# **Postgraduate Seminar**

The aim of the seminar is to give postgraduate students an opportunity to socialise and share their interests with their peers. As such the seminar is a mix of afternoon teas and talks.
The seminar is coordinated by grads for grads. We welcome expressions of interest in giving a talk. If you have any comments on how we can make the most of the series, then we are all ears!
Please use the contact form here to leave feedback or get in touch with us. If you provide contact details, we’ll get back to you as soon as possible.
## **Upcoming talks/afternoon tea**

Date | Time | Location | Description |
---|---|---|---|

Monday 20 November 2023 | 2pm | CO431 and Zoom on request | Speaker: Dr Louise McMillan Title: How I learned to stop worrying and love statistics: Why it's ok to leave academia and come back to it later Abstract: I am a statistician now, but I started as a mathematician who hated stats. I had no interest in biology when I was 17, but now I'm an ecological statistician. I'm an academic with a PhD, but as late as a year before the start of my PhD I had decided that I would never do a PhD. How did I get here? I've worked in industry as well as academia, and I've enjoyed both. What did I learn from them? Come along if you don't know what to do with your life and are worried people will judge you for it. Come along if you, like me, find it hard to choose between all the different topics that interest you. Come along if you want to hear about rats and whales, Mars landers and space telescopes, and the delights of life as a consultant. |

Monday 27 November 2023 | Cancelled because of upcoming NZMASP conference the same week. |

## ** Calendar**

## ** Previous talks**

Date | Description |
---|---|

Monday 13 November 2023 | Speaker: Timotheus Keanu Title: Mordell-Weil Groups over Elliptic Curves Abstract: Elliptic curves is a type of algebraic curves that has many applications in number theory due to its arithmetic nature. They have lots of interesting properties. For example, the Mordell-Weil theorem tell us that an elliptic curve over the rational numbers is a finitely generated abelian group. In this talk, we will explore this result in more detail and discuss the challenges faced in the current research. |

Monday 30 October 2023 | Speaker: Aleksa Vujičić Title: Academic Experience Post-VUW Desription: Q&A to discuss differences between graduate life at VUW and overseas. |

Monday 16 October 2023 | Speaker: Grace Jacobs Corban Title: Hexaflexagons are the bestagons Description: Lets flexigate our way to happiness with the most fun you can have on six sides! In this seminar we will introduce the most influential mathematical object of the 20th century, that some call Feynman's greatest contribution to humanity, and go over how to make a 3-sided, 6-sided, and maybe 12-sided hexaflexagon. Hexaflexagons are paper folding toys, where 'flexing' reveals a new side and a new colour. How does this work? Where do the colours come from?!?! Come along to find out 🙂 |

Monday 2 October 2023 | Speaker: Grace Jacobs Corban Title: Maths anxiety; a proof by induction Abstract: Maths anxiety is an issue that affects New Zealand students' math performance and wellbeing and can have long-lasting effects throughout their lives, impacting career choices. What is clear is that maths anxiety is a societal issue, and that it continues to be perpetuated (by parents, teachers, and media). What is less clear is where it originated. In this we will explore the current state of maths anxiety, as well as how it continues to be an issue, and investigate its potential origin. |

Monday 18 September 2023 | Speaker: Sam Bastida Title: The Mathematics of Juggling Abstract: A mathematical model of juggling, known nowadays as site swaps, was developed around 1985 independently by groups in both the United States and United Kingdom. In this model numbers are assigned to the common throws jugglers use and a sequence of such numbers describes a juggling pattern or site swap. Not all site swaps are valid and the sequences that are valid correspond to affine permutations which are relevant for the construction of affine symmetric groups. In this talk I will discuss some of the mathematical ideas behind site swaps as well as exploring the various ways they have been extended to apply to different juggling tricks, I will also give some live demonstrations. |

Monday 4 September 2023 | Speaker: Mashfiqul Huq Chowdhury Title: Variational Inference, Latent Representation and Clustering Abstract: In unsupervised learning, we aim to find the hidden structure within the data. In this talk, I will concentrate on modeling the distribution of unsupervised data using the variational inference technique. I will explain, how to simultaneously learn latent representation and clustering by leveraging a deep neural network architecture. The variational autoencoder (VAE) can be used effectively for both of these purposes. I will conclude my discussion after demonstrating an application of the VAE model. |

