Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

Archived seminars in Mathematics

Seminars 51 to 100
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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.
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.
Jacques Hadamard and his matrices

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.
**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.
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.
$L^{p}$ 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.
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.
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.
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.
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.
Project presentations

Honours and PGDip students

Department of Mathematics and Statistics

Date: Friday 25 May 2018

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

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~~
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.

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

Markus Antoni

Department of Mathematics and Statistics

Date: Tuesday 15 May 2018

In this talk we discuss an approach to obtain maximal regularity estimates for solutions of stochastic Volterra integral equations driven by a multiplicative Gaussian noise. To achieve that, we mainly focus on suitable estimates for deterministic and stochastic convolution operators. Starting with the scalar-valued case, we use functional calculi results to lift the corresponding estimates to the operator-valued setting. Once maximal regularity estimates for convolutions are obtained, appropriate Lipschitz and linear growth assumptions on the nonlinearities will lead to unique mild solutions with Hölder continuous trajectories. This is joint work with Petru Cioica-Licht and Boris Baeumer.
Asymptotics of black hole(s) in a parabolic-hyperbolic formulation of the constraints

Joshua Ritchie

Department of Mathematics and Statistics

Date: Tuesday 8 May 2018

We consider a parabolic-hyperbolic formulation of the Einstein constraint equations that allows them to be solved as an IVP. By adapting a black hole superposition method, we construct a family of initial data sets that can be interpreted as binary black hole systems. Furthermore, by numerical means, we provide evidence that suggests that there exist asymptotically flat initial data sets in this family.
Modelling sea ice

Lettie Roach

Victoria University Wellington and NIWA

Date: Tuesday 1 May 2018

Sea ice plays an integral role in the global climate system, determining absorption of solar radiation and influencing atmospheric and oceanic circulation. In this talk, I will give a brief overview of current sea ice models and discuss how lack of representation of sea ice floes may contribute to a poor simulation of the marginal ice zone. We have recently developed a comprehensive model of the sea ice floe size distribution, and results will be presented that give insight into the relative importance of different small-scale processes to overall behaviour. Development of this model also motivated an observational study of sea ice growth processes, which had not previously been quantified in the field. We find the additional physics modelled with inclusion of sea ice floes impacts sea ice concentration and thickness. This opens up new opportunities to study the interaction of sea ice with the climate system.
Modelling spatial-temporal processes with applications to hydrology and wildfires

Valerie Isham, NZMS 2018 Forder Lecturer

University College London

Date: Tuesday 24 April 2018

Mechanistic stochastic models aim to represent an underlying physical process (albeit in highly idealised form, and using stochastic components to reflect uncertainty) via analytically tractable models, in which interpretable parameters relate directly to physical phenomena. Such models can be used to gain understanding of the process dynamics and thereby to develop control strategies.

In this talk, I will review some stochastic point process-based models constructed in continuous time and continuous space using spatial-temporal examples from hydrology such as rainfall (where flood control is a particular application) and soil moisture. By working with continuous spaces, consistent properties can be obtained analytically at any spatial and temporal resolutions, as required for fitting and applications. I will start by covering basic model components and properties, and then go on to discuss model construction, fitting and validation, including ways to incorporate nonstationarity and climate change scenarios. I will also describe some thoughts about using similar models for wildfires.
Epidemic modelling: successes and challenges

Valerie Isham, NZMS 2018 Forder Lecturer

University College London

Date: Monday 23 April 2018

##Note time and venue of this public lecture##
Epidemic models are developed as a means of gaining understanding about the dynamics of the spread of infection (human and animal pathogens, computer viruses etc.) and of rumours and other information. This understanding can then inform control measures to limit spread, or in some cases enhance it (e.g., viral marketing). In this talk, I will give an introduction to simple generic epidemic models and their properties, the role of stochasticity and the effects of population structure (metapopulations and networks) on transmission dynamics, illustrating some past successes and outlining some future challenges.
Hamiltonian cycles in 5-connected planar triangulations

Robert Aldred

Department of Mathematics and Statistics

Date: Tuesday 17 April 2018

Many problems modelled by graphs have Hamiltonian cycles representing the desired outcome. As such, the existence and numbers of Hamiltonian cycles are widely studied in different classes of graphs. In this talk we will look at some interesting constructions that have produced graphs with various Hamiltonian properties.

