Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

## Archived seminars in Mathematics

Seminars 1 to 50

Next 50 seminars
Bayesian sequential inference (filtering) in a functional tensor train representation.

### Colin Fox

Physics University of Otago

Date: Tuesday 17 November 2020

Colin Fox (Physics, Otago), joint work with Sergey Dolgov (Mathematics, Bath) Bayesian sequential inference, a.k.a. optimal continuous-discrete filtering, over a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker--Planck equation, in alternation with Bayes’ conditional updating. We present a numerical grid-method that represents density functions on a mesh, or grid, in a tensor train (TT) representation, equivalent to the matrix product states in quantum mechanics. By utilizing an efficient implicit PDE solver and Bayes' update step in TT format, we develop a provably optimal filter with cost that scales linearly with problem size and dimension. This ability to overcome the curse of dimensionality' is a remarkable feature of the TT representation, and is why the recent introduction of low-rank hierarchical tensor methods, such as TT, is a significant development in scientific computing for multi-dimensional problems. The only other work that gets close to the scaling we demonstrate in high-dimensional settings is due to Stephen S.T. Yau and his Fields-medallist brother Shing-Tung Yau. We give a gentle introduction to filtering, functional tensor train representations and computation, present some examples of filtering in up to 80 dimensions, and explain why we can do better than a Fields medallist.
201110160229
Toward building a conceptual model on wave transformation over shore platforms and Wave transformation on a fringing reef system with spurs and grooves structures

### Raphael Krier and Cesar Acevedo Ramirez

Department of Geolgraphy University of Otago

Date: Tuesday 20 October 2020

This research focuses on wave transformation over intertidal shore platforms. Intertidal shore platforms act as natural buffers dissipating wave energy in rocky shore environments. In the context of sea level rise and increasing storminess, cliff erosion and over wash hazards are rising issues in rocky shore environments. The understanding of wave transformation represents a first step toward the characterisation of rock coast erosion mechanisms in response to a changing climate. Unlike sandy beaches, characterised by a seasonal or inter-annual erosion/accretion cycle, cliff erosion is an irreversible process threatening rock coasts which represent 80% of the world’s coastline and 23% of New Zealand’s coastline. Wave transformation on shore platforms has mostly been studied over linear transects which limited previous observations to two dimensions. The aim of my research is to redefine shore platform hydrodynamic processes in three dimensions and assess the effects of platform morphology on wave transformation. This will allow to establish conceptual models coupling morphology and hydrodynamics. These types of models are well developed on sandy beaches, but currently lacking in rocky shore environments and the information they provide can be useful to assess the level of hydrodynamic forcing acting on different rocky shores and, subsequently, cliff erosion rates.
$\cdots$
Spurs and Grooves (SAG) structures can be found on coral reefs around the world. However, there are few studies that relate SAG morphology with wave transformation process. Using an array of pressure sensors and current meters deployed over 10 days we present observations of wave dissipation over SAG structures at Xahualxol, Quintana Roo, Mexico. This site has a different morphology of SAG compared to previous studies. Our results indicate that SAG structures are more important in wave transformation than has previously been reported. The rate of dissipation (up to 80W/m2) and the wave dissipation friction factor, fw, (1.1) found, are also high compared to previous values found on other reefs. We also found that the wave dissipation rate over the spurs can be up to three times higher than in the adjacent grooves. This study demonstrates that SAG morphologies play a discernable role in wave dissipation over the forereef.
201012121910
Reconstructing the evolution of bayesian brains by computational modelling

### Mike Paulin

Department of Zoology University of Otago

Date: Tuesday 13 October 2020

Placozoans are the smallest, simplest animals on Earth. They are microscopic patches of bacteria-eating epithelium, which evolved about 560 million years ago, were the first animals that moved, and may be ancestors of all modern animals with nervous systems, i.e. of all animals except sponges and placozoans themselves. I will show, using realistically constrained biophysical models and simulations, how placozoan’s ability to perceive bacteria by contact chemosensing (taste) can be extended in a remarkably simple way to perceive salient aspects of the world beyond their body surface, by constructing a map in their epithelial margin of the Bayesian posterior density of external signal sources given receptor states. This leads to an elegant and possibly true model of statistically optimal perception, decision-making and movement control at the algorithmic level, in a ridiculously simple animal without a nervous system. The algorithm can be ported to spiking neural networks in a straightforward way. This may be a basic computational motif which has evolved by duplication, specialization and recursion (i.e. observer modules passing inferences to each other as priors and data) to produce nervous systems capable of dynamical statistical inference, planning and control in animals like us, with complex anatomy living in complex modern ecosystems.
201009130855

### Dr Hamza Bennani

Department of Computer Science at Otago

Date: Tuesday 6 October 2020

This research investigates an accurate method for three dimensional (3D) reconstruction of the human spine from bi-planar radiographs with comparable results to CT scans or MRI. In this work, we generated a publicly available dataset which corresponds to the training data used. We subsequently solved the problem of correspondences using a landmark-free algorithm applied on the vertebrae. Finally, we developed a semi automatic method based on simulated radiographs for the reconstruction of the human lumbar spine in 3D from bi-planar radiographs.
We validated the results in vitro on radiographs of dried vertebrae with models constructed from a laser-scanner, then in vivo on radiographs of living patients with models extracted from CT scans or MRI.
The results show the feasibility of generating personalised models of patients from bi-planar radiographs.
The contributions are:
- Evaluation of the methods for creating 3D models of vertebrae and estimation of the errors in comparison with ground truth data. These methods are applicable to other free-form shapes;
- Creation of landmark free ASMs of lumbar vertebrae;
- Definition and evaluation of a process for estimating the shape and position of lumbar spine from uncalibrated bi-planar radiographs.
200930131300

