# Denison Summer Scholarships - Mathematics and Statistics

**Project**:** (MATH1)** **Dynamical systems on randomly evolving networks and applications in neuroscience**

The brain is naturally described as a collection of a large number of electrical potentials evolving on a network of interconnected neurons. In the course of the brain activity, some connection between neurons are switched off and new connections are created and this switching process is inherently random. We will consider simple mathematical models for those processes, for which some experience in graph theory would be helpful. Suitable for up to two students two work as a team.

**Supervisor: ****Professor**** Ben Goldys**

**Secondary Supervisor: **** **

**Dates:** Feb-Early Mar

**Prerequisites: **MATH2962 and MATH2965 and STAT291

**Project**:** (MATH2)** **Can we make money using the games theory?**

Mathematical theory of games was invented by John von Neumann who expected it would provide a universal language for Economics, Finance and Social Sciences. The aim of the project is to learn the basic concepts of games theory and some of its applications in Finance and hazard games. We will also try to examine the question if people really play mathematical games in real life.

**Supervisor: ****Professor**** Ben Goldys**

**Secondary Supervisor: **** **

**Dates:** Feb-Early Mar

**Prerequisites: **STAT2911

**Project**:** (MATH3)** **Fourier series, (non)uniqueness and set theory**

In 1870 Georg Cantor proved that a periodic complex-valued function of a real variable coincides with the values of at most one trigonometric series. This result has lead to a number of fascinating results about Fourier series and deep properties of the subsets of real line. Many interesting questions still remain open. We will learn about classical results and explore some open problems.

**Supervisor: ****Professor**** Ben Goldys**

**Secondary Supervisor: **** **

**Dates:** Feb-Early Mar

**Prerequisites: **MATH2962

**Project**:** (MATH4)** **Statistical Laws in Complex Systems**

The distribution of frequency of words shows a striking similar pattern regardless of the book or textual database one looks. The same is true for the distribution of population of cities in different countries. These statistical regularities have long been collected in form of "laws" and play an important role in the field of complex systems. The goal of this project is to explore the properties, origin, and validity of these laws using new databases and methods. Suitable for up to two students.

**Supervisor: ****Dr**** Eduardo Altmann**

**Secondary Supervisor: **** **

**Dates:** Nov-Dec, Jan-Feb, Feb-Early Mar

**Prerequisites: **MATH1005 Statistics (or similar) and some programming

**Project**:** (MATH5)** **Functional data analysis**

Functional data is where we observe a curve for each sample. Examples of functional data include growth curves, brain electrical activity and colour spectra measurements. It is important to be able to identify any unusual sample curves that do not align closely with other observations so that they can be dealt with appropriately in any subsequent analysis of the data. This project will explore approaches to outlier identification in functional data obtained from meat colour spectra measurements. We are also interested in classifying functional measurements into multiple categories and assessing the accuracy of such automated approaches to colour measurement calibrated against a trained human assessor. Suitable for up to two students.

**Supervisor: ****Dr**** Garth Tarr**

**Secondary Supervisor: **** **

**Dates:** Jan-Feb, Feb-Early Mar

**Prerequisites: **STAT2XXX

**Project**:** (MATH6)** **Collision orbits in the restricted three body problem**

The restricted three body problem describes a massless satellite in the gravitational field of say the Earth and Moon revovling around each other. Collision orbits can be thought of as transfer orbits of a spacecraft starting at the earth and ending at the moon. The goal of the project is to find such collision orbits by numerical continuation from the integrable 2-centre problem, where a complete classification of such orbits is known. Suitable for up to three students.

**Supervisor: ****Professor**** Holger Dullin**

**Secondary Supervisor: **** **

**Dates:** Nov-Dec or Jan-Feb

**Prerequisites: **MATH3977

**Project**:** (MATH7)** **Reduction of resonant Harmonic Oscillators**

The harmonic oscillator with integer frequencies provides an important class of examples of integrable systems that can be reduced to integrable systems on compact symplectic manifolds (or generally orbifolds). The goal of the project is to study some particular cases in this class and use invariant theory to perform the reduction explicitlely. The action variables of the reduced system are confined to a convex polytope with special properties that will be explored in this project, possibly with the help of computer algebra, so some programming experience is desirable. Suitable for up to three students.

**Supervisor: ****Professor**** Holger Dullin**

**Secondary Supervisor: **** **

**Dates:** Nov-Dec or Jan-Feb

**Prerequisites: **MATH3977

**Project**:** (MATH8)** **Totally disconnected locally compact groups**

Topological group theory is the study of groups that have a concept of closeness.Some classes are less well understood than others. Totally disconnected locally compact groups are not as well understood as others and there is still much to be done. Examples include the group of symmetries of an finite but locally finite graph. Projects can involve looking at properties of specific examples or the general theory, depending on the background of the student.