Several attempts at proving the famous 4-colour theorem involved the existence of Hamiltonian cycles in planar graphs related to triangulations of the plane. We will discuss some of these and outline a proof that that the number of Hamiltonian cycles in a 5-connected planar triangulation on $n$ vertices grows exponentially with $n$.
Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation

Mihály Kovács

Chalmers University of Technology and Gothenburg University

Date: Tuesday 10 April 2018

We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension $d\le 3$. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.
Anomalous superdiffusive transport, trapping on scale-free networks and human activity

Sergei Fedotov

The University of Manchester

Date: Tuesday 27 March 2018

In this seminar I will discuss Levy walks and Levy flights that are the fundamental notions in physics and biology with numerous applications including T-cell motility in the brain and active transport within living cells. I will present a new single integro-differential wave equation for a Levy walk. I will also present a theory of anomalous trapping and aggregation of individuals on scale-free network. Using the empirical law of cumulative inertia and fractional analysis, I will show that "anomalous inertia" overpowers highly connected nodes in attracting network individuals. This fundamentally challenges the classical result that individuals tend to accumulate in high-order nodes.
Graph-related problems for networking on Optical Network-on-Chips

Yawen Chen

Department of Computer Science

Date: Tuesday 20 March 2018

In this talk, I would like to introduce my recent graph-related research problems on Optical Network-on-Chips, which need to be investigated with the knowledge of graph theory and combinatorial optimisation. Feedback and suggestions would be much appreciated from the math department. Below is the background of this research.

Nowadays microprocessor development has moved into a new era of many-­core on-­chip design, with tens or even hundreds of cores fitting within a single processor chip to speed up computing. However, conventional electrical interconnect for inter-­core communication is limited by both bandwidth and power density, which creates a performance bottleneck for microchips in modern computer systems -­ from smartphones to supercomputers, and to large-­scale data centers. Optical Network-­on-­Chip (ONoC), a silicon-­based optical interconnection among cores at the chip level, overcomes the limitations of conventional electrical interconnects by supporting greater bandwidth with less energy consumption, and opens the door to bandwidth-­ and power-­hungry applications. This talk will introduce graph-related research problems on ONoCs from a networking perspective and present our current results and challenging problems for designing efficient multicast routing schemes specific for ONoCs.
Besov regularity of parabolic and hyperbolic PDEs

Cornelia Schneider

University Erlangen-Nuremberg

Date: Tuesday 13 March 2018

This talk is concerned with the regularity of solutions to linear and nonlinear evolution equations on ~~nonsmooth~~ domains. In particular, we study the smoothness in the specific scale of ~~Besov~~ spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the ~~Besov~~ regularity is high enough to justify the use of adaptive algorithms.
Stability in water waves and asymptotic models

Bernard Deconinck

University of Washington

Date: Tuesday 6 March 2018

The water wave problem is the problem of understanding the dynamics of surface water waves such as tsunamis and rogue waves. It has been at the forefront of applied mathematics for over 200 years, and it continues to be of great interests. Throughout its history, its study has led to the development of many new areas of applied mathematics that are now commonly used in other areas, such as infinite-dimensional bifurcation theory, asymptotics, numerical techniques, etc.

I will provide an overview of what is known about the stability of spatially periodic water waves, discussing historically significant results, skipping all mathematical details. Then I will introduce different asymptotic models that are used to describe water waves in different regimes (long waves in shallow water, modulated waves in deep water, etc) and I will discuss how the stability results in this context do or do not make sense compared to those in the context of the full water wave problem.

