School of Mathematics and Statistics - Pure Maths Seminar
https://www.maths.unsw.edu.au/category/school-seminar-series/pure-maths-seminar
enPure Math Honours presentations
https://www.maths.unsw.edu.au/seminars/2021-11/pure-math-honours-presentations-1
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">-</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">UNSW Sydney</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Fri, 19/11/2021 - 2:00pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/87228629401</div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><p>Honours presentations, in order, of Elie Sikh, Abdellah Islam, Daniel Czapski, and Godfrey Wong. </p>
<p><strong>Note that we changed the order of the talks from the previously announced program.</strong></p>
<p>Each talk is 20 minutes long followed by 5 minutes of questions and by 5 minutes break.</p>
<p>We will start at 2:05.</p>
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<p>Speaker: Elie Sikh</p>
<p>Title: Intersection Theory from a Differential Viewpoint</p>
<p>Abstract: Intersection theory is one of the main branches of algebraic geometry and refers to the study of the intersections of subvarieties. We choose to showcase intersection theory through the lens of differential topology, allowing us to develop a geometric intuition for the underlying mathematics. Through this approach we see how local differential properties are ultimately linked to the global topological properties of the object being studied. We also explore the interesting case of analytic subvarieties, where the complex structure restricts the possible types of intersections. We focus on presenting results of Hirsch, and Griffith and Harris in a manner approachable for undergraduate students seeking to get a feel for the topic.</p>
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<p>Speaker: Abdellah Islam</p>
<p>Title: Toric Varieties</p>
<p>Abstract: The study of algebraic geometry investigates properties of varieties such as their singularities and their (co)homologies. The family of varieties which contain an embedded torus as an open dense subset, named toric varieties, give explicit examples of varieties to test hypotheses and theorems. We outline the construction of toric varieties and give geometric examples.</p>
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<p>Speaker: Daniel Czapski</p>
<p>Title: Asymptotic eigenvalue distributions of Gaussian Hermitian random matrices</p>
<p>Abstract: Early elements of random matrix theory can be seen in the works of Wishart on random covariance matrices, dating to the early 1920s. The study of random matrices, however, began in earnest with the works of Eugene Wigner from the 1950s on the application of random Hermitian matrices to problems in atomic and nuclear physics. The statistical behaviour of the eigenvalues of randomly chosen matrices – spectral statistics – is a key problem that has seen significant development since that time and remains an active area of research. The first three major results, proved primarily by Dyson and Wigner in the 1950s and 1960s, relate to the “average” distribution of eigenvalues of N-dimensional Hermitian matrices, their asymptotic distribution as N increases without bound, and the distribution of eigenvalues of Hermitian matrices of very large dimension. We review these three major results, briefly discuss a classical method of proof and touch on both generalisations and current research directions.</p>
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<p>Speaker: Godfrey Wong</p>
<p>Title: Permutation polynomials over finite fields: Construction and statistics</p>
<p>Abstract: Permutation polynomials are polynomials over the finite field F_q of q elements that permute F_q. Every permutation polynomial of degree at most 6 has been classified. However, there is no coherent theory on the construction of permutation polynomials of any degree yet. For example, finding the exact and even an asymptotic number of permutation polynomials of a given degree n < q-1 still an open problem.</p>
<p>We will study the construction of permutation polynomials and then give a new lower bound on the number of permutation polynomials of large degree. We will then extend the result to permutation rational functions, which are rational functions over F_q that permutes F_q union infinity.</p></div></div></section><section class="field field-name-taxonomy-vocabulary-4 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">School Seminar Series: </h2><ul class="field-items"><li class="field-item even"><a href="/category/school-seminar-series/pure-maths-seminar">Pure Maths Seminar</a></li></ul></section>Thu, 04 Nov 2021 03:58:49 +0000z35245655782 at https://www.maths.unsw.edu.auPure Math Honours presentations
https://www.maths.unsw.edu.au/seminars/2021-11/pure-math-honours-presentations-0
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">-</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">UNSW Sydney</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Tue, 16/11/2021 - 2:30pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/87456341337</div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><p>Honours presentations, in order, of Aditya Ganguly, Dilshan Wijesena, Ethan Brown, Chesta Wu, and Michael Horton. </p>
<p>Each talk is 20 minutes long followed by 5 minutes of questions and by 5 minutes break.</p>