**Supervisor: ****Professor**** Jacqui Ramagge**

**Secondary Supervisor: **** **

**Dates:** Jan-Early Mar

**Prerequisites: **For second-years, MATH2962 and MATH2968, For third-years, MATH3961 and MATH3962

**Project**:** (MATH9)** **Operator algebras**

Operators are linear maps from (typically infinite-dimensional) vector spaces to themselves. They are important in modelling systems in physics. Their mathematical theory provides the basis of quantum mechanics and statistical mechanics. We will be looking at problems involving states of C*-algebras.

**Supervisor: ****Professor**** Jacqui Ramagge**

**Secondary Supervisor: **** **

**Dates:** Jan-Early Mar

**Prerequisites: **For second-years, MATH2962 and MATH2968, For third-years, MATH3961 and MATH3962

**Project**:** (MATH10)** **PDE models for the distribution of ingested lipids in macrophages in atherosclerotic plaques**

Atherosclerotic plaques in artery walls are accumulations of lipid (fat) loaded cells and necrotic (dead) cellular debris in artery walls. We have written and done some analysis on a PDE advection model for the dynamics of cells and lipids in plaques where the number of active cells in the plaque is a function of both time t and accumulated lipid a. The aim of this project is to produce numerical solutions to the model when cell behavior is a function of a, the accumulated lipid inside each cell. This project is particularly suitable for students who are interested in applications of mathematics to biomedical problems, have completed a third year unit on PDEs and have at least some experience in coding in Matlab, C, Python or similar. Suitable for up to two students.

**Supervisor: ****Professor**** Mary Myerscough**

**Secondary Supervisor: **** **

**Dates:** Jan-Early Mar

**Prerequisites: **MATH2065/2965

**Project**:** (MATH11)** **Mathematics in Medicine: Using Optimisation to Improve Cancer Treatment.**

Effective radiotherapy is dependent on being able to (i) visualise the tumour clearly, and (ii) deliver the correct dose to the cancerous tissue, whilst sparing the healthy tissue as much as possible. In the presence of motion, both of these tasks become increasingly difficult to perform accurately – increasing the likelihood of incorrect dose delivered to cancerous tissue and exposure of healthy tissue to unnecessary radiation, causing adverse effects. This project will develop mathematical optimisation tools to improve the quality of diagnostic images and treatment accuracy, and requires some programming experience.

**Supervisor: ****Dr**** Michelle Dunbar**

**Secondary Supervisor: **** **

**Dates:** Nov-Dec or Jan-Feb

**Prerequisites: **MATH1002/1902 and MATH2061/2961 and some programming experience

**Project**:** (MATH12)** **Bifurcations of differential equations**

In this project, systems of differential equations depending on parameters will be studied. Such systems naturally occur in modelling chemical reactions, biological systems, physical phenomena. The student will investigate changes that occur in the behaviour of the solutions by changing the parameters and interpret them in the corresponding model. Suitable for up to three students.

**Supervisor: ****Dr**** Milena Radnovic**

**Secondary Supervisor: **** **

**Dates:** Nov-Dec or Feb-Early Mar

**Prerequisites: **

**Project**:** (MATH13)** **Pencils of conics**

In this project, we will study the pencils of second order curves in the plane. The aim is to find which types of conics (ellipses, hyperbolas, parabolas, or degenerate conics) appear in a pencil and to classify the pencils according to that.

**Supervisor: ****Dr**** Milena Radnovic**

**Secondary Supervisor: **** **

**Dates:** Nov-Dec or Feb-Early Mar

**Prerequisites: **MATH1002/1902

**Project**:** (MATH14)** **Minimization of finite sums with stochastic gradient methods**

There has been an explosion of interest in stochastic gradient methods for computing a minimizer of a finite sum of functions measuring misfit over a large number of data points. The goal of this project is to get a good understanding of this very quickly-evolving area and explore many possible variants on the existing algorithms for further improvements. This project benefits from new creative ideas and good coding skills. Suitable for up to two students.

**Supervisor: ****Dr**** Ray Kawai**

**Secondary Supervisor: **** **

**Dates:** Nov-Dec, Dec-Jan, Feb-Early Mar (NOT Jan-Feb)

**Prerequisites: **MATH3976 or STAT3911, prefer both

**Project**:** (MATH15)** **Exact simulation of SDE**

The exact method enables one to simulate a hitting time, and other functionals of a one-dimensional jump diffusion with state-dependent drift, volatility, jump intensity, and jump size. This acts as an alternative to the discretization-based approximate methods and eliminates the need to control the bias of a discretization-based simulation estimator. In this project, we will explore a variety of exact simulation methods with a view towards applications, including unbiased estimation of security prices, transition densities, hitting probabilities, and other quantities arising in jump-diffusion models. Strong numerical experiment skill is essential.

