Jonathan Michael Borwein (20 May 1951 - 2 Aug 2016) had many talents, among which were his abilities to make discoveries in mathematics, to seek tenaciously for proofs of these, and to do both of those things in collegial concert with other workers. In this colloquium I shall give three examples of situation in which I had the pleasure of seeing those talents in action. They concern multiple zeta values, walks on lattices, and modular forms. In each case I shall give a notable identity, comment on its proof, and indicate further work that was provoked by the discovery. The identities in question are chosen to be comprehensible to anyone with an undergraduate education in mathematics and also to people, like myself, who lack that particular qualification.
Mathscraft is a workshop for junior high school students that aims to give them the experience of doing maths the way research mathematicians do. It is coordinated and sponsored by the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), and sessions are conducted by Anthony Harradine, (Prince Alfred College, Adelaide).
In a Mathscraft session there are up to 10 groups, each comprising three students (years 7-10), one teacher and one mathematician. The teams are given mathematical problems and are guided through a problem-solving process. The problems and the process are designed to mimic the mathematics that is done by research mathematicians - exploring, noticing patterns, making conjectures, proving them, figuring out why, and thinking of ways to extend the problem.
In this talk I'll describe the design of the problems and process (with examples), and explain the motivations behind them. I'll also talk about a Professional Development workshop that we ran for teachers in November last year, which had the aim of training them to run Mathscraft sessions in their own local areas. This workshop was sponsored by ACEMS and MATRIX.
In this talk, we discuss our new approach to design reverse logistics models for dairy industries, in particular whey products. Whey is a by-product of cheese making with many applications spanning from dairy and meat to pharmaceuticals. We develop a hierarchical location-routing model for a whey recovery network design. In this class of models, the location and routing decisions are made simultaneously. As the problem is NP-hard, it may not be possible to solve even the small-size instances efficiently. We suggest different approaches such as adding valid inequalities and improving lower and upper bounds to solve the problem in a reasonable amount of time.
In 1975, culminating more than 40 years of published work by Paul Erdos on the problem, he and John Selfridge proved that the product of consecutive integers cannot be a nonzero perfect power. Their proof was a remarkable combination of elementary and graph theoretic arguments. Subsequently, Erdos conjectured that this result can be generalized to a product of consecutive terms in an arithmetic progression, under certain basic assumptions. In this talk, we discuss joint work with Samir Siksek in the direction of proving Erdos' conjecture. Our approach is via techniques based upon the modularity of Galois representations, bounds for the number of supersingular primes for elliptic curves, and analytic estimates for Dirichlet character sums.
The talk will include general information on the current state of plasma fusion as an energy source and some more detailed aspects of this research area.
Gamiﬁcation refers to the use of elements of games in non-game contexts and has been applied in workplace, marketing, health programs and other areas, with mounting evidence of increased interest, involvement, satisfaction and performance of the participants. More recently gamiﬁcation has been emerging as a teaching method that has a great potential to improve students’ motivation and engagement. Gamiﬁcation in education should not be confused with playing educational games, as it only uses concepts such as points, leader boards, etc, rather than computer games themselves. In this talk we describe the gamiﬁcation of a theoretical computer science course we performed in 2014/2015/2016 as well as our experience with two other STEM courses.
There is to date no overarching classification theorem for C*-algebras, which means the theory of C*-algebras is an example-driven field of mathematics. Perhaps the most important class of examples are group C*-algebras, which are as old as the field itself. An analogous construction of C*-algebras associated to semigroups has been an active area of research among operator algebraists since Coburn’s Theorem regarding the universality of the C*-algebra generated a single isometry appeared in the 1960s. In July this year, Newcastle will host the AMSI/AustMS sponsored event "Interactions between operator algebras and semigroups". In this talk I will give a gentle introduction to the theory of semigroup C*-algebras and perhaps it will convince some of you to come along and take part in the meeting.
In this talk, I will describe the conditional value at risk (CVaR) measure used in modelling risk aversion in decision making problems.