Time permitting, I will give more detail on recent work to understand the stability of modulated waves in deep water with respect to so-called subharmonic perturbations.
Modelling wind flows over complex topographies

Sarah Wakes

Centre for Materials Science and Technology

Date: Tuesday 27 February 2018

Numerical modelling of wind flow over complex topography is an ambitious prospect. There is an increasing need to understand such wind flows for land planning purposes, wind farm power predictions and prediction of sediment erosion and deposition. New surveying techniques permit the development of digital terrain models, however a stumbling block is the ability of Computational Fluid Dynamic simulations (CFD) to emulate the wind flow over such landscapes. To overcome these difficulties, it is important to establish the parameters within which such simulations can operate. One long-term aim is to instil confidence in numerical techniques so that such tools can be used for predictive purposes. This seminar will cover aspects of simulation work Sarah has done over complex vegetated dune systems, the design of a sand trap, wind turbine blade stalling behaviour and optimization of wind farms.

Sarah has a BSc (Jt Hons) in Mathematics and Physics and a PhD in Theoretical Mechanics from the University of Nottingham (UK), a chartered engineer in the UK and a chartered member of Engineering NZ.
What are Earth System models and how can they be useful?

Jonny Williams

National Institute of Water and Atmospheric Research (NIWA)

Date: Wednesday 15 November 2017

##Note day and time of this seminar##
##A joint seminar with the Department of Physics##
Earth System models are able to produce the most advanced computational representations of our planet that we have. They are able to simulate the properties of the atmosphere, ocean and cryosphere as well as biogeochemical processes in the air and in the water. I will give a tour of Earth System models and will discuss their strengths and weaknesses since all models are wrong but some are useful! These models require a lot of computational power and right now we are in the process of replacing New Zealand's supercomputers so I'll discuss these too.
The Dirichlet-to-Neumann operator on rough domains

Tom ter Elst

University of Auckland

Date: Tuesday 31 October 2017

We consider a bounded connected open set $\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite $(d-1)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form methods. The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on $L_2(\Gamma)$. We show that the asymptotic behaviour of $S_t$ as $t \to \infty$ is related to properties of the trace of functions in $H^1(\Omega)$ which $\Omega$ may or may not have. We also show that they are related to the essential spectrum of the Dirichlet-to-Neumann operator.

The talk is based on a joint work with W Arendt (Ulm).
Why does the stochastic gradient method work?

Matthew Parry

Department of Mathematics and Statistics

Date: Tuesday 24 October 2017

The stochastic gradient (SG) method seems particularly suited to the numerical optimization problems that arise in large-scale machine learning applications. In a recent paper, Bottou et al. give a comprehensive theory of the SG algorithm and make some suggestions as to how it can be further improved. In this talk, I will briefly give the background to the optimization problems of interest and contrast the batch and stochastic approaches to optimization. I will then give the mathematical basis for the success of the SG method. If time allows, I will discuss how the SG method can also be applied to sampling algorithms.
The changing face of undergraduate mathematics education: a US perspective

Rachel Weir

Allegheny College, Pennsylvania

Date: Monday 16 October 2017

##Note day and time of this seminar##
A common theme in the United States in recent years has been a call to increase the number of graduates in STEM (science, technology, engineering, and mathematics) fields and to enhance the scientific literacy of students in other disciplines. For example, in the 2012 report Engage to Excel, the Obama administration announced a goal of "producing, over the next decade, 1 million more college graduates in STEM fields than expected under current assumptions." Achieving these types of goals will require us to harness the potential of all students, forcing us to identify and acknowledge the barriers encountered by students from traditionally underrepresented groups. Over the past few years, I have been working to understand these barriers to success, particularly in mathematics. In this talk, I will share what I have learned so far and how it has influenced my teaching.
Investigating the neighbourhood of a heterodimensional cycle