Monday 26 June 2023 | Speaker: Mashfiqul Huq Chowdhury Title: EM Algorithm and Gaussian Mixture Model Abstract: We usually aim to fit a suitable probability distribution to the observed data using maximum likelihood estimation (MLE). In practice, the complex structure of the data motivates us to include latent variables in our model. However, finding the MLE of parameters, in this case, is challenging because it involves non-convex optimization. In such a setting, the EM algorithm provides an efficient approach for finding the MLE of the parameters by constructing the lower bound (ELBO) of the likelihood of the data. The EM algorithm iteratively optimizes ELBO objective function until convergence. The Gaussian mixture models (GMM) are widely used for unsupervised clustering tasks in Machine learning. The application of the EM algorithm on the GMM will also be discussed. This talk will finish by showing an illustration of GMM for clustering purposes. |

Tuesday 20 June 2023 | Speaker: Shonaugh Wright Title: Topological properties of graded groupoids Abstract: A groupoid is a generalisation of a group. I will introduce the definition of a groupoid with some examples, mainly looking at second-countable, locally compact, Hausdorff, étale groupoids. Then I will discuss the identity graded groupoid and some of the properties that it inherits from the original groupoid. I will finish by discussing how to construct a C*-algebra from a groupoid. |

Monday 1 May 2023 | Speaker: Mark Bishop Title: Effects of observed projections on turbulence statistics in the intracluster medium Abstract:The total mass of a cluster is one of its most fundamental properties. Measurements of the galaxy cluster mass often relies upon assuming hydrostatic equilibrium. However, this is often invalidated as the intracluster medium (ICM) is continuously disturbed by mergers, feedback processes, and motions of galaxies. These processes generate gas motions that contribute nonthermal pressure; typically turbulence, that leads to an underestimation of the mass by as much as 30\%. We can measure turbulence through indirect probes that come in the form of fluctuations in the X-ray surface brightness and Sunyaev-Zeldovich effect maps. These are projected characteristics, encoding the 3D structure of the turbulence in the ICM. This project will be an analysis of the 3D to 2D projections of the intracluster medium and its effect on the retrieved statistical measures commonly used in turbulence analysis like the power spectrum by using numerical simulations appropriate to galaxy clusters. |

Tuesday 18 April 2023 | Special Event: Dr. Brendan Harding on MathJobs and more. Abstract: I'll give an overview of mathjobs and how to use it to find and apply for jobs. I'll also give some tips on putting together an application, including things to focus on in your cover letter and some suggestions for building an academic CV. I'll show some examples of applications I have submitted in the past. |

Monday 13 March 2023 | Speaker: Jago Edyvean Title: Hermite Transforms and Plasma Turbulence Abstract: Transforms are a useful and powerful method for investigating and understanding systems. In collisionless plasma turbulence, the governing equation. the Vlsov Equation, is very hard to solve numerically and so transforms reduce the complexity of this. Transforming the Vlasov equation also yields interesting physical interpretations of the system. The Hermite transform is one such method. |

Monday 13 February 2023 | Speaker: Daniel Wrench Title: Intro to Bayesian Abstract: The Bayesian approach is a very popular way of doing statistical inference and rests upon a simple theorem of conditional probability. Daniel will be discussing the basics of this mathematical method of updating your beliefs in light of new information. |

Monday 30 January 2023 | Speaker: Sapir Ben-Shahar Title: Introduction to Computable Metric Spaces Abstract: This talk will begin with an introduction to metric spaces and some notions of computability, including presentations of spaces, and computably enumerable (c.e) sets. |

Monday 14 November 2022 | Speaker: Ellen Hammatt Title: Punctually 1-Decidable Boolean Algebras Abstract: In my talk I will discuss my recent work on online (punctual) presentations of boolean algebras. The main result is: There exists a 1-decidable boolean algebra such that is not computably isomorphic to any punctually 1-decidable boolean algebra. All terms and definitions will be clarified in the talk. |

