Summer Research Projects Mathematics 2021

MATH01

Garth Tarr

Title: Modelling consumer data from the red meat industry

Summary: The beef industry in Australia is worth $13 billion annually and the sheep meat industry is worth another $4 billion. A key question concerning the red meat industry is the ability to predict the eating quality of cuts of meat. Doing this well has major financial implications for the industry. This project would focus on the statistical issues associated with analysing consumer sensory data to predicting meat eating quality. Examples of possible projects include: the analysis of consumer sensory data which often contains many outliers; determining the relative importance of eating quality factors such as flavour, tenderness and juiciness; looking at the importance of "link product" as a common starting benchmark across consumers; and developing methods to evaluate new objective grading technologies.

Time: Jan-Feb

Max number of students: 2

Prerequisites: DATA2002 or DATA2902

MATH02

Ellis Patrick

Title: Data-intensive science to understand the molecular aetiology of disease.

Description: Biotechnological advances have made it possible to monitor the expression levels of thousands of genes and proteins simultaneously promising exciting, ground-breaking discoveries in complex diseases. This project will focus on the application and/or development of statistical and machine learning methodology to analyse a high-dimensional biomedical experiment. Our lab works on projects spanning multiple diseases including melanoma, ovarian cancer, acute myeloid leukemia, Alzheimer's disease, multiple sclerosis and HIV. We also work with various high-throughput technologies including single-cell RNA-Seq, SWATH-MS, flow cytometry, CyTOF, CODEX imaging and imaging mass cytometry.

Availability: I am flexible with dates.

Max number of students: 4

Prerequisites: DATA2X02

MATH03

Alexander Fish

Title: Polynomial method in additive combinatorics - Extension of CAP sets problem to more than three points

Description: The project with focus on the new polynomial method in (additive) combinatorics which allowed to resolve long standing problems such as Kakeya problem over finite fields, cap set problem and Erdos distance problem. We will study the proof by Croot-Pach-Lev-Ellenberg-Gijswijt-Tao of the sub exponential bound for cap-set problem concerning the maximal cardinality of a set A in F_q^n which does not contain distinct x,y,z satisfying x+y = 2z, or in other words sets which do not contain non-trivial three term arithmetic progressions in vector spaces over a finite field. We will also study the work of Lovett on lower bounds of a slice rank of a tensor. Achieving a good lower bound on the latter allowed to resolve the CAP-set problem. The ultimate goal will be to extend the polynomial method to resolve the following conjecture: Show that a set A in F_q^n which does not contain a non-trivial 4-term arithmetic progression is exponentially small in size with respect to q^n (the size of F_q^n).

Time: Flexible for me.

Max number of students: preferably 2 students, but can be also 1 student.

UoS prerequisites: Linear Algebra, Some sort of abstract algebra (a weak requirement), this project is suitable for students with a minimal background (after year 1)

MATH04

Alexander Fish

Title: Sum-Product Phenomenon for infinite sets

Description: Erdos and Szemeredi proved that there exists \eps > 0 such that for any finite set A in the integers we have |A+A| or |AA| is O(|A|^{1+\eps}) (for instance, the number of different results in the multiplication table is much bigger than the number of different results in the addition table). We will study the following analog of this result to infinite sets in the integers - namely, if A is a set of integers of positive density then there exists k >=1 such that (A-A)*(A-A) contains k\Z. The ultimate goal is to prove its two-dimensional analogue: Assume that a set E in the two dimensional integer lattice has positive density, then there exists k >= 1 such that the set A = \{ xy | (x,y) in E-E \} contains k\Z, where the set E-E = \{ u - v | u,v in E \}.

Time: Flexible for me.

Max number of students: preferably 2 students, but can be also 1 student.

UoS prerequisites: Measure theory, Metric spaces (preferable).

MATH05

Beniamin Goldys

Title: Set theory and uniqueness for orthogonal expansions of functions.

Abstract: It is well known that every function square-integrable on a bounded set has unique Fourier series expansion, but Fourier coefficients can be associated to any function integrable on a bounded set. A famous question going back to Riemann is: is such a function uniquely determined by its Fourier coefficients? This question leads to many beautiful and deep problems in the set theory and harmonic analysis. In this project we will learn about the methods used to study such problems, about some open questions and will explore the possibility of extending this theory to other orthogonal series.

Time: I am available after January 15.

UoS prerequisites: This project is for students after the 2nd year, who did the analysis course.

MATH06

Stephan Tillmann

Title: Convex projective surfaces

Abstract: The basic objects of geometric topology are curves and surfaces. This project studies them using techniques from geometry, algebra and combinatorics. The focus will be on projective structures on surfaces -- this includes spherical, euclidean and hyperbolic structures, but also many more! Some basic questions that may be addressed are: How do you put a projective structure on a surface? How do you tell two projective structures on a surface apart? Can you compute the lengths of the shortest curves on a projective surface? The most interesting projective structures are called "convex". The set of all convex projective structures on a surface will be parameterised using the concept of a moduli space. Questions about this moduli space include: What is a natural concept of distance between points in the moduli space? What characterises a shortest curve between two points in the moduli space?

Availability: Anytime.

Max number of students: 3

UoS prerequisites: Essential: MATH2922 (or MATH2022). Desirable: MATH2921 (or MATH2021).

MATH07

Nalini Joshi

Title: Elliptic curves

Description: Imagine curves given by $f(x, y)=0$, where $f(x,y)$ in a polynomial in 2 variables. Two families of curves are famous: (i) when $f$ is cubic in $x$ and quadratic in $y$; and (ii) when $f$ is quadratic in each of $x$ and $y$ - because in these cases, the general curves are parametrized by doubly periodic functions called elliptic functions. In this project, we focus on transformations between these two families, using computer algebra when needed.