CVaR is a highly consistent risk measure for modelling risk aversion.
I will then present two applications of CVaR. The first application considers all problems that are representable by decision trees. In this application, I show that these problems under the CVaR criterion can be solved efficiently by solving a linear program. In the second application, I consider a basic problem in the area of production planning with random yield. For this problem, I present a risk aversion model. The model is nonconvex. I present an efficient locally optimal solution method and then provide a sufficient optimality condition.
In this talk, we discuss a new approach to demand forecasting in supply chains. Demand forecasting is an inevitable task in supply chain management. Due to the endogenous and exogenous factors impact a supply chain, the regime of the supply chain may vary significantly. Such changes in the regime can bring a high volatility to demand time series and consequently, a single statistical model may not suffice to forecast the demand with a desirable level of precision. We develop a nexus between stochastic processes and statistical models to forecast the demand in supply chains with regime switching. The preliminary results on real world time series data sets are promising.
This talk gives an outline of (mostly unfinished) work done collaboratively while on sabbatical in semester 2 last year. Join me as we travel through the USA, Germany, Belgium and Austria. Your guide will share off-the-beaten-track highlights such as quaternionic splines, prolate shift systems, higher-dimensional Hardy, Paley-Wiener and Bernstein spaces, the Clifford Fourier transform, multidimensional prolates, and a Jon Borwein-inspired optimization-based approach to the construction of multidimensional wavelets. Breakfast not included.
In recent joint work on equilibrium states on semigroup C*-algebras with Afsar, Brownlowe, and Larsen, we discovered that the structure of equilibrium states admits an elegant description in terms of substructures of the original semigroup. More precisely, we consider two almost contrary subsemigroups and related features to obtain a unifying picture for a number of predating case studies. Somewhat surprisingly, all the examples from the case studies satisfy a list of four abstract properties (and are then called admissible). The nature and presence of these properties is yet to be fully understood. In this talk, I will focus on a class of examples arising as Zappa-Szép products of right LCM semigroups which showcases some interesting features. No prerequisites in operator algebras are required to follow this talk.
Totally disconnected, locally compact (t.d.l.c.) groups are a large class of topological groups that arise from a few different sources, for instance as automorphism groups of a range of algebraic and combinatorial structures, or from the study of isomorphisms between finite index subgroups of a given group. A general theory has begun to emerge in recent years, based on the interaction between small-scale and large-scale structure in t.d.l.c. groups. I will give a survey of some ways in which these groups arise and some of the tools that have been developed for understanding them.
In a way, mathematics can be seen as a language game, where we use symbols, together with some rewriting rules, to represent objects we are interested in and then ask what can be said about the sequences of symbols (languages) that capture certain phenomena. For example, given a group G with generators a and b, can we recognise (using a computer) the sequences of generators that correspond to non-trivial elements of G? If yes, how strong computer do we need, i.e. how complicated is the language we are studying?
There is a natural duality between various types of computational models and classes of languages that can be recognised by them. Until recently most problems/languages in group theory were classified within the Chomsky hierarchy, but there are more computational models to consider. In the talk I will briefly introduce L-systems, a family of classes of languages originally developed to model growth of algae, and show that the co-word problem in Grigorchuk's group, a group of particularly nice transformations of infinite binary tree, can be seen as a language corresponding to a fairly simple L-system.
Problem solving, communication and information literacy are just a few graduate attributes that employers value, yet it commonly appears that students upon graduating show only limited improvement in these areas. For instance, 3rd year students can still be thrown by relatively simple unfamiliar problems, even after working actively on numerous related exercises and problems throughout their degree. I will discuss some of the things I have implemented in my teaching to specifically target the development of student graduate attributes. My experience is with teaching mathematics, physics and engineering students, however much of my discussion will be non-discipline-specific.
I am going to look at three unsolved graph theory problems for which the same family of graphs presents a barrier to either solving or making substantial progress on the problems. The graphs in this family are called honeycomb toroidal graphs. The three problems are not closely related.