### Anindya Sen

Department of Accountancy and Finance University of Otago

Date: Tuesday 29 September 2020

Students of mathematics learn early on that infinity comes in at least two sizes - countable and uncountable. However, that is merely the tip of the iceberg. In this lecture, I will talk about how we can generate the "ladder of infinity" - an infinite sequence of ever larger infinities that stretches to unimaginable heights. On the way we shall meet ordinal numbers, the well ordering principle and ZFC axioms. If time permits, we will touch on advanced topics like inaccessible cardinals which are the subject of modern research.
200923160556
The combinatorics of ‘capturing’ a phylogenetic tree from discrete data or distances

### Mike Steel

University of Canterbury

Date: Tuesday 11 August 2020

We consider two versions of the following question: What is the smallest amount of ‘data’ required to uniquely determine a phylogenetic (evolutionary) tree? In the first version, the ‘data’ consists of a sequence of discrete states observed at the leaves of a tree, and these states are assumed to have evolved from an unknown ancestral state; either with or without homoplasy. For the second version, the data consists of leaf-to-leaf distances between certain pairs of leaves in the tree. Both questions give rise to some interesting combinatorial subtleties.
200807154023
Initial data sets from parabolic-hyperbolic formulations of the Einstein vacuum constraints

### Joshua Ritchie

Mathematics and Statistics Department University of Otago

Date: Tuesday 21 July 2020

The constraint equations are a subset of the full Einstein equations that describe the universe at a fixed moment of time'. In this talk, we investigate the generic asymptotic behaviour of vacuum initial data sets constructed as solutions of an evolutionary formulation of the constraint equations. Our main focus here is on the construction of physically relevant' solutions of the constraint equations. To do this we introduce a method that allows us to construct binary black hole solutions.
200717103938
Inner Products on Convex Sets

### David Bryant

Mathematics and Statistics, University of Otago

Date: Tuesday 17 March 2020

I've recently been trying to model how ecological niches change over evolutionary time. This lead to a bunch of thorny mathematical problems related to random convex sets, like how should we think about a random process when the state space is the set of (bounded, closed, convex) random sets? how do we do statistics on these things? and how can we compute with them? It turns out that progress can be made on some of these questions by framing the space of bounded, closed, convex sets as a kind-of vector space. We can add sets, and we can scale them. But X - X doesn't equal zero when |X|>1. I'll talk about some of the mathematics of these spaces and about our attempts to define an appropriate inner product. This is joint work with Petru Cioica-Licht, Lisa Orloff-Clarke and Rachael Young.
200310161353
Adapting analysis/synthesis pairs to pseudodifferential operators

### Melissa Tacy

Mathematics and Statistics Department University of Otago

Date: Tuesday 10 March 2020

Many problems in harmonic analysis are resolved by producing an analysis/synthesis of function spaces. For example the Fourier or wavelet decomposition. In this talk I will discuss how to use Fourier integral operators to adapt analysis/synthesis pairs (developed for the constant coefficient PDE case) to the pseudodifferential setting. I will demonstrate how adapting a wavelet decomposition can be used to prove $L^{p}$ bounds for joint eigenfunctions.
200304104113
Supporting cooperation via agreement equilibrium

### Ronald Peeters

Economics Department University of Otago

Date: Tuesday 3 March 2020

We introduce 'agreement equilibrium' as a novel solution concept that can explain the abundance of cooperative behavior that is often observed in laboratory experiments in various contexts. The main idea of the agreement equilibrium is to identify behaviors that individuals can (tacitly) agree on while being ambiguous about their opponents' intentions to respect or to betray this (tacit) agreement. We investigate properties of the agreement equilibrium, including in comparison to other equilibrium concepts, and illustrate the agreement equilibrium in a series of famous applications.​
200302134134
Nonlinear Dirac equations

### Sebastian Herr

Bielefeld University

Date: Tuesday 11 February 2020

In this talk I will start by introducing the Dirac equation as a dispersive PDE and draw a connection to the Restriction Problem in Harmonic analysis. Then, I will turn to nonlinear Dirac equations, such as the Soler model and the Dirac-Klein-Gordon system. Finally, I will describe some recent progress on the regularity theory of the corresponding Cauchy problems and discuss some open questions.
200207111115
Dynamical low-rank approximation