Gemma Mason

University of Auckland

Date: Tuesday 10 October 2017

We investigate a four-dimensional ordinary differential equation model for intracellular calcium dynamics. This model exhibits a heterodimensional cycle, which is a heteroclinic connection between two saddle-periodic orbits whose corresponding stable manifolds are of different dimensions. This heterodimensional cycle is a hallmark of nonhyperbolic dynamics, an exciting frontier in theoretical dynamical systems. Observing it in a model with practical applications opens up new connections between applied mathematics and this emerging theoretical knowledge. We wish to examine how the heterodimensional cycle in the flow of this differential equation affects the nearby dynamics. To observe the behaviour of this practical example, we require computational tools and visualisation techniques for invariant manifolds of periodic orbits in four dimensions. We present a three-dimensional Poincaré section, which provides an overview of the model's behaviour. Projections of the four-dimensional flow into three dimensions are then used to explain the behaviour of objects in the Poincaré section in more depth.
Project presentations

Honours and PGDip students

Department of Mathematics and Statistics

Date: Friday 6 October 2017

Jodie Buckby : ~~Model checking for hidden Markov models~~
Jie Kang : ~~Model averaging for renewal process~~
Yu Yang : ~~Robustness of temperature reconstruction for the past 500 years~~

Sam Bremer : ~~An effective model for particle distribution in waterways~~
Joshua Mills : ~~Hyperbolic equations and finite difference schemes~~
Gems of Ramanujan and their lasting impact on mathematics

Ken Ono

Emory University; 2017 NZMS/AMS Maclaurin Lecturer

Date: Thursday 5 October 2017

##Note venue of this public lecture##
Ramanujan’s work has had a truly transformative effect on modern mathematics, and continues to do so as we understand further lines from his letters and notebooks. In this lecture, some of the studies of Ramanujan that are most accessible to the general public will be presented and how Ramanujan’s findings fundamentally changed modern mathematics, and also influenced the lecturer’s work, will be discussed. The speaker is an Associate Producer of the film ~~The Man Who Knew Infinity~~ (starring Dev Patel and Jeremy Irons) about Ramanujan. He will share several clips from the film in the lecture.

Biography: Ken Ono is the Asa Griggs Candler Professor of Mathematics at Emory University. He is considered to be an expert in the theory of integer partitions and modular forms. He has been invited to speak to audiences all over North America, Asia and Europe. His contributions include several monographs and over 150 research and popular articles in number theory, combinatorics and algebra. He received his Ph.D. from UCLA and has received many awards for his research in number theory, including a Guggenheim Fellowship, a Packard Fellowship and a Sloan Fellowship. He was awarded a Presidential Early Career Award for Science and Engineering (PECASE) by Bill Clinton in 2000 and he was named the National Science Foundation’s Distinguished Teaching Scholar in 2005. In addition to being a thesis advisor and postdoctoral mentor, he has also mentored dozens of undergraduates and high school students. He serves as Editor-in-Chief for several journals and is an editor of The Ramanujan Journal. He is also a member of the US National Committee for Mathematics at the National Academy of Science.
Jensen polynomials for Riemann's Xi-function and suitable arithmetic sequences

Ken Ono

Emory University; 2017 NZMS/AMS Maclaurin Lecturer

Date: Thursday 5 October 2017

##Note day and time of this seminar##
In 1927 Pólya proved that the Riemann Hypotheses is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$ which allows us to prove the hyperbolicity of 100% of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.
This is joint work with Michael Griffin, Larry Rolen and Don Zagier.
Gravitational collapse, cosmic censorship and the Penrose inequality: A geometrical perspective

Jörg Frauendiener

Department of Mathematics and Statistics

Date: Tuesday 26 September 2017

In 1965 Penrose proved the first singularity theorem for gravitational fields which states that under rather general conditions too much gravitational matter and energy confined into a small volume will contract and ultimately collapse to a singularity. He formulated the cosmic censorship hypothesis according to which such singularities have no influence on the surrounding world, they are hidden behind a horizon. Assuming the validity of his hypothesis he was later able to derive an inequality relating the area of the horizon with the total mass contained in space. If there were situations in which the Penrose inequality was violated then it would show that there is no cosmic censorship. Until today, no violations were found, but it has also not been possible to give an independent proof of the Penrose inequality in full generality.
In this talk the origin of the Penrose inequality and some attempts and special cases of its proof will be discussed in more detail.
The nonlinear stability of Minkowski spacetime for self-gravitating massive field