In this talk we consider a class of monotone operators which are appropriate for symbolic representation and manipulation within a computer algebra system. Various structural properties of the class (e.g., closure under taking inverses, resolvents) are investigated as well as the role played by maximal monotonicity within the class. In particular, we show that there is a natural correspondence between our class of monotone operators and the subdifferentials of convex functions belonging to a class of convex functions deemed suitable for symbolic computation of Fenchel conjugates which were previously studied by Bauschke & von Mohrenschildt and by Borwein & Hamilton. A number of illustrative computational examples utilising the introduced class of operators will be provided including computation of proximity operators, recovery of a convex penalty function associated with the hard thresholding operator, and computation of superexpectations, superdistributions and superquantiles with specialization to risk measures.
In this talk we discuss a new approach for the Hamilton cycle problem (HCP). The HCP is one of the classical problems in combinatorial mathematics. It can be stated as given a graph G, find a cycle that passes through every single vertex exactly once, or determine that such a cycle does not exist. In 1994, Filar and Krass developed a new model for HCP by embedding this problem into a Markov decision process. This approach was the motivation of a new line research which was extended by several other people afterwards. In this approach, a new polytope corresponding to a given graph G was constructed and searching for Hamiltonian cycles in a given Hamiltonian graph G was converted to searching for particular extreme points (called Hamiltonian extreme points) among extreme points of that polytope. In this research, we are going to design a Markov chain with certain properties to sample Hamiltonian extreme points of that polytope. More precisely, we would like to study a specific class of input graphs, the so-called random graphs. Some preliminary theoretical results are presented in this talk.
Part of my 2016 SSP included completion of a semi-historical review on the mathematics of W.N. Bailey, a familiar name in some combinatorics circles in relation with the "Bailey lemma" and "Bailey pairs." My personal encounters with the mathematician from the first half of the 20th century were somewhat different and more related to applications of special functions to number theory—the subject Bailey had never dealt with himself. One motivation for my writing was the place where I spent my SSP—details to be revealed in the talk. There will be some formulas displayed, sometimes scary, but they will serve as a background to historical achievements. Broad audience is welcome.
Control Lyapunov functions (CLFs) for the control of dynamical systems have faded from the spotlight over the last years even though their full potential has not been explored yet. To reactivate research on CLFs we review existing results on Lyapunov functions and (nonsmooth) CLFs in the context of stability and stabilization of nonlinear dynamical systems. Moreover, we highlight open problems and results on CLFs for destabilization. The talk concludes with ideas on Complete CLFs, which combine the concepts of stability and instability. The results presented in the talk are illustrated and motivated on the examples of a nonholonomic integrator and Artstein's circles.
We determine the Borel complexity of the topological isomorphism problem for profinite, t.d.l.c., and Roelcke precompact non-Archimedean groups, by showing it is equivalent to graph isomorphism.
For oligomorphic groups we merely establish this as an upper bound.
Joint work with Kechris and Tent.
The research interest in pattern avoiding permutations is inspired by Donald Knuth’s work in stack-sorting. According to Knuth, a permutation can be sorted by passing through a single infinite stack if and only if it avoids a sub-permutation pattern 231. Murphy extended Knuth’s research by using two infinite stacks in series and found out that the basis for generated permutations is infinite but Elder proved that the basis is finite when one of the stack is limited to depth two and the permutations are algebraic. My research is to investigate the permutations generated by a stack of depth 3 and an infinite stack in series. It is to determine the basis and nature of the permutations in term of formal language.
Lyapunov's second or direct method provides an easy-to-check sufficient condition for stability properties of equilibria. The converse question - given a stability property, does there exist an appropriate Lyapunov function? - has been fundamental in differentiating and classifying different stability properties, particularly with regards to "uniform" stability.
In this talk, I will review the usual textbook definitions for Lyapunov functions for time-varying systems and describe where they are deficient. Some interesting new sufficient (and probably necessary) conditions pop up along the way.