<p>We will start at 2:35.</p>
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<p>Speaker: Aditya Ganguly</p>
<p>Title: Existence of 2-factors in random uniform regular hypergraphs</p>
<p>Abstract: In the theory of random graphs, it is a common objective to prove that almost all graphs of a certain type contain a specified subgraph. To this end, the second moment method is an elementary tool from probability theory which can sometimes show that, with 'high probability', the discrete random variable counting the number of occurrences of a given subgraph in a random graph is strictly positive. When this technique fails, however, the Small Subgraph Conditioning Method (SSCM), introduced by Robinson and Wormald in 1992, has been used to prove the existence of subgraphs such as perfect matchings, Hamilton cycles and spanning trees in almost all regular graphs.</p>
<p>Our paper extends a result of Robalewska (1996), that almost all regular graphs contain a 2-factor (2-regular subgraph), to the world of uniform regular hypergraphs by using the SSCM. We also explain the consequences of this result for a version of qualitative equivalence between random hypergraph models, known as 'contiguity'.</p>
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<p>Speaker: Dilshan Wijesena</p>
<p>Title: A New Continuous Class of Irreducible Representations of R. Thompson’s Groups using Jones’ Machinery</p>
<p>Abstract:Richard Thompson’s groups F, T and V are one of the most fascinating discrete groups for their several unusual properties and their analytical properties have been challenging experts for many decades. Most notably, it was conjectured by Ross Geoghegan in 1979 that F is not amenable and thus another rare counterexample to the von Neumann problem. However, surprisingly despite many attempts, the question about amenability remains unanswered along with even more elementary questions such as Cowling-Haagerup weak amenability.</p>
<p>Surprisingly, these discrete groups were recently discovered by Vaughn Jones while working on the very continuous structures of conformal nets and subfactors. In this talk, I will explain how Jones’ work provided a new method for constructing unitary representations of Thompson’s groups and provided easy new proofs of certain known analytical properties of Thompson’s groups. In particular, I will talk about a specific family of Jones’ representations called Pythagorean representations. With my supervisor, Arnaud Brothier, we constructed a new continuous class of Pythagorean representations. These representations were proven to be almost always irreducible and pairwise non-isomorphic. Further, we proved that they are not the induction of a finite representation of any subgroup of F.</p>
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<p>Speaker: Ethan Brown</p>
<p>Title: Permutation Automorphisms of the Cuntz Algebras</p>
<p>Abstract: We will discuss automorphisms of the Cuntz algebras - specifically, those which arise from permutations acting upon products of generators. We will follow work by Conti and Szymański in finding such automorphisms, by turning the problem into an equivalent combinatorial problem, and see how this process can be implemented computationally. Finally, we will briefly discuss further directions, and how this method could be used to efficiently determine substructures of the automorphism groups.</p>
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<p>Speaker: Chesta Wu</p>
<p>Title: TBA</p>
<p>Abstract: TBA</p>
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<p>Speaker: Michael Horton</p>
<p>Title: Relative position of subspaces in a Hilbert space</p>
<p>Abstract: In this talk, we will look at the ways in which you can classify systems of subspaces in a Hilbert space by their relative position to each other. My talk briefly covers the work by Halmos regarding 2 subspaces and how their relative position can be classified up to unitary equivalence. I then go over some of the more recent work by Enemoto and Watatani, who develop a classification of n subspaces by relaxing the equivalence relation to isomorphism classes, looking more specifically at the cases for 3 and 4 subspaces.</p></div></div></section><div class="field field-name-field-seminar-latex field-type-text-long field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><div class="field field-name-field-seminar-url field-type-link-field field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div>Thu, 04 Nov 2021 03:54:39 +0000z35245655781 at https://www.maths.unsw.edu.auPure Math Honours presentations
https://www.maths.unsw.edu.au/seminars/2021-11/pure-math-honours-presentations
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">-</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">UNSW Sydney</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Fri, 12/11/2021 - 2:00pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/81215637331</div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><p>Honours presentations, in order, of Joshua Graham, Edmond Gao, Kevin Tran, and Madhav Padmakumar.</p>
<p>Each talk is 20 minutes long followed by 5 minutes of questions and by 5 minutes break.</p>
<p>We will start at 2:05.</p>
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<p>Speaker: Joshua Graham</p>
<p>Title: Higher Algebraic K-theory of Rings</p>