Philippe LeFloch

Université Pierre et Marie Curie

Date: Tuesday 19 September 2017

##A joint seminar with the Department of Physics##
This lecture will present recent work on a class of partial differential equations arising in mathematical physics in the context of Einstein's theory of gravity. Specifically, I will consider the question of the nonlinear stability of Minkowski spacetime and I will review the global evolution problem for self-gravitating massive matter. The presentation will kept at an introductory level, accessible to students and non-experts.
Waves around us. An applied mathematician's perspective

Philippe LeFloch

Université Pierre et Marie Curie, visiting William Evans fellow

Date: Wednesday 13 September 2017

##Note time and venue of this public lecture##
Waves surround us, and many technological advances were made possible only because engineers, physicists, and applied mathematicians worked together in order to understand these phenomena. Understanding shock waves was essential to design the modern airliners, which we use to travel. Understanding electro-magnetic waves propagating in space (and time!) was essential to design the GPS global navigation system and allow us to use cell phones. In this lecture, from the perspective of an applied mathematician, I will illustrate with examples the role of mathematics in overcoming practical problems using pioneering works by Leonhard Euler, James Maxwell, and Albert Einstein. Blog:
The random wave model in quantum chaos

Melissa Tacy

Department of Mathematics and Statistics

Date: Tuesday 12 September 2017

As the energy of a quantum system increases we are able to see echoes of the behaviour of the analogous classical system. Of particular interest are those systems whose classical analogue is chaotic. It is conjectured that such systems cannot display any phase space concentration. In the 70s Berry introduced the random waves as a model of quantum chaotic billiards. In this talk I will discuss the conjectures for general quantum systems and some of the recent results on random waves.
A computable error bound for MCMC with a proxy

Richard Norton

Department of Mathematics and Statistics

Date: Tuesday 5 September 2017

The aim is to find the expectation of a function of a random variable, given an unnormalized density function for the random variable.

Markov chain Monte Carlo (MCMC) methods compute a sequence of correlated samples of the random variable, and estimate the expectation by an average over the samples. The error of computing finitely many samples is estimated by computing estimates of the integrated autocorrelation time and variance. Generally, to be accurate, MCMC requires cheap evaluations of the density function.

When the density function is costly (in CPU time) to evaluate then we can try replacing it with a 'proxy' that is cheap to evaluate and perform MCMC on the proxy. But what error does this introduce?

I will present a computable upper bound for this error and apply it to a couple of simple examples.
Mudflow resuspension in estuaries: from shear instabilities to mixing layer

Alice Harang

Department of Marine Science

Date: Tuesday 22 August 2017

Mudflow ressuspension is a key process for sediment transport in estuaries. This numerical study focuses on the behavior of the lutocline (mudflow-water interface) in a shear flow and aims to better understand the mechanism of resuspension of cohesive sediment. Mud flow, or mud partially consolidated, is modeled by an equivalent homogeneous fluid miscible in water, with newtonian or viscoplastic rheology. The focus of the study is on the stability of the interface and the transition to a turbulent mixing layer through shear instabilities. The specificity of this flow lies on the large viscosity of the mud and its threshold to be put in motion, when presenting a viscoplastic feature. A linear stability study assesses the influence of the various parameters of the flow. Then, the non-linear evolution of the flow is studied by using direct numerical simulations (DNS).
Regular random field solutions for stochastic evolution equations