### Christian Lubich

Mathematisches Institut Universitaet Tuebingen

Date: Tuesday 12 November 2019

This talk reviews differential equations on manifolds of low-rank matrices or tensors or tree tensor networks. They serve to approximate, in a data-compressed format, large time-dependent matrices and tensors that are either given explicitly via their increments or are unknown solutions of high-dimensional differential equations, such as multi-particle time-dependent Schr\"odinger equations or kinetic equations such as Vlasov equations. Recently developed numerical time integrators are based on splitting the projector onto the tangent space of the low-rank manifold at the current approximation. In contrast to all standard integrators, these projector-splitting methods are robust with respect to the unavoidable presence of small singular values in the low-rank approximation. This robustness relies on geometric properties of the low-rank manifolds. The talk is based on work done intermittently over the last decade with Othmar Koch, Bart Vandereycken, Ivan Oseledets, Emil Kieri and Hanna Walach.
191108135544
Numerical simulation of slow slip events

### Yiming Ma

Mathematics and Statistics, University of Otago

Date: Tuesday 8 October 2019

Slow slip events (SSEs), a type of slow earthquakes, play an important role in releasing strain energy in subduction zones, where a tectonic plate bends and slides under another one. Observations of their occurrence patterns can be used to infer the probability of triggering a damaging earthquake at the interface between the two plates. However, the underlying geophysical mechanisms governing SSEs are still not well understood. In this talk, I will introduce a physical model to simulate periodic SSEs, based on dislocation theory and rate- and state-dependent friction (RSF) law. I will further discuss the sensitivity of the model to the parameters (e.g. consititutive parameters, geometry of the fault) and some computational issues associated with the numerical scheme implemented.
191001110246
Whakatipu te Mohiotanga o te Ira: Growing Māori capability and content in genetics-related education

### Phillip Wilcox

Mathematics and Statistics, University of Otago

Date: Thursday 3 October 2019

This Seminar will focus on recent efforts at the University of Otago to increase (a) Māori content in statistics, genetics and biochemistry courses, and (b) Māori involvement in genetics-based research and applications.
191001110542
Spherical Splits

### Tom McCone

Department of Mathematics and Statistics

Date: Tuesday 1 October 2019

Suppose we have some points on the surface of a sphere, and a plane passing through the sphere (but through none of the points). Naturally, the points will be partitioned into two sets. When we consider the collection of all such possible partitions, an interesting question arises: How does the structure of the collection relate to the positions of the points (and vice versa)? Motivated by problems in data analysis, the idea of such a collection leads us to investigate connections through a range of mathematical fields, including convex geometry and graph theory, and leaves us with a handful of intriguing questions requiring further thought.
190925125536
Meta-benchmarking: what can we learn by comparing benchmarks?

### Paul Gardner

Department of Biochemistry, Otago

Date: Tuesday 24 September 2019

In the field of bioinformatics, software proliferation has become a significant issue for researchers in the field. There are hundreds of software tools for addressing problems in phylogenetics, genome assembly and protein structure prediction. As a consequence, just selecting a tool and parameter settings is a significant challenge for researchers. Consequently, neutral software benchmarks are the gold-standard for determining what software tools are optimal for addressing specified problems. Yet, are benchmarks themselves reliable, and could these be driving suboptimal practices in software development?
190918094034
Making waves discretely by putting balls into boxes and using crystals

### Travis Scrimshaw

University of Queensland

Date: Tuesday 17 September 2019

In August, 1834, John Scott Russell followed a wave traveling through a narrow channel and noticed that as the wave propagated, it did not change shape nor speed. This observation was then given a mathematical theory starting with Boussinesq in 1871, and is now known as the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation. In particular, the KdV equation admits solutions for such waves, which are called solitons. We first can make the time steps discrete, which was done by Hirota in 1977, and we will make the height and position of a wave discrete following Takahashi and Satsuma. Indeed, by using boxes that can hold at most one ball in a simple discrete dynamical system, they relate the size of a wave to a coupled collection of balls. In this talk, we will discuss the Takahashi-Satsuma box-ball system and how it can be described using Kashiwara's crystal bases, a combinatorial interpretation of representation theory arising from mathematical physics (specifically, quantum groups). This allows the system to be generalized and more tools from mathematical physics to be applied, which will also be described as time permits.
190911145730
Multiscale Methods for Modelling Intracellular Processes

University of Oxford

Date: Tuesday 10 September 2019

: I will discuss mathematical and computational methods for spatio-temporal modelling in molecular and cell biology, including all-atom and coarse-grained molecular dynamics (MD), Brownian dynamics (BD), stochastic reaction-diffusion models and macroscopic mean-field equations. Microscopic (BD, MD) models are based on the simulation of trajectories of individual molecules and their localized interactions (for example, reactions). Mesoscopic (lattice-based) stochastic reaction-diffusion approaches divide the computational domain into a finite number of compartments and simulate the time evolution of the numbers of molecules in each compartment, while macroscopic models are often written in terms of mean-field reaction-diffusion partial differential equations for spatially varying concentrations. In the first part of my talk, I will discuss connections between these different modelling frameworks, considering chemical reactions both at a surface and in the bulk. In the second part of my talk, I will discuss the development, analysis and applications of multiscale methods for spatio-temporal modelling of intracellular processes, which use (detailed) BD or MD simulations in localized regions of particular interest (in which accuracy and microscopic details are important) and a (less-detailed) coarser model in other regions in which accuracy may be traded for simulation efficiency. I will discuss error analysis and convergence properties of the developed multiscale methods, their software implementation and applications of these multiscale methodologies to modelling of intracellular calcium dynamics, actin dynamics and DNA dynamics. I will also discuss the development of multiscale methods which couple MD and coarser stochastic models in the same dynamic simulation.
190902160311
Time inconsistency in games