Monday 7 November 2022 | Speaker: Ben Roberts Title: The Khmaladze Transformations Abstract: The empirical process is the central object of goodness of fit statistical testing, with test statistics being chosen as functionals of this process. Clever construction of statistics, such as the chi-squared statistic, have the additional property of being asymptotically distribution free; their limit distribution does not depend on the particular hypothesis under consideration. An intriguing alternative is to instead transform the empirical process into some other asymptotically distribution free process. A broad class of test statistics (functionals) of this new process will then be automatically asymptotically distribution free. Such a transformation was introduced in 1981 through the Khmaladze transformation, we will consider this and another recent alternative from 2016, the Khmaladze-2 transformation. We will report on some recent work extending the applicability of the Khmaladze transformation and using the Khmaladze-2 transformation for distribution-free goodness of fit testing for discrete distributions. |

Monday 7 November 2022 | Speaker: Sam Bastida Title: List Colouring and Maximal Local Edge Connectivity |

Monday 10 October 2022 | Speaker: Diamant Pireva Title: An introduction to typed set theory Abstract: Sets are often purported to be “predicates in extension”, in that for any property, there is (or should be) a set containing all and only those things with that property as its elements. Actually formalizing this principle can quickly lead to contradiction and so one has to be careful in how they incorporate it into a theory of sets. The most common way this is done is by using a restricted axiom scheme of comprehension, so that we can only collect together sets satisfying some property as subsets of already existing sets. A less common way to do this is by introducing what are known as types. Rather than restrict comprehension by means of quantification, typed set theories restrict what formulae are considered to be well-formed in defining properties. This talk introduces the typed theories TST, TZT, and NF. We start by looking at how these theories overcome the standard paradoxes of set theory. We then look at models of said theories and comparing them. Finally, we look at consistency results coming from the scheme of typical ambiguity. |

Monday 12 September 2022 | Speaker: Flynn Owen Title: Finite Mixture Models - A Divide and Conquer Approach Abstract: Within the field of data clustering, methods are commonly referred to as either ‘distance-based’ or ‘model-based’, with each having certain limitations. Distance-based methods are fast, but lack an underlying parametric model. Model-based methods are interpretable and generalise easily to distributions other than Gaussian, but are slow to estimate, especially when the number of clusters is large. In this thesis we take inspiration from divisive hierarchical clustering and develop four ‘divide and conquer’ model-based algorithms to rapidly estimate mixture models with large numbers of components. Our algorithms are: 1) The Hierarchical No Turning Back (HNTB) algorithm, 2) The Hierarchical Single Reallocation (HSR) algorithm, 3) The Hierarchical Reallocation Loop (HRL) algorithm, 4) The Hierarchical algorithm that Reallocates During execution (HRD). Using Akaike Information Criterion (AIC) as our measure of model quality, our algorithms work by recursively fitting a data matrix with R = 1 and R = 2 components, and then partitioning this data-matrix dependent on cluster membership, halting when no sub-components can be partitioned further. We then adapt the ‘E-step’ from the Expectation-Maximisation (EM) algorithm to reallocate data points across each estimated sub-component of the data matrix. We perform a series of Monte Carlo simulation studies on a series of datasets generated from the Bernoulli distribution, and compare our results against an adapted version of well known ‘Wolfe’s method’ (Wolfe, 1971). We establish that ‘divide and conquer’ mixture modelling methods are able to identify mixtures with large numbers of components at a much faster rate than our adapted version of Wolfe’s method, and approximate both the true simulated data, as well as the results from our adapted method to a good degree. Algorithms introduced in this thesis provide suitable methods to address scalability with the ever increasing magnitude of data being stored and processed, while ensuring interpretability with the increasing demand for data-driven insights. |