The theory of minimal surfaces (a.k.a. soap films) goes back to Euler’s discovery in 1741 that the catenoid is area-minimising. It is still a remarkably vibrant area of research. I will describe recent joint work with Franc Forstneric of the University of Ljubljana, Slovenia. We assemble all minimal surfaces with a given shape into a space. It is an infinite-dimensional space. What does it look like? We have been able to determine its "rough shape". I will explain what we mean by "rough shape" and describe the ingredients from complex analysis, differential topology, and homotopy theory that go into our result.
This presentation will outline my research into fitness for purpose of tertiary algebra textbooks used in Iraq in the teaching of undergraduate algebra courses with regard to the training of pre-service teachers. The project draws on work done in textbook analysis, and work done into the teaching and learning of abstract algebra and the nature of proof.
It is well recognised that for many students learning abstract algebra and the nature of proof is difficult (Selden, 2010). Courses in abstract algebra are central to many tertiary pre-service mathematics teacher programs, including in Iraq. Capaldi (2012) suggests that abstract algebra textbooks can lay the foundation for a course and greatly influence student understanding of the material. However, it has been found that there can be large differences in textbooks used, at the school level at least, in different cultures. (Alajmi A. H., 2012, Fan and Zhu, 2007, Pepin and Haggarty, 2001). For instance, Mayer and Sims, (1995) Japanese texts feature many more worked out examples than texts used in the United States for mathematics.
I will be examining the textbooks in light of theories by Harel and Sowder, and Stacey and Vincent, regarding types and proof and modes of reasoning (Stacey and Vincent, 2009) and Capaldi (2012) regarding reader's relationships with books.
The textbooks will also be examined to try to infer the underlying assumptions about pedagogies and knowledge made by the author(s). Baxter-Magolda's theory, linking forms of assessment to underlying theories of knowledge (Baxter-Magolda, 1992) will be helpful in this pursuit.
The aim of this workshop is to bring together the world's foremost experts on the theory of semigroups and their relationships to other fields of mathematics such as operator algebras and totally disconnected locally compact groups. This workshop will allow the international leaders in the field to come to Australia to teach young Australian ECRs, and to forge new collaborations with Australian mathematicians.
Details are available on the conference website.
Operator algebras associated to semigroups can be traced back to a famous theorem of Coburn from the 1960s. The theory has recently been reinvigorated through Xin Li's construction of semigroup C*-algebras. Li's construction has introduced new and interesting classes of C*-algebras, which have deep connections to number theory and dynamical systems. One connection that will be thoroughly explored through this meeting is that to the representation theory of totally disconnected locally compact groups.
I will discuss how to relate regular origami tilings to vertex models in statistical mechanics. The Miura-ori origami pattern has found many uses in engineering as an auxetic metamaterial. I analyze the effect of crease assignment defects on the long-range order properties of the Miura-ori and 4 other foldable lattices. These defects are known to affect the material's compressibility properties, so my exact results help to understand how easy it is to tune an origami metamaterial to have desired compressibility properties by introducing a set density of defects. I have found that certain origami patterns are more easily tunable than others, and conversely, the long-range ordering of some are more stable with respect to defect formation. I have also found analytical expressions for the locations of phase transition points with respect to crease assignment ordering as well as layer ordering.
Colour images are represented by functions of 2 variables that output 3 variables, and analysing them requires tools that can handle these dimensions. One method is to use Clifford Algebras and their recently discovered Fourier Transform. We prove the Clifford Fourier Transform has a Hardy Space, and that it's Paley-Wiener Space and Bernstein Spaces are identical. Another method is to find 2 dimensional wavelets that are non-separable. We achieve this through the use of the Douglas-Rachford Projection Algorithm, and hope to achieve it through the use of Proximal Alternating Linear Methods. This talk briefly overviews these methods and the path to completion.
In this talk I will briefly introduce the mixed finite element method and show their applications. I consider Poisson, elasticity, Stokes and biharmonic equations for the applications of the mixed finite element method. The mixed finite element method also arises naturally in Stokes flow, multi-physics problems as well as when we consider non-conforming discretisation techniques. I will also present my recent works on the mixed finite element method for biharmonic and Reissner-Mindlin plate equations.