Markus Antoni

Department of Mathematics and Statistics

Date: Tuesday 15 August 2017

In this talk we give an introduction to the theory of stochastic integrals with respect to a Brownian motion β and stochastic evolution equations in $L^p$ spaces. Starting with the stochastic heat equation we uplift the setting to an abstract level and elaborate strategies to find solutions for stochastic equations. More precisely, for a closed operator $A$ on $L^p(U)$ (e.g. the Laplace operator) we consider equations of the form
d X(t) + A X(t) d t = F(t,X(t)) d t + B(t,X(t)) d β(t)
for random fields $X \colon \Omega \times [0,T] \times U \to \mathbb{R}$, where $[0,T]$ is a time interval, $(\Omega,\mathcal{F},\mathbb{P})$ a measure space representing the randomness of the system, and $U$ is typically a domain in $\mathbb{R}^d$ (or again a measure space). We reduce the existence and uniqueness of solutions to a fixed point equation in certain fixed point spaces. To be more precise, we look for mild solutions so that $X(\omega,\cdot,\cdot)$ has values in $L^p(U;L^q[0,T])$ for almost all $\omega \in \Omega$ under appropriate Lipschitz and linear growth conditions on the nonlinearities $F$ and $B$. In contrast to the classical semigroup approach, which gives $X(\omega,\cdot,\cdot) \in L^q([0,T];L^p(U))$, the order of integration is reversed. In combination with concrete examples of stochastic partial differential equations we show that this new approach leads to strong regularity results in particular for the time variable of the random field $X(\omega,t,u)$, e.g. pointwise Hölder estimates for the paths $t \mapsto X(\omega,t,u), \mathbb{P}$-almost surely.
Applications of generalized Polynomial Chaos in ocean wave / sea ice interactions

Johannes Mosig

Department of Mathematics and Statistics

Date: Tuesday 8 August 2017

In this talk I will present a group of computational techniques known as "generalized polynomial chaos" (gPC). These methods can be used to estimate the response of a physical model when parameters are uncertain. Specifically, when some input parameter R of a system is known only in terms of its probability distribution, then gPC methods can be used to efficiently calculate the moments (e.g. expectation and variance) of the output quantities of the model. The results of the gPC methods typically converge exponentially with the number of single model runs / system size.

I will present my Mathematica package which makes it easy to work with gPC methods on any physical model, and demonstrate a few applications in the area of ocean wave/sea ice interactions.
Governing equations for two-state semi-Markov processes: modelling reactive transport in a river with hyporheic exchange

Boris Baeumer

Department of Mathematics and Statistics

Date: Tuesday 1 August 2017

We model the movement of particles through two zones (say river and river bed), where the (random) time spent in one zone (river bed) is not exponentially distributed (the longer a particle is in the zone, the less likely it is that it will be released within the next ten minutes). We take a general probabilistic approach to deduce a system of deterministic partial differential equations (Fokker-Planck) modelling the evolution of the concentration profile. We then take great care to extend this system to incorporate reaction and stochasticity and provide numerics to approximate the solutions.
Singular solutions of the Einstein-Euler equations. PDE structure and physical interpretation

Florian Beyer

Department of Mathematics and Statistics

Date: Tuesday 23 May 2017

Einstein’s equations can be considered as a particular geometric evolution system where the underlying PDEs are essentially of nonlinear wave equation type. Similar to other famous geometric evolution problems, there has been a lot of effort over the last decades to study the formation of singularities of solutions. In general this turns out to be a formidable (and essentially unsolved) task due to the complexity of Einstein’s equations. In this talk I will discuss a recent result by P LeFloch and myself regarding singular solutions of the coupled Einstein-Euler equations. I will explain the analytic PDE aspects of this result and link it to the physical problem of modelling the very early universe.
Theory, modeling, and impact of the floe size distribution of sea ice

Chris Horvat

Harvard University

Date: Tuesday 2 May 2017

Earth's sea ice cover is composed of a myriad of distinct pieces, known as floes, with sizes ranging over many orders of magnitude. The evolution of Arctic climate and ecology is strongly tied to the multi-scale heterogeneity of sea ice, in particular the distribution of these floe sizes. Yet modern climate models still do not simulate the evolution of floes, their size distribution, or important quantities derived from this distribution.