<p>Abstract: The focus of this talk will be looking at how lower K-groups of rings are defined by algebraic means and how we extend the definition to higher K-groups in a functorial manner. I will also briefly talk briefly about K-theory of the integers and of finite fields.</p>
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<p>Speaker: Edmond Gao</p>
<p>Title: Futoshiki - A Latin Square Puzzle Variant</p>
<p>Abstract: The act of filling in the empty cells of a partial Latin square to form a Latin square has become a popular logic puzzle over the last few decades. In that time, many variants with additional conditions have emerged, the most popular being the Sudoku puzzle. Both Latin squares and Sudoku have been studied extensively by combinatorialists, however other variants of Latin square puzzles remain mathematically untouched. One such variant is the Futoshiki puzzle, which incorporates inequalities into Latin squares.</p>
<p>It is surprising that a puzzle that is so mathematical in nature is barely present in mathematical literature. I aim to change this, exploring Futoshiki puzzles from a combinatorial perspective. I define and introduce features and characteristics of the variant, and extend results about Latin squares to this special case.</p>
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<p>Speaker: Kevin Tran</p>
<p>Title: Combinatorial and arithmetic properties of Hilbert cubes</p>
<p>Abstract: In the area of additive combinatorics and number theory, Hilbert cubes are sets defined to have a strong additive structure. Following the standard paradigm of additive combinatorics, our goal is to show the absence of any multiplicative structure. Results in this form include those about the intersection of Hilbert cubes in a ring with multiplicative subgroups of this ring, or those about powers in and prime divisors of elements of Hilbert cubes of integers. We also show that the results in "Hilbert cubes meet arithmetic sets" by Hegyvári and Pach (2020), which aim to show that Hilbert cubes in finite fields are "uncorrelated" with reciprocals of sumsets, are essentially void, as they are no stronger than a trivial bound. Instead, we suggest a different approach which allows us to get nontrivial versions of these results.</p>
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<p>Speaker: Madhav Padmakumar</p>
<p>Title: Measuring and Computing Optimal Generators for Homology Groups</p>
<p>Abstract: The field of topological data analysis typically concerns itself with imposing a topological space on a dataset, then finding topological invariants. This talk will define the problem of "Optimal" generators for those topological features, as well as how to compute them. I will outline the Smith Normal Form reduction algorithm for computing homology groups and the persistent homology algorithm for extracting topological features from datasets. I will then explain the algorithm for computing optimal generators, with examples and potential applications.</p></div></div></section><div class="field field-name-field-seminar-latex field-type-text-long field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><div class="field field-name-field-seminar-url field-type-link-field field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><section class="field field-name-taxonomy-vocabulary-4 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">School Seminar Series: </h2><ul class="field-items"><li class="field-item even"><a href="/category/school-seminar-series/pure-maths-seminar">Pure Maths Seminar</a></li></ul></section>Thu, 04 Nov 2021 03:48:20 +0000z35245655780 at https://www.maths.unsw.edu.auFrom Thompson's groups to holographic toy models
https://www.maths.unsw.edu.au/seminars/2021-10/thompsons-groups-holographic-toy-models
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Deniz Stiegemann</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">University of Queensland</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Thu, 07/10/2021 - 12:00pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/81777644468</div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"><span style="font-size: 11pt; font-family: Arial; color: #000000; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; vertical-align: baseline; white-space: pre-wrap;">Almost all algebraic structure in common use have some sort of associative binary operation ("multiplication"), but we don't always require there to exist all, or even any, inverses with respect to the operation. For instance, the definition of a group and a monoid are identical except that in a group every element has an inverse, whereas in a monoid this need not be the case. </span></p>
<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"><span style="font-size: 11pt; font-family: Arial; color: #000000; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; vertical-align: baseline; white-space: pre-wrap;">However, through a procedure called 'localisation', we can add inverses constructively (thereby turning e.g. a monoid into a group). This type of construction can be applied to other algebraic structures, such as rings, modules, and categories.</span></p>
<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"><span style="font-size: 11pt; font-family: Arial; color: #000000; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; vertical-align: baseline; white-space: pre-wrap;">V. F. R. Jones has described how forming the localisation of an algebraic structure automatically also gives 'localisations' for representations of (i.e. functors from) that algebraic structure.</span></p>