**Supervisor: ****Dr**** Ray Kawai**

**Secondary Supervisor: **** **

**Dates:** Nov-Dec, Dec-Jan, Feb-Early Mar (NOT Jan-Feb)

**Prerequisites: **MATH3969 or MATH3976 or STAT3911, prefer all

**Project**:** (MATH16)** **Geometric topology**

The basic objects of geometric topology are curves and surfaces. This project studies them using techniques from geometry, algebra or combinatorics. Some basic questions that may be addressed are: How do you tell two knots apart? Can you quantify how complicated a given knot is? How do you tell two surfaces apart? How do you knot surfaces in four dimensions? Suitable for up to six students.

**Supervisor: ****Associate Professor**** Stephan Tillmann**

**Secondary Supervisor: **** **

**Dates:** Jan-Early Mar

**Prerequisites: **MATH2061/2961 or MATH2069/2969 or MATH2962 or MATH2065/2965 or MATH2968

**Project**:** (MATH17)** **Research in false discoveries...**

The multiple testing problem arises when we wish to test many hypotheses at once. Initially people tried to control the probability that we falsely reject at least one true null hypothesis. However, in a ground breaking paper Benjamini and Hochberg suggested that alternatively we can control the false discovery rate (FDR): the expected percentage of true null hypotheses among all the rejected hypotheses. Shortly after its introduction FDR became the preferred tool for multiple testing analysis with the original 1995 paper garnering over 35K citations. There are several related problems in the analysis of false discoveries that would be intriguing to explore. Suitable for up to three students.

**Supervisor: ****Associate Professor**** Uri Keich**

**Secondary Supervisor: **** **

**Dates:** Nov-Early Mar

**Prerequisites: **STAT2911

**Project**:** (MATH18)** **FDR in mass spectrometry**

In a shotgun proteomics experiment tandem mass spectrometry is used to identify the proteins in a sample. The identification begins with associating with each of the thousands of the generated peptide fragmentation spectra an optimal matching peptide among all peptides in a candidate database. Unfortunately, the resulting list of optimal peptide-spectrum matches contains many incorrect, random matches. Thus, we are faced with a formidable statistical problem of estimating the rate of false discoveries in say the top 1000 matches from that list. The problem gets even more complicated when we try to estimate the rate of false discoveries in the candidate proteins which are inferred from the matches to the peptides. We will look at some of these interesting statistical questions that are critical to correct analysis of the promising technology of shotgun proteomics.

**Supervisor: ****Associate Professor**** Uri Keich**

**Secondary Supervisor: **** **

**Dates:** Nov-Early Mar

**Prerequisites: **STAT2911

**Project**:** (MATH19)** **Fast exact tests**

Exact tests are tests for which the statistical significance is computed from the underlying distribution rather than, say using Monte Carlo simulations or saddle point approximations. Despite their accuracy exact tests are often passed over as they tend to be too slow to be used in practice. We recently developed a technique that fuses ideas from large-deviation theory with the FFT (Fast Fourier Transform) that can significantly speed up the evaluation of some exact tests. In this project we would like to explore new ideas that we allow us to expand the applicability of our approach to other tests. Suitable for up to three students.

**Supervisor: ****Associate Professor**** Uri Keich**

**Secondary Supervisor: **** **

**Dates:** Nov-Early Mar

**Prerequisites: **STAT2911

**Project**:** (MATH20) Characterisation of the fractional Laplacian on open sets**

The linear heat equation is a typical model for a local diffusion process, for instance, in a heat conducting rod. The term in the heat equation modelling the diffusion is given by the Laplace operator. However, there are many diffusion processes of non-local character, as, for instance, in crystal dislocation, or in phase transition. The quantity in mathematical equation modelling these non-local phenomena is given by the fractional Laplace operator. On the Euclidean space, it known that the fractional Laplace operator can be defined in many different ways. In this research project, we want to investigate these characterisation of the fractional Laplace operator restricted on a bounded open set when the operator is equipped with some boundary conditions.

**Supervisor: ****Dr Daniel Hauer**

**Dates:** Nov-Dec, Jan-Feb, Feb-Early Mar

**Prerequisites: **MATH2961, MATH2962, MATH2065 or MATH2965