I'll discuss how melt ponding on sea ice floes has dramatically shifted the ecological status quo in the Arctic. Using a combination of simple modeling techniques, observations, and reanalysis products I'll demonstrate that the thinning of sea ice in the past several decades allows for extensive and frequent under-ice phytoplankton blooms, which can have a significant effect on the ecological and carbon cycle in the high latitudes.

I'll also discuss how the thermodynamic evolution of sea ice is determined through the interaction of sea ice floes and ocean eddies, and how these are determined by the floe size distribution. I'll then present a predictive model for the joint statistical distribution of floe sizes and thicknesses (FSTD) which is tested under different forcing scenarios to establish its conservation properties and demonstrate its usability in future climate studies.

Suggested reading:
Horvat, Rees Jones, Iams, Schroeder, Flocco, & Feltham (2017). The frequency and extent of sub-ice phytoplankton blooms in the Arctic Ocean. Science Advances.
Horvat, Tziperman, & Campin (2016) Interaction of sea ice floe size, ocean eddies, and sea ice melting. Geophys. Res. Lett.
Fully pseudospectral time evolution

Jörg Hennig

Department of Mathematics and Statistics

Date: Tuesday 11 April 2017

We demonstrate how time-dependent PDE problems can numerically be solved with a fully pseudospectral scheme that uses spectral expansions with respect to both spatial and time directions. We obtain highly accurate numerical solutions for a moderate number of grid points and observe spectral (i.e. exponential) convergence, a feature known from solving elliptic PDEs with spectral methods. We briefly discuss several applications of this method that we studied in the past couple of years: numerical verification of a universal black hole formula, simulation of Newtonian stellar oscillations, and computation of Gowdy-symmetric cosmological models. Afterwards we look in more detail at an ongoing joint project with Jörg Frauendiener, namely an investigation of the conformally invariant wave equation on a black hole background. This can be considered as a toy model for the propagation of gravitational waves. It turns out that certain logarithmic singularities form at the future boundary, but we still obtain solutions of good accuracy. We also describe how initial data can be chosen that avoid the leading-order logarithmic terms, which further improves the numerical accuracy.
Freedom for Modules

John Clark

Department of Mathematics and Statisitcs

Date: Tuesday 4 April 2017

Let $R$ be a ring with multiplicative identity, but not necessarily commutative. An additive abelian group $M$ is said to be a (left) $\textit{R-module}$ if there is a (left) multiplication map $R \times M\longrightarrow M$ (the product of $r$ from $R$ with $m$ from $M$ is denoted by $rm$) provided, for all $r, s\in R$ and all $m, n\in M$, we have
$$r(sm) = (rs)m, \; (r+s)m = rm + sm,\; \text{ and }\; r(m+n) = rm + rn.$$
Note that this definition generalises that of a vector space $M$ over a field $R$.

In the special vector space setting, linear algebra introduces the important concepts of linearly (in)dependent subsets and generating subsets of a vector space $M$. The same definitions carry over to modules. A pivotal result in linear algebra says that every vector space $M$ has a ${\bf basis}$, i.e., $M$ has a linearly independent subset $B$ which also generates $M$. Moreover any two bases of $M$ have the same size. This unique size, say $n$, is called the ${\bf dimension}$ of $M$ and we write $\dim(M) = n$.