### Ronald Peeters

Economics Department

Date: Tuesday 3 September 2019

Decision makers with time-inconsistent preferences have been studied in great detail in recent decades. Addressing the dearth of literature on time inconsistency in a strategic context, our work provides the foundations for dealing with two-player games with naive and sophisticated players. We model various types' beliefs explicitly, and then proceed to define equilibrium notions based on these belief hierarchies. Second-order beliefs of naive players turn out to be crucial, and lead to different equilibria even in relatively simple games. We provide several examples, including models of bargaining and bank runs, showing the applicability of our framework.
190822101839
Conditions of Wilf equivalence

### Jinge Lu

Otago Computer Science

Date: Tuesday 20 August 2019

Two classes of combinatorial structures are said to be Wilf-equivalent if they contain the same number of structures of each size. In this talk we examine Wilf equivalences in permutation classes from two different directions. First, given some small classes, we determine the conditions of Wilf equivalence among its principal subclasses. Second, we determine what a class would look like, if we assume the maximum (or near maximum) extent of Wilf equivalences among its principal subclasses.
190813095405
Modelling the coupled ocean waves/sea ice system while making sense of the data: what’s the challenge?

### Fabien Montiel

Department of Mathematics and Statistics

Date: Tuesday 13 August 2019

Observations of ocean waves breaking up sea ice floes in the ice-covered Southern Ocean date back to the early 20th century and Sir Ernest Shackelton’s famous Endurance expedition. Recent evidence now suggests that this very process could be a key driver of sea ice extent and morphology, and therefore impact the global climate system. Describing, let alone modelling, the range of physical processes governing the coupled ocean waves/sea ice system is not an easy task, mainly due to the difficulty of collecting data in such a harsh environment. This is therefore the perfect playing field for applied mathematicians to propose highly idealised models, ranging from sea ice as a homogeneous viscoelastic material to more sophisticated models of wave scattering by large arrays of perfectly circular floes. Due to its relevance to the climate system as well as to the shipping industry, theoretical research on ocean waves/sea ice linkages has been burgeoning in recent years and has attracted much funding worldwide. This talk is an attempt to reflect on these recent developments (including my own work), which in some cases are driven by the need to incorporate some kind of representation of the system in large scale forecasting models as opposed to trying to understand the underlying physics. I will further discuss results from recent field work data that seem to challenge our current modelling approaches.
190807154223
Nodal sets and conformal geometry

### Dmitry Jakobson

McGill University

Date: Tuesday 6 August 2019

I will explain how nodal sets of Laplace eigenfunctions give rise to conformal invariants. The talk will be self-explanatory. Movies will be shown.
190805134854
computer vision for culture and heritage

### Steven Mills

Department of Computer Science

Date: Tuesday 30 July 2019

In this talk I will present some of our recent and ongoing work, with an emphasis on cultural and heritage applications. These include historic document analysis, 3D modelling for archaeology and recording the built environment, and tracking for augmented spectator experiences. I will also outline some of the outstanding issues we have where collaboration with mathematicians and statisticians might be valuable.
190724111803
Paraconsistent logic and inconsistent mathematics

### Zach Weber

Philosophy Department University of Otago

Date: Tuesday 23 July 2019

There are nowadays many different well-understood systems of logic: classical, intuitionistic, and paraconsistent, to name a few. This introductory talk will explain some of the motivations for studying paraconsistent logic—systems of formal logic developed since the 1970s that make it possible to have some local inconsistency without global absurdity. We will look at some of the basic details of how a paraconsistent logic works in practice, and apply it to some elementary foundational mathematics, in particular the original ‘naive’ set theory of Cantor and Dedekind, and some point-set topology. I’ll conclude with a brief discussion of the place of non-classical logic and prospects for the wider inconsistent mathematics program as it stands today.
190716134613
Condorcet Domains Satisfying Arrow's Single-Peakedness

Department of Mathematics. University of Auckland

Date: Tuesday 9 July 2019

Condorcet domains are sets of linear orders with the property that, whenever the preferences of all voters belong to this set, the majority relation of any profile with an odd number of voters is transitive. Maximal Condorcet domains historically have attracted a special attention. We study maximal Condorcet domains that satisfy Arrow's single-peakedness which is more general than Black's single-peakedness. We show that all maximal Black's single-peaked domains on the set of m alternatives are isomorphic but we found a rich variety of maximal Arrow's single-peaked domains. We discover their recursive structure, prove that all of them have cardinality 2^{m-1}, and characterise them by two conditions: connectedness and minimal richness. We also classify Arrow's single-peaked Condorcet domains for up to 5 alternatives.
190626143334
Image reconstruction and unique continuation properties

### Leo Tzou

University of Sydney

Date: Tuesday 11 June 2019

A classical result of Jerison-Kenig showed that the optimal assumption for unique continuation properties for elliptic PDE. In this talk we will explore its connection to image reconstruction with impedance tomography. We will develop an analogous theory in the context of partial data inverse problems to obtain the same sharp regularity assumption as Jerison-Kenig. The method we use involves explicit microlocal construction of the Dirichlet Green's function which on its own may be of interest for partial data image reconstruction.
190529141321
Taming the beast of the cosmological big bang singularity: Dynamics and degrees of freedom