Monday 01 August 2022 | Speaker: Malcolm Jones Title: A filter approach to groupoids Abstract: Filters simplify the construction of groupoids used to study C*-algebras. Groupoids and inverse semigroups are both generalisations of groups. They model partial symmetry (Wagner-Preston) as opposed to groups, which model total symmetry (Cayley). Given an inverse semigroup, one can construct groupoids. Two constructions are common: the germ approach pioneered by Paterson, and the filter approach developed by Kellendonk, Lenz, Lawson, Margolis and Steinberg. We describe the equivalence between them studied in collaboration with Armstrong, Clark, an Huef and Lin. This had been known for some time by the experts, but details in the literature were scarce. The aim of the talk is to convey the elegance of a filter approach compared to the classical alternative and give some intuition as to how defining properties of filters capture the same information. |

Monday 18 July 2022 | Speaker: Matthew Askes Title: Strongly Online Graph Colouring Abstract: An online algorithm is an algorithm that processes its input in pieces and must process the current piece of data before it gets the next. Online algorithms have wide-ranging applications from emergency vehicle dispatching, scheduling, and processing large data sets. In effect, any algorithm that needs to work on partial information can be represented as an online algorithm. However, online algorithms do not fully capture the nature of certain problems. Often you can see slightly ahead, or you may be able to delay an action temporarily until more data is available. In these situations the problem is not fully online. To this end, we introduce the notions of locally strongly online and strongly online presentations. In particular, we will discuss strongly online colourings of graphs and touch on the idea of strongly online path decompositions. |

Monday 4 July 2022 | Speaker: Aleksa Vujičić Title: The Problem with Braid Abstract: Braid is a puzzle game released in 2008 whose central mechanic revolves around time manipulation. The goal of any level is to manipulate different level elements so that you can reach the exit - but is this always possible to do? It turns out that answering this question in general is impossible to do, and we look at why this is the case. |

Tuesday 15 June 2021 | Speaker: Joseph Wilson Title: Geometric algebra and an application to special relativity Abstract: Real Clifford algebras are affectionately called “geometric algebras” by physicists for their utility and generality. They provide an elegant framework for describing rotations in arbitrary dimensions, including 3+1 dimensional spacetime. This leads to a simple and novel formula for the composition of Lorentz transformations in terms of their infinitesimal generators. |

Monday 17 May 2021 | Speaker: Linus Richter Title: On Homological Algebra, Group Extensions, and Descriptive Set Theory Abstract: One way of thinking about group extensions is via short exact sequences in the category of groups. I will outline, using simplicial homology, how algebraic topology associates algebraic objects as topological invariants. Using cohomology and descriptive set theory, I will then briefly explain how the class of group extensions whose bonding morphisms are Borel is trivial in certain circumstances. This extends results by Kanovei and Reeken (2000). |

Monday 16 November 2020 | Speaker: Linus Richter Title: When the search for proofs turns futile Abstract: It came as a shock when in 1931 Kurt Gödel proved that mathematics is not complete. We can find mathematical statements that are "independent"; they can neither be proven nor refuted. Surprisingly, while the original independent "Gödel sentence" is a somewhat contrived construction, meaningful mathematical questions can be independent, too. For example, a problem long confounded set theorists: is there a subset of the real numbers that is larger than the set of integers yet smaller than the real numbers themselves? The assertion that such intermediate sets do not exist is called the "Continuum Hypothesis" (CH), and many set out to prove CH. In 1963, Paul Cohen showed that those attempts had been futile -- he proved that CH is independent. Cohen did so by inventing the versatile method of "forcing", which has since been used to yield many more independence proofs. I will introduce some mathematical logic, the notion of independence, and why we care about all this. Using set theory, I will then outline how forcing yields a proof of the independence of CH. |

Monday 2 November 2020 | Speaker: Rafael Pereira Lima Title: KMS states on groupoid C*-algebras Abstract: The theory of C*-algebras started in the study of quantum mechanics. Since then, the subject has evolved and now it interacts with several areas of mathematics. Many important examples in the theory can be described as groupoid C*-algebras. In this talk, we will introduce these concepts and see a theorem due to Neshveyev, which gives a formula for the KMS states on groupoid C*-algebras. This theorem is an example of how topological properties of the groupoid help us understand C*-algebras in more detail. Slides: KMS-groupoid-C-algebras.pdf |

