear map π : Z n → Z k such that π(A) is a linearly independent set. Caley structures on A are highly relevant for properties of a certain geometric object XA called the toric variety associated to A. In particular, linear subspaces of XA are determined by Cayley structures of A, as is the so-called dual defect. The goal of this project is to develop algorithms to study Cayley structures for a given subset A ⊂ Z n , in particular, to develop an algorithm classifying Cayley structures of A up to equivalence. This will have important applications for explicit computations involving dual defects and linear subspaces of toric varieties. In addition to theoretical work, the student will be expected to implement their algorithm in the computer algebra system Macaulay2. In carrying out this project, the student will be exposed to a variety of ideas from algebraic and discrete geometry

**Supervisor: Dr Jonathan Jedwab**

**Email: ****jed@sfu.ca**

**PROJECT:**__ Combinatorial Problems Arising in Digital Communications__

The project will involve the study of combinatorial problems arising in digital communications. See http://people.math.sfu.ca/~jed/research.html for background on the general area of research, and http://people.math.sfu.ca/~jed/students.html for examples of previous USRA projects.

**Requirements:** Students should have completed MACM 201 (Discrete Mathematics II), and preferably have some programming experience. However, the most important attributes are enthusiasm, persistence, and a willingness to learn new skills.

**Supervisor: Dr Cedric Chauve**

**Email: ****cedric. chauve@sfu.ca**

**PROJECT: Improving the Efficiency of Third Generation Sequencing Algorithm Through Pseudoalignment**

Third Generation Sequencing (TGS) technologies (Oxford Nanopore, Pacific Biosciences) generate long reads (which is great for downstream applications that are quite noisy (>10% error rate, which is a problem for downstream applications). As a consequence, the problem of correcting TGS data using short but accurate Illumina reads (error rate < 1%) is an important bioinformatics problem. This approach is called hybrid correction, and is the only accurate approach for data sets characterized by a low to moderate depth of coverage of long reads.

In a recent paper (https://doi.org/10.1093/bioinformatics/btw463) we proposed the combinatorial algorithm CoLoRMap for hybrid correction based on a combination of graph-theoretical methods (shortest paths in a graph) and local-assembly of short reads. This method is very accurate but the initial phase of mapping the short reads onto the long reads is time consuming and impacts the applicability of the method on desktop computers.

Recently, the approach of pseudoalignment has been shown to speed-up enormously applications requiring the mapping of short-reads (see the blog post http://nextgenseek.com/2015/05/kallisto-paper-summary-near-optimal-rna-seq-quantification/). But so far, pseudoalignment has been used only with accurate short reads, and not with noisy long reads.

The first goal of this project is to modify CoLoRMap to replace the time-consuming short-reads mapping phase by pseudoalignment. In parallel, the intern will be integrated to a group of researchers (from Computer Science at SFU and the Vancouver Prostate Centre) working on the use of Oxford Nanopore TGS sequencing data in cancer genomics.

**Requirements:** good background in discrete mathematics, excellent coding skills in C++, a decent background in probabilities is a plus.

**Supervisors: Drs Ben Adcock & Simone Brugiapaglia**

**PROJECT: Deep Learning Techniques for Inverse Problems in Imaging**

In the last few years, deep neural networks have outperformed state-of-the-art methods in challenging computer-vision tasks such as classification and segmentation. More recently, they have also shown promising results when applied to inverse problems in computer imaging such as deblurring, denoising, super-resolution, and the efficient reconstruction of medical images from linear measurements (e.g., Magnetic Resonance Imaging and Computer Tomography).

Instead of fixing a statistical prior (such as the sparsity of the image) to regularize the inverse problem, these new approaches seek to learn the reconstruction algorithm itself by taking advantage of a preliminary learning phase where the neural network is optimized based on a large dataset of training examples.

In this project, we will explore these new techniques and compare them with classical tools for inverse regularization, with the aim of understanding how to fully exploit their potential for inverse problems in computer imaging.

**Requirements:**Candidates should have a strong mathematical background. Analysis, linear algebra, numerical analysis and Matlab experience are essential. Previous knowledge of optimization and image processing are beneficial, but not strictly necessary.