Available times: Flexible

Max number of students: 2

UoS prerequisites: available for anyone who has done first-year mathematics courses in calculus and algebra.

MATH08

Simon Luo

Title: Developing Machine Learning Models for Anomaly Detection in a Sensor Network

Description: This project involves developing machine learning models to be used for anomaly detection. More specifically, this project aims to develop anomaly detection algorithms for time series data from a sensor network. The sensors could be the same type of sensors, or they could come from different types of sensors that are looking for the same type of events. The objective is to look into developing novel algorithms for this problem. Projects are available for all levels of students, from website development, coding to implement the model and developing the mathematical theory. For research projects, this will involve a literature review of the techniques currently used and implementing the state-of-the-art techniques in python. For more details, please contact s.luo@sydney.edu.au.

Time: Available all times

Max number of students: The project is intended to be a group project. But students that are more competent can choose to work individually. The maximum number of student is 5.

Prerequisites: Know how to code in python, have an understanding of machine learning and statistics.

MATH09

Wanchuang Zhu

Title: Contrastive learning for discrete Markov random field

Summary: The parameter inference of discrete Markov random field has the so-called normalizing constant problem. This project aims to investigate learning procedures using a contrastive learning framework. This will involve literature review and conducting experiments in R/ Python.

Time: Any.

Max number of students: 2.

Prerequisites: Experience with R or Python.

MATH10

Jonathan Spreer

Title: Graph encoded manifolds

Description: It is quite easy to visualise orientable surfaces such as the sphere or the torus (the surface of a donut) embedded in three-dimensional space. Surfaces are two-dimensional manifolds. It is a challenging task to visualise manifolds of higher dimensions. One general - and perhaps surprising - way of achieving this is by representing a manifold by a graph with coloured edges. Such graph encoded manifolds, or gems, can always be drawn on a sheet of paper while containing all the information about the surface or manifold. While some of this information is very hard (or impossible) to access, some information can be read off the graph quite easily and other bits and pieces can be recovered by simple combinatorial rules. This project is about using these simple combinatorial rules to deduce interesting facts about manifolds, to construct large families of such gems satisfying some given properties (which is interesting for all kinds of reasons), to design a method to randomly generate such gems in certain settings (which is important for even more kinds of reasons), or to do more theoretical work.

Max number of students: 5

MATH11

Milena Radnovic

Title: Unusual billiards

Description: Everybody knows about the game of billiards on the standard rectangular desk. How would it be to play that game on a desk of another shape: circle, oval, triangle? Can we imagine billiard on a three-dimensional desk or on some surface? This project will aim to answer to those questions.

Available times: Any, except in January.

The project can be run as individual or group project, for 1-3 students.

Prerequisites: first year linear algebra.

MATH12

Milena Radnovic

Title: Fractal geometry

Description: Fractals are intriguing geometrical objects that appear in mathematics, but also are found in nature. This project aims to explore those beautiful sets, from the points of geometry, dynamics, topology.

Available times: Any, except in January.

The project can be run as individual or group project, for 1-3 students.

Prerequisites: first year linear algebra and calculus

MATH13

Milena Radnovic

Title: The nature of Platonic and Archimedean solids

Description: The aim is to study regular and semi-regular polyhedra, and search for their applications in mathematics and/or occurence in nature.

Available times: Any, except in January.

This project is suitable for one student or a pair.

Prerequisite: First year linear algebra.

MATH14

Jean Yang

Title: Data science methods for cardiovascular precision medicine

Description: Over the past decade, new and more powerful -omic tools have been applied to the study of complex disease such as cancer and generated a myriad of complex data. However, our general ability to analyse this data lags far behind our ability to produce it. This project is to develop computational methods that helps to deliver better prediction of cardiovascular outcome. In collaboration with Prof Gemma Figtreeā€™s team who will provide access to the BioHEART dataset. We will use a unique large scale multi-omics data with over 1,000 samples in lipidomics, proteomics and metabolomics data to by integrating these multi-layered and multi-omics data. There are three different aspects to this project, (i) use machine learning to establish the patient or sample specific accuracy; (ii) use a sequential, machine learning approach to build a multi-level clinical diagnostic tree with multi-omics data and (iii) established a transferable biomarker model for multi-omics data.

Times: Nov-Dec, Late Jan-Feb, Feb-Mar

Max number of students: 4

UoS prerequisites: DATA2002 or equivalent

MATH15

Jean Yang

Title: Data analytics with multi-omics COVID19 data

Description: Over 30 public COVID19 multi-omics datasets are currently available, including ten single-cell RNA-seq datasets containing hundreds of individuals. The purpose of this project is to curate and process COVID-19 multi-omics datasets in order to construct a multi-modality risk prediction model for risk severity.

Times: Nov-Dec, Late Jan-Feb, Feb-Mar

Max number of students: 2

UoS prerequisites: DATA2002 or equivalent

MATH16

Holger Dullin

Title: The 3-body problem in dimension 4

Description: The 3-body problem is one of the most important problems in dynamical systems. The perenial question on the (in)stability of Earth's orbit is unsolved to this day. It turns out that certain problems concerning the stability of solutions are easier to solve in spatial dimension 4 then in dimension 3. The goal of this project is to study the particular case where the angular momentum tensor has degenerate eigenvalues in 4D. A combination of numerical and analytical tools will be used.

Times: flexible

Max number of students: 3

UoS prerequisites: ideally MATH3977