### Florian Beyer

Mathematics and Statistics, University of Otago

Date: Tuesday 28 May 2019

The history of the universe, in particular its very beginning at the big bang'', is one of the great unsolved mysteries in science. It is modelled mathematically by solutions of Einstein's equations, the complex equations of general relativity first envisioned by Albert Einstein in 1915. Despite recent successes to prove certain stability results for the singular dynamics, there are many open issues whose resolutions would require a much stronger control of the asymptotics than so far possible with rigorous PDE techniques. One of these outstanding problems is to understand the relationship between asymptotics and degrees of freedom for singular hyperbolic PDE systems. To address this we have recently introduced a rigorous matching technique which can yield a topological characterisation of how the degrees of freedom are encoded in the asymptotics. In the particular case of Einstein's equations, this could eventually answer fundamental questions: What are the general degrees of freedom to create a universe''? How large were the chances for our universe to turn out exactly the way it has?
190522134923
Rational points on curves

### Brendan Creutz

School of Mathematics and Statistics, University of Canterbury

Date: Tuesday 14 May 2019

Many interesting problems in number theory are related to finding rational solutions to polynomial equations, a famous example being Fermat's Last Theorem. The real or complex solutions to such equations yield familiar geometric objects (curves, surfaces, etc) and in many cases the qualitative nature of the set of rational solutions is determined by the geometry. In this talk I will give a gentle introduction to this perspective in the case of polynomials of two variables.
190507135722
Wilf-equivalence and Wilf-collapse.

### Michael Albert

University of Otago Computer Science

Date: Tuesday 7 May 2019

Enumerative coincidences abound in combinatorics -- perhaps the most famous being the huge collection of different classes which are enumerated by the Catalan numbers. While some have dismissed these coincidences as arising from nothing more than the human penchant for simplicity it does seem reasonable to ask: `Are there contexts in which a large number of coincidences should be expected?''. Enumerative coincidences that occur between collections of structures avoiding some particular substructure have been called ~~Wilf-equivalences~~. For instance, the collection of permutations avoiding the pattern 312 is one of those enumerated by the ubiquitous Catalan numbers whose growth is $\Theta(n^{-3/2} 4^n)$. What about permutations that avoid 312 and one additional pattern of size $n$? There are only $o(2.5^n)$ distinct Wilf-equivalence classes -- a ~~Wilf-collapse~~. A more thorough investigation of this phenomenon leads to the conclusion that the combination of at least one non-trivial symmetry and a greedy algorithm for detecting the occurrence of a pattern leads to Wilf-collapse quite generally.
190502124540
The geometry behind MRD codes

### Geertrui van de Voorde

School of Mathematics and Statistics University of Canterbury

Date: Tuesday 16 April 2019

Rank-metric codes are widely seen in a variety of applications ranging from storing information in the cloud to public-key cryptosystems. For over thirty years, the only optimal rank-metric codes known were Gabidulin codes. This changed when Sheekey recently constructed a new family of optimal rank-metric codes (MRD codes) using objects from finite geometry called linear sets on a projective line. In this talk, I will explain the interplay between rank-metric codes and linear sets.
190403114112
Optimising the performance of wind turbines using computational fluid dynamics

### Sarah Wakes

Department of Mathematics and Statistics

Date: Tuesday 9 April 2019

Two cases studies are presented that look at optimising the performance of two scales of wind turbines. The first is work undertaken with local business PowerHouse Wind to understand the flow behaviour over their unique one blade small scale horizontal axis wind turbine. Soft stall on the blade is applied through varying the speed of the rotor with an electric brake and is used to regulate power output and mitigate against damaging winds. Two- and three- dimensional air flow simulations were undertaken as well as visualisation of stall patterns on a working blade. This work allowed prediction of power output of the blade over a range of wind and rotor speeds. A larger next generation blade has also been studied to aid in the optimisation of the power output and blade design. The second case study is work undertaken with University of Waikato using machine learning techniques to predict the wake from a large-scale wind turbine and wind flow over a complex topography. The ultimate aim is to use Computational Fluid Dynamics with machine learning to optimise wind farm layouts over complex topographies.
190327131340
Dispersive PDE and the restriction problem

### Tim Candy

Department of Mathematics and Statistics

Date: Tuesday 2 April 2019

A dispersive equation is a partial differential equation (PDE) for which solutions at different wavelengths propagate at different velocities (or directions). An important consequence of this is that the amplitude of solutions decays, while the energy or mass can remain conserved. Important examples of dispersive PDE include the wave equation, the KdV equation, and the Schrödinger equation. In the 70's and 90's it was observed that dispersion implies global space time estimates known as Strichartz estimates, these estimates are closely connected to the restriction problem in harmonic analysis. In this talk we will review this connection, explain how these estimate can applied to study nonlinear dispersive PDE, and cover some recent developments on bilinear restriction estimates and the wave maps equation.
190328140937
Projective Characters of Metacyclic p-Groups