**Supervisor: Drs Ben Adcock & Simone Brugiapaglia**

**PROJECT: **** Stochastic collocation methods for uncertainty quantification of physical models**

Uncertainty is ubiquitous in science and engineering. Uncertainty quantification is the process of deriving quantitative characterizations of uncertainties in both computational models and real-world applications. One of the most common problems in UQ focuses on determining the parameter dependence of solutions of parametrized systems of PDEs. Although this can be recast as the problem of approximating a high-dimensional function from sampled data, a major challenge lies with the fact that the amount of data is often small, since gathering data is time-consuming. Efficient uncertainty quantification requires techniques from not only approximation theory, but also numerical analysis, statistics, and sparse recovery.

This project will investigate a recent class of techniques for high-dimensional approximation based on sparse approximation. Its focus will firstly be on the improvement of these techniques, through, for instance, the design of novel adaptive sampling procedures, and the development of learning procedures to reduce the dimensionality. It will also consider their application to parametric PDEs.

**Requirements**: Analysis, numerical analysis, linear algebra and Matlab experience are essential. Optimization is beneficial but not mandatory.

**SUPERVISORS: Drs Sandy Rutherford and JF Williams**

**EMAIL**: arruther@sfu.ca and jfwillia@sfu.ca

**PROJECT: Mathematical Modelling of HIV Testing, Treatment, and Care**

This project will combine mathematical modelling of the HIV epidemic in Vancouver with operations research analysis of HIV testing, engagement in care, and retention in care. The models are defined as systems of ordinary differential equations. Data for the project has been provided by Vancouver Coastal Health and the BC Centre for Excellence in HIV/AIDS. Key questions to be addressed are:

1. What are the optimal allocations of resources across treatment programs to minimize HIV incidence, morbidity, and mortality?

2. What are the optimal allocations of public health resources between treatment and testing programs to minimize HIV incidence, morbidity, and mortality?

These questions can be framed as “black box” optimization problems, in which the objective function is obtained by numerically solving the ordinary equation models. However, these optimization problems are computationally challenging, because the objective function is nonconvex and high-dimensional. The focus of this research will be on developing parallelized code for solving these problems using metaheuristic optimization methods. The computations will be done on Compute Canada’s high-performance computing cluster at SFU.

**Supervisor: Dr Paul Tupper**

**Email: ****pft3@sfu.ca**

**PROJECT: A diversity view on clustering algorithms**

Clustering is a common used procedure in which a collection of objects is broken up into "natural" groupings. For example, given a finite collection of points in a metric space, you assign points to one of two clusters such that points within clusters tend to be closer together than points in different clusters. Clustering in metric spaces is a well-established field. In this project the student will study clustering in the context of diversities, a generalization of metric spaces where a positive number is assigned to every subset of points, and not just pairs as in a metric space.

**Requirements:** Discrete math and/or optimization.

**Supervisor: Dr Paul Tupper**

**Email: ****pft3@sfu.ca**

**PROJECT: Tight Spans for Metric Structures**

Given a metric space, its tight span is the set of all minimal one-point extensions of the space. The tight span can be equipped with its own metric, and viewed as its own metric space, so that the original metric space is naturally embedded in its tight span. The theory of tight spans arose in phylogenetics but has become of interest in its own right. In this project the student will work to see how much of the theory of tight spans can be generalized to metric

structures, which are basically metric spaces with extra structure (one example being normed spaces). We see already how this generalization works in some cases, but we'd like to push it further.

**Requirements:** Analysis, model theory.

**Supervisor: Dr Imin Chen**

**Email: **ichen@sfu.ca

**PROJECT: The LWE (learning with errors) problem**

The LWE (learning with errors) problem involves finding a vector s, given that s lies on a list of hyperplanes known only up to some error. The objective of this project is to study and give an independent exposition of a result of Regev [1] which shows that if there exists an efficient algorithm that solves LWE, then there exist an efficient quantum algorithm that approximates the decision version of the shortest vector problem. This hardness result, and later variants such as RLWE, are often used to motivate approaches to post-quantum cryptography and homomorphic encryption.

[1] On Lattices, Learning with Errors, Random Linear Codes, and Cryptography, Oded Regev, Journal of the ACM 56(6), article 34, 2009.

**Supervisor: Dr Weiran Sun**

**Email: **weirans@sfu.ca

**PROJECT: To study the well-posedness of the famous Boltzmann equation with so-called non-cutoff kernels**

Due to the singular nature of the interaction operator, one expects there will be regularizing effect for any positive time. Existing results have shown such regularization starting for initial data with sufficient smoothness. Our goal is to lower the assumption on the regularity of the initial data and pursue regularization of weak solutions. Any student interested in this project is required to have taken measure theory and functional analysis courses.