I will present a brief survey of some recent results that deal with the characterization of hyperbolic dynamics in terms of the existence of appropriate Lyapunov functions. The main novelty of these results lies in the fact that they consider noninvertible and infinite-dimensional dynamics. This is a joint work with L. Barreira, C. Preda and C. Valls.
Given a sequence of integers, one would like to understand the pattern which generates the sequence, as well as its asymptotics. If the sequence is viewed as the coefficients of the series expansion of a function, called its generating function, many questions regarding the sequence can be answered more easily. If the generating function satisfies a linear ODE or a nonlinear algebraic DE, the differential equation can be found if enough terms in the sequence are given. In this talk I'll discuss my implementation in C of such a search, applications, and a systematic search of the entire Online Encyclopedia of Integer Sequences (OEIS) for generating functions.
Schoenberg’s polynomial cardinal B-splines of order $n$ provide a family of compactly supported $C^{n-2}$-functions. We present several generalizations of these B-splines, discuss their properties, and relate them to fractional difference and differentiation operators. Potential applications are mentioned.
We consider variations on the commutative diagram consisting of the Fourier transform, the Sampling Theorem and the Paley-Wiener Theorem. We start from a generalization of the Paley-Wiener theorem and consider entire functions with specific growth properties along half-lines. Our main result shows that the growth exponents are directly related to the shape of the corresponding indicator diagram, e.g., its side lengths. Since many results from sampling theory are derived with the help from a more general function theoretic point of view (the most prominent example for this is the Paley-Wiener Theorem itself), we motivate that a closer examination and understanding of the Bernstein spaces and the corresponding commutative diagrams can—via a limiting process to the straight line interval [−A,A]—yield new insights into the Lp(R)-sampling theory. This is joint work with Gunter Semmler, Technische Universität Bergakademie Freiberg, Germany.
One of the most contentious areas in Indigenising Curriculum is the Maths and Sciences. This presentation considers how Maths and Statistics can provide a solid and meaningful response to the Indigenising imperative that will fulfil the two criteria of socially just education:
Suggestions on both content areas and student recruitment, retention and success will be discussed. Examples will be based on the presenters experiences as cultural facilitators in education from Foundations to the tertiary sector.
Associate Professor Kathy Butler and Ms Tammy Small are employed in the Office of the Pro Vice-Chancellor Indigenous Education and Research at the University of Newcastle. With Professor Steve Larkin, Tammy and Kathy are currently examining ways for the University to provide cultural competency training as a whole-of-university initiative.
This presentation is to assist academics consider how to adapt programs and course content and delivery to incorporate, be mindful of and better appeal to people with Indigenous backgrounds and interests.
We present $h$ and $p$-versions of the time domain boundary element method for boundary and screen problems for the wave equation in $\mathbb{R}^3$. First, graded meshes are shown to recover optimal approximation rates for solution in the presence of edge and corner singularities on screens. Then an a posteriori error estimate is presented for general discretizations, and it gives rise to adaptive mesh refinement procedures. We also discuss preliminary results for $p$ and $hp$-versions of the time domain boundary element method. Numerical experiments illustrate the theory. Joint with H. Gimperlein and D. Stark, Heriot-Watt University, Edinburgh.
The Chebyshev conjecture is a 59-year-old open problem in the fields of analysis, optimisation, and approximation theory, positing that Chebyshev subsets of a Hilbert space must be convex. Inspired by the work of Asplund, Ficken and Klee, we investigate an equivalent formulation of this conjecture involving Chebyshev subsets of the unit sphere. We show that such sets have superior structure and use the Radon-Nikodym Property to extract some local structural results about such sets.
This presentation will discuss the megatrends, both technological and societal, that are impacting the modern supply chain. In particular, the balance between people and machines will be explored in the context of future of work within supply chains. What are the appropriate roles for robotics within the supply chain of the future? What is the future for people in the supply chain? Examples of existing and emerging technologies will be presented to show that the future supply chain is close at hand.