### Conor Finnegan

University College Dublin

Date: Tuesday 26 March 2019

The projective characters of a group provide us with important information regarding the structure and properties of the group. The purpose of this research was to find the projective character tables of metacyclic p-groups. This aim was achieved for metacyclic p-groups of positive type, but not in the negative type case. In this talk, I will give an introductory overview of some of the fundamental methods and results in projective representation theory. I will then discuss the application of these methods to metacyclic p-groups of positive type, using the previously understood abelian case as an example.
190318153651
CEBRA: mathematical and statistical solutions to biosecurity risk challenges

### Andrew Robinson

University of Melbourne

Date: Thursday 21 March 2019

CEBRA is the Centre of Excellence for Biosecurity Risk Analysis, jointly funded by the Australian and New Zealand governments. Our problem-based research focuses on developing and implementing quantitative tools to assist in the management of biosecurity risk at national and international levels. I will describe a few showcase mathematical and statistical projects, underline some of our soaring successes, underplay our dismal failures, and underscore the lessons that we've learned.
190311141043
Folding, surprise and playing games: deep learning at the CS department

### Lech Szymanski

Department of Computer Science, University of Otago

Date: Tuesday 19 March 2019

This talk will give an overview of the research done by the deep learning group at the Department of Computer Science. Specifically, I will talk about the work in three different areas: theoretical analysis of deep architectures using folding transformations, reinforcement learning with surprise, and teaching a deep network to play Atari games without catastrophic forgetting.
190214131558
Pattern formation in reaction-diffusion systems on time-evolving domains

### Robert van Gorder

Department of Mathematics and Statistics, University of Otago

Date: Tuesday 12 March 2019

The study of instabilities leading to spatial patterning for reaction-diffusion systems defined on growing or otherwise time-evolving domains is complicated, since there is a strong dependence of spatially homogeneous base states on time and the resulting structure of the linearized perturbations used to determine the onset of stability is inherently non-autonomous. We obtain fairly general conditions for the onset and persistence of diffusion driven instabilities in reaction-diffusion systems on manifolds which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and the growth rate functions, which result in sufficient conditions for diffusive instabilities phrased in terms of differential inequalities.
These conditions generalize a variety of results known in the literature, such as the algebraic inequalities commonly used as sufficient criteria for the Turing instability on static domains, and approximate or asymptotic results valid for specific types of growth, or for specific domains.
190214131405

### Russell Higgs

School of Mathematics and Statistics, University College Dublin

Date: Tuesday 5 March 2019

This will be a survey talk discussing three open conjectures concerning the degrees of irreducible projective representations of finite groups. First a review of ordinary representations will be given with illustrative examples, before considering projective representations. A projective representation of a finite group $G$ with 2-cocycle $\alpha$ is a function $P:G \rightarrow GL(n, \mathbb{C})$ such that $P(x)P(y) = \alpha(x, y)P(xy)$ for all $x, y\in G$, where $\alpha(x, y)\in \mathbb{C}^*.$ One of the conjectures is can one conclude that $G$ is solvable given that the degrees of all its irreducible projective representations are equal.}
190214131235
The geometry and combinatorics of phylogenetic tree spaces

### Alex Gavryushkin

Department of Computer Science University of Otago

Date: Tuesday 16 October 2018

The space of phylogenetic (aka evolutionary) trees is known to have a unique and non-trivial geometry with complicated combinatorial properties. Despite the recent major advances in our understanding of the tree space, a number of gaps remain. In this talk I will concentrate on a specific instance of phylogenetic trees called time-trees (aka dated trees), where internal nodes of the tree are ranked with respect to their time. This class of trees inherits some of the properties of classic, non-ranked, trees. However, some of the fundamental properties of the space (seen as a metric space), including its curvature, computational complexity, and neighbourhood growth function, are significantly different. These differences call for further investigations of these properties, which have a potential to become a stepping stone for new efficient phylogenetic inference methods. In this talk I will introduce all necessary background, present some of our results in this direction, and conclude with the exciting opportunities this area has to offer in computational geometry, combinatorics, and complexity theory.
181012104548
Inferring species trees for many species from Allele frequency spectra

### Marnus Stoltz

Department of Mathematics and Statistics

Date: Tuesday 9 October 2018

In this talk we describe an algorithm for efficiently computing the likelihood of a species tree from unlinked binary markers or allele frequency data. The model assumptions are similar to those implemented in SNAPP however, unlike SNAPP, the method can handle hundreds or even thousands of individuals. Our approach is based on a diffusion approximation of gene dynamics. However we work directly with backwards processes to compute the probability of every marker individually, bypassing the need to compute the entire joint allele frequency across all species. We point out some of the challenges encountered along the way such as boundary conditions for the backwards diffusion to ensure uniqueness and existence, computational bottlenecks and parameter mappings between models.
180815154042
Electric Impedance Tomography