Monday 12 October 2020 | Speaker: Josh Baines Title: The Painlevé-Gullstrand Form of Various Spacetimes Abstract: Every spacetime is defined by its metric, the mathematical object which further defines the spacetime curvature. From the relativity principle, we have the freedom to choose which coordinate system to write our metric in. Some coordinate systems, however, are better than others. In this talk, we begin with a brief introduction into general relativity, Einstein's masterpiece theory of gravity. We then discuss some physically interesting spacetimes and the coordinate systems that the metrics of these spacetimes can be expressed in. More specifically, we discuss the existence of the rather useful Painlevé-Gullstrand coordinate system in these spacetimes. Using this useful coordinate system then allows us to conduct further analysis of these spacetimes, which we discuss. |

Monday 28 September 2020 | Speaker: Meenu Mariya Jose Title: Recognising Principal Transversal Matroids Abstract: Lattice path matroids are a well-behaved subclass of transversal matroids - they are closed under minors and duality and can easily be understood geometrically. Principal transversal matroids are another subclass of transversal matroids that is closed under duality but not minors. We investigated the intersection of these classes and found that a polynomial time algorithm can determine when a lattice path matroid is also principal. This is quite the contrast to the fact that it requires a tedious process to prove that a transversal matroid is principal. No prior knowledge of matroids is assumed. |

Monday 31 August 2020 | Speaker: Michal Salter-Duke Title: Tangles in networks and their application to describing communities Abstract: Communities in networks are structurally or functionally distinct subgraphs. They can be disjoint or overlapping, depending upon definition; there are several definitions of communities, and many algorithms to identify them. We present a new definition based on the graph-theoretic concept of tangles, and investigate how well it identifies subgraphs that represent cohesive parts of the network, using two protein-protein interaction networks. This requires the development of an algorithm for finding tangles in graphs. We compare our results with standard methods from the literature based on metrics that use metadata to establish communities. Our results show that tangles provide a different view of communities that complements other methods, although they are computationally expensive to identify. Slides: MichalSalter-DukeGradTalk.pdf |

Monday 10 August 2020 | Speaker: Liam Jolliffe Title: FI modules and their submodules Abstract: The theory of FI modules was introduced by Church, Ellenburg and Farb in 2014. These modules occur in many different places in the wild, but our interest in them is to gain some insight into the representation theory of the symmetric group. In this talk we will briefly revisit the representation theory of the symmetric group, before defining FI modules and seeing some examples. We will conclude by looking at some families of submodules of representable FI modules and describing some future work. |

Monday 27 July 2020 | Speaker: Malcolm Jones Title: An algebraic model in public health: the groupoid of bubbles Abstract: Groupoids are a generalisation of groups that arose in the early 20th century in work on quadratic forms by H. Brandt. Since their inception groupoids have provided a fruitful framework in a variety of fields, including fundamental work by J. Renault in operator algebras in the 80s. Groupoids have even appeared in epidemiology as recently as 2014 as models of cognitive processes in biology. In this talk, we will review the elementary theory of groupoids, and we will discuss a widely applicable groupoid model of interactions in populations. |

Monday 29 June 2020 | Speaker: Jordan Mitchell Barrett Title: Ramsey theory of semigroups Abstract: Many fundamental results in Ramsey theory concern the structure of certain semigroups (sets with an associative binary operation). These include the Hales–Jewett theorem, the Graham–Rothschild theorem, Gowers' FINₖ theorem, and Hindman's theorem. In this talk, we will discuss and aim to understand these fundamental results. Time permitting, we will see a common generalisation of many of these theorems to the setting of arbitrary layered semigroups, as introduced by Farah, Hindman and McLeod, and further generalised in ongoing work of Barrett and Lupini. Follow the link to see the talk: https://www.youtube.com/watch?v=FuFWwWeeu-Q |

I | Attachment | Action | Size | Date | Who | Comment |
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KMS-groupoid-C-algebras.pdf | manage | 971 K | 03 Nov 2020 - 15:19 | Main.jonesmalc | ||

MichalSalter-DukeGradTalk.pdf | manage | 1 MB | 16 Sep 2020 - 10:35 | Main.jonesmalc |