In this talk we’ll look at how the transfer of these and some related results fail for some rings.
Effect of distributed surface roughness on the laminar-turbulent transition

Maciej Floryan

University of Western Ontario

Date: Thursday 30 March 2017

Note venue for this seminar
A joint seminar by the Departments of Mathematics and Statistics, and Physics

It has been recognized since the pioneering experiments of Reynolds in 1883 that surface roughness plays a significant role in the dynamics of shear layers. This is a classical problem in fluid dynamics but, nevertheless, its resolution is still lacking. Most of the efforts have been focused on experimental approaches that have resulted in a number of correlations but have failed to uncover the mechanisms responsible for the flow response. Theoretical analyses have also failed to provide a consistent explanation of the flow dynamics. As there are an uncountable number of possible geometrical roughness forms, the problem formulation represents a logical contradiction as it might not be possible to find a general answer to a problem that has an uncountable number of variations. The recent progress towards the theoretical resolution of this apparent contradiction will be discussed and recent results dealing with the problem of distributed surface roughness will be presented. The progress has hinged on the development of the immersed boundary conditions method and the reduced geometry concept. It will be shown that it is possible to propose a rational definition of a hydraulically smooth surface by invoking flow bifurcations associated with the presence of roughness. Successful resolution of roughness problems gives access to the design of surface roughness for passive flow control where drag reduction can be achieved either directly, through re-arrangement of the form of the flow that results in the reduction of the shear stress, or indirectly, through delay of the laminar-turbulent transition.
The geometry of diversities

David Bryant

Department of Mathematics and Statisitcs

Date: Tuesday 21 March 2017

The field of metric geometry was wrenched from mathematical obscurity 20 years ago by the discovery that some really hard optimisation problems in graph theory could be solved, at least approximately, by converting them into metric embedding problems. Recently, we have found that many celebrated results for metric embeddings can be generalised, in a satisfying way, to diversities. A diversity is like a metric except that it defines values for all finite subsets, not just all pairs. The mathematics of diversities may still be obscure, but we've found it to be rich and multifaceted, with connections and applications in all sorts of places.
Mass-loss due to gravitational waves

Vee-Liem Saw

Department of Mathematics and Statisitcs

Date: Tuesday 28 February 2017

The theoretical basis for the energy carried away by gravitational waves that an isolated gravitating system emits was first formulated by Hermann Bondi during the 1960s. Recent findings from looking at distant supernovae revealed that the rate of expansion of our universe is accelerating (Nobel Prize in Physics, 2011), which may be well-explained by sticking in a positive cosmological constant into the Einstein field equations for general relativity. By solving the Newman-Penrose equations (which are equivalent to the Einstein field equations), we generalise this notion of Bondi-energy and thereby provide a firm theoretical description of how an isolated gravitating system loses energy as it radiates gravitational waves, in a universe that expands at an accelerated rate. This is in line with the observational front of LIGO's first announcement in February 2016 that gravitational waves from the merger of a binary black hole system have been detected.
An entropy-based measure for comparing distributions

Rajeev Rajaram

Kent State University, Ohio

Date: Tuesday 7 February 2017

NOTE venue is not our usual
In this talk, I will develop an entropy-based measure called case-based entropy which can be used to compare the diversity of distributions. The measure is based on computing the support of a Shannon-equivalent equi-probable distribution. It also has the capacity to compare whole or parts of distribution in a scale-free manner. I will develop the main idea from scratch and will keep the talk accessible to graduate students and researchers alike. The utility of the measure is still being explored, but one of the latest uses that I found is its use in economics as a better method to compare income or wealth inequality than the Gini index, for example. I have also used the measure to compare the diversity of complexity in a variety of distributions from the velocities of galaxies to the energy distribution of Maxwell Boltzmann, Bose-Einstein and Fermi-Dirac distributions.
3D-Dynamic Geometry Systems for learning and instruction

Olaf Knapp

University of Education Weingarten, Germany

Date: Wednesday 25 January 2017

Note day, time and venue of this special seminar
3D-dynamic geometry systems (3D-DGS) with graphical user interfaces can open up a wide range of creative activities for mathematics education. Examples will be given from a research project into their use for didactic presentations, visualization, synthetic geometry, 3D modelling, morphing and mapping, design, and analogization. They allow interactive exploration of mathematical concepts. I will show their potential to become an integral part of mathematics education and to modify the curriculum. There is also the opportunity for hands-on experience with a 3D-DGS.