### Nikola Stoilov

University of Burgundy

Date: Tuesday 25 September 2018

Electric Impedance Tomography (EIT) is a medical imaging technique that uses the response to voltage difference applied outside the body to reconstruct tissue conductivity. As different organs have different impedance, this technique allows to produce images of the inner body without exposing the patient to potentially harmful radiation. In mathematical terms, EIT is as an inverse problem, whereby data inside a given domain is recovered from data on its boundary. In contrast with techniques like X-ray tomography (based on a linear problem), the particular inverse problem employed in EIT is non-linear - it reduces to a so-called D-bar problem. Such problems also find application in the area of Integrable Systems, specifically in the inverse scattering problem associated with 2+1 dimensional integrable equations such as the Davey - Stewartson and Kadomtsev-Petviaschvili equations. I will discuss the design of numerical algorithms based on spectral collocation methods that address D-bar problems found in both integrable systems and medical imaging. Successfully implementing these methods in EIT should allow us to achieve images with much higher resolutions at reduced processing times. We take advantage of the fact our approach is highly parallelisable by implementing on graphical processing units (GPUs) to gain efficiency and speed without increasing the cost of the process. Finally I will describe the route towards the full development of the technology, and the hope that EIT will emerge as an effective, fast, convenient and less intrusive and distressing form of medical imaging.
180914084320

### Mike Hendy

Department of Mathematics and Statistics

Date: Tuesday 18 September 2018

Can we give an upper bound on the value of the determinant det $(A)$ for all $n×n$ real matrices $A=(a_{ij})$? This question is no, unless we bound the entries $a_{ij}$. In 1893 the French mathematician Jacques Hadamard (1865 - 1963) showed that
$|a_{ij} |≤1,∀i,j⟹|det⁡(A) |≤n^{(n⁄2)}$,
and found matrices satisfying this bound for many values of $n$.

An $n×n$ real matrix $A=(a_{ij})$ with entries $|a_{ij}|≤1$ satisfying
$| det⁡(A)|=n^{(n∕2)}$
is now referred to as a Hadamard matrix. We will see that for $n≥4$, Hadamard matrices can exist only for $n≡0$ (mod 4). Although there are Hadamard matrices for an infinite number of multiples of 4, the Hadamard conjecture that postulates there exists a Hadamard matrix of order $n$, for each positive integer multiple of 4, has remained unresolved for 125 years.

In this talk I will present some practical applications of Hadamard matrices, including my own discovery of their application in phylogenetics, and reveal a personal encounter with Hadamard's ghost.
180822085908
**POSTPONED - NEW DATE TBC** The geometry and combinatorics of phylogenetic tree spaces

### Alex Gavryushkin

Department of Computer Science

Date: Tuesday 11 September 2018

The space of phylogenetic (aka evolutionary) trees is known to have a unique and non-trivial geometry with complicated combinatorial properties. Despite the recent major advances in our understanding of the tree space, a number of gaps remain. In this talk I will concentrate on a specific instance of phylogenetic trees called time-trees (aka dated trees), where internal nodes of the tree are ranked with respect to their time. This class of trees inherits some of the properties of classic, non-ranked, trees. However, some of the fundamental properties of the space (seen as a metric space), including its curvature, computational complexity, and neighbourhood growth function, are significantly different. These differences call for further investigations of these properties, which have a potential to become a stepping stone for new efficient phylogenetic inference methods. In this talk I will introduce all necessary background, present some of our results in this direction, and conclude with the exciting opportunities this area has to offer in computational geometry, combinatorics, and complexity theory.
180815153938
A new algorithm for characteristic extraction and matching in numerical relativity

### Chris Stevens

Rhodes University, South Africa

Date: Tuesday 4 September 2018

We are now in the exciting new era of gravitational wave astronomy, where we can study the universe through the gravitational waves emitted by massive events such as coalescing black holes or neutron stars.

An important part of gravitational wave astronomy is the numerical simulations that compute the emitted gravitational radiation, which are non-trivial since the simulations are on a physical domain of finite extent but gravitational waves are unambiguously defined only at future null infinity (scri+). There are a number of methods for waveform estimation, but only in characteristic extraction is the waveform calculated at scri+.

We present a new algorithm and implementation of characteristic extraction. It has the key feature of being simply extendable to characteristic matching, in which the characteristic evolution provides outer boundary data for the "3+1" simulation. The key advantage of characteristic matching is that it would lead to a significant speed-up in the time required to complete a numerical simulation.
180905124424
$L^{p}$ estimates for joint eigenfunctions

### Melissa Tacy

Department of Mathematics and Statistics

Date: Tuesday 21 August 2018

Consider the problem of the $L^{p}$ growth of joint eigenfunctions of a set of pseudodifferential operators $\Delta=P_{1},\dots,P_{r}$ that satisfy a suitable non-degeneracy assumption. In the special case of symmetric spaces of rank $r$ Marshall obtained $L^{p}$ estimates that indicated that the $L^{p}$ growth of $u$ behaves like that of products $u(x)=u_{1}(x_{1})\dots u_{r}(x_{r})$ where each $u_{i}(x_{i})$ is a Laplace eigenfunction in $n/r$ variables. In this talk I will discuss a more general case where we only assume that the normals to the characteristic sets of each $P_{i}$ are linearly independent. In this case we are able to obtain $L^{p}$ results which are as good as the symmetric space results for some $p$ and examples to show that in this general setting we cannot improve the results.
180813142130
Comparative probability orders and noncoherent initial ideals of exterior algebras

### Dominic Searles

Department of Mathematics and Statistics

Date: Tuesday 14 August 2018

Term orders on monomials in exterior algebras coincide with comparative probability orders on subsets. In the context of comparative probability, it has long been known that there exist orders that cannot be represented by a system of weights. Such orders are called noncoherent. In 2000, D. Maclagan asked whether there exists an ideal in an exterior algebra that has an initial ideal with respect to some noncoherent term order that is unequal to any initial ideal with respect to a coherent term order. In joint work with A. Slinko, we use ideas from the theory of comparative probability to construct such an ideal, answering Maclagan's question in the affirmative.
180807151416
A faster algorithm for updating the likelihood of a phylogeny

### David Bryant

Department of Mathematics and Statistics

Date: Thursday 9 August 2018

##Note day and time. A joint Mathematics and Statistics seminar taking place in the usual slot for Statistics seminars## Both Bayesian and Maximum Likelihood approaches to phylogenetic inference depend critically on a dynamic programming algorithm developed by Joe Felsenstein over 35 years ago. The algorithm computes the probability of sequence data conditional on a given tree. It is executed for every site, every set of parameters, every tree, and is the bottleneck of phylogenetic inference. This computation comes at a cost: Herve Philippe estimated that his research-associated computing (most of which would have been running Felsenstein's algorithm) resulted in an emission of over 29 tons of $CO_2$ in just one year. In the talk I will introduce the problem and describe an updating algorithm for likelihood calculation which runs in worst case O(log ~~n~~) time instead of O(~~n~~) time, where ~~n~~ is the number of leaves/species. This is joint work with Celine Scornavacca.
180731153252
Amenability of quasi-lattice ordered groups

### Ilija Tolich

Department of Mathematics and Statistics

Date: Tuesday 31 July 2018

Amenability of groups has been an interesting topic for a long time and there are still many groups that cannot be classified as either amenable or non-amenable. Nica adapted the notion of amenability when introduced a class of partially ordered groups called quasi-lattice ordered groups and studied their C*-algebras. We say a quasi-lattice ordered group is amenable if its universal algebra and reduced algebra are isomorphic.

The Baumslag-Solitar group is an example of an amenable quasi-lattice ordered group. In particular, it is an HNN-extension of the integers. Studying the Baumslag-Solitar group gave us the insight to prove a new means of detecting amenability in quasi-lattice ordered groups and also to construct new examples of amenable quasi-lattice ordered groups.
180730102106
Kohnert polynomials

### Dominic Searles

Department of Mathematics and Statistics

Date: Tuesday 29 May 2018

In 1990, Kohnert introduced an algorithmic operation on box diagrams in the positive quadrant. Kohnert proved that when the diagrams are left-justified, a weighted sum over such diagrams yielded a formula for the key polynomials, important in representation theory. He also conjectured that applying the same algorithm to another specific class of box diagrams, the Rothe diagrams of permutations, gave a formula for the geometrically-important Schubert polynomials.

In joint work with Assaf, we consider the application of Kohnert's algorithm to arbitrary box diagrams in the positive quadrant; we call the resulting polynomials Kohnert polynomials. We establish some structural results about Kohnert polynomials, including that their stable limits are quasisymmetric. Certain choices of box diagrams yield bases of the polynomial ring in a natural way; as an application, we use these results to introduce a new basis of polynomials whose stable limit is a new basis of quasisymmetric functions that contains the Schur functions. Some further conjectures regarding Kohnert polynomials will be presented.
180409101045
Project presentations

### Honours and PGDip students

Department of Mathematics and Statistics

Date: Friday 25 May 2018

STATISTICS
Qing Ruan : ~~Bootstrap selection in kernel density estimation with edge correction~~
Willie Huang : ~~Autoregressive hidden Markov model - an application to tremor data~~

MATHEMATICS
Tom Blennerhassett : ~~Modelling groundwater flow using Finite Elements in FEniCS~~
Peixiong Kang : ~~Numerical solution of the geodesic equation in cosmological spacetimes with acausal regions~~
Lydia Turley : ~~Modelling character evolution using the Ornstein Uhlenbeck process~~
Ben Wilks : ~~Analytic continuation of the scattering function in water waves~~
Shonaugh Wright : ~~Hilbert spaces and orthogonality~~
Jay Bhana : ~~Visualising black holes using MATLAB~~
180515110323
Numerical scalar curvature deformation in the $ð$-formalism

### Boris Daszuta

Department of Mathematics and Statistics

Date: Tuesday 22 May 2018

The Einstein field equations of general relativity may be decomposed and viewed as an evolutionary (PDE) system in time. Initial data must hence be provided on a spatial manifold and consequently non-trivial constraint equations must be solved.

In particular, assuming a moment in time symmetry in vacuum reduces the problem of solving the constraints to a restriction of zero scalar curvature associated with the initial data set. A result due to [1] at the analytical level provides a technique for local control on the aforementioned set and may be used to engineer initial data with well-defined asymptotics. In short, one may glue together distinct, known solutions from differing regions in a controlled manner forming new data.

The aim of this talk is to demonstrate how a numerical scheme may be fashioned out of the above and present results pertaining to a numerical gluing construction.

Ref:
[1]: (Corvino, J.) Scalar Curvature Deformation and a Gluing Construction for the Einstein Constraint Equations. ~~Communications in Mathematical Physics~~ 214, 1 (2000), 137-189.
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