<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"><span style="font-size: 11pt; font-family: Arial; color: #000000; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; vertical-align: baseline; white-space: pre-wrap;">I'm going to explain how this can be used to construct discrete toy models in physics using representations of Thompson's groups. (There're called *toy* models because our spacetime is [probably] not discrete.) I'll also talk about some fascinating purely mathematical questions around Thompson's groups.</span></p>
<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"><span style="font-size: 11pt; font-family: Arial; color: #000000; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; vertical-align: baseline; white-space: pre-wrap;">I will assume zero knowledge of any of the physics involved – everything will be explained and illustrated.</span></p></div></div></section><div class="field field-name-field-seminar-latex field-type-text-long field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><div class="field field-name-field-seminar-url field-type-link-field field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><section class="field field-name-taxonomy-vocabulary-4 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">School Seminar Series: </h2><ul class="field-items"><li class="field-item even"><a href="/category/school-seminar-series/pure-maths-seminar">Pure Maths Seminar</a></li></ul></section>Tue, 05 Oct 2021 02:15:31 +0000z35245655746 at https://www.maths.unsw.edu.auThe metric geometry of subsets of the Hamming cube
https://www.maths.unsw.edu.au/seminars/2021-10/tba-0
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Ian Doust</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">UNSW Sydney</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Thu, 28/10/2021 - 12:00pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/82230444349</div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><div class="tex2jax"><p dir="ltr">All the distance data for a finite metric spaces $X = \{x_1,\dots,x_n\}$ is stored in its distance matrix $D = (d(x_i,x_j))_{i,j=1}^n$. A common theme in distance geometry is to try to link linear algebraic properties of this matrix with more geometric properties, such as whether you can isometrically embed $X$ into Euclidean space (or perhaps some other nice space).</p>
<p dir="ltr">Two much-studied classes of finite metric spaces are trees with the path metric, and spaces formed by taking collections of bit-strings of length $n$ and applying the Hamming metric. The distance matrices for such spaces have some quite surprising properties. In this talk we shall start by introducing Graham and Pollak's 1971 formula for the determinant of the distance matrices of tree, and progress to some much more recent formulas. No special knowledge beyond second year linear algebra will be assumed!</p>
<p dir="ltr">This is joint work with Gavin Robertson, Alan Stoneham, Tony Weston and Reihard Wolf.</p></div></div></div></section><section class="field field-name-taxonomy-vocabulary-4 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">School Seminar Series: </h2><ul class="field-items"><li class="field-item even"><a href="/category/school-seminar-series/pure-maths-seminar">Pure Maths Seminar</a></li></ul></section>Sun, 26 Sep 2021 22:22:02 +0000z35245655739 at https://www.maths.unsw.edu.auGroups acting on trees with prescribed local actions
https://www.maths.unsw.edu.au/seminars/2021-09/groups-acting-trees-prescribed-local-actions
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Colin Reid</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">University of Newcastle</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Thu, 23/09/2021 - 12:00pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/88385879800</div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><p dir="ltr" style="line-height: 1.6667; margin-top: 9pt; margin-bottom: 0pt;">Actions on trees are ubiquitous in group theory. The standard approach to describing them is known as Bass–Serre theory, which presents the group acting on the tree as assembled from its vertex and edge stabilizers. However, a different approach emerges if instead of considering vertex and edge stabilizers as a whole, we focus on local actions, that is, the action of a vertex stabilizer only on the immediate neighbours of that vertex. Groups acting on trees defined by their local actions are especially important as a source of examples of simple totally disconnected locally compact groups, with a history going back to a 1970 paper of Tits. I will go through some highlights of this theory and then present some recent joint work with Simon Smith: we develop a counterpart to Bass–Serre theory for local actions, which describes all possible local action structures of group actions on trees. The talk is partly based on the following preprint of Simon Smith and the speaker: <a style="font-size: 0.9em;" href="https://arxiv.org/abs/2002.11766"><span style="font-size: 12pt; font-family: Calibri; color: #006580; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; text-decoration-line: underline; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;">https://arxiv.org/abs/2002.11766</span></a><span style="color: #000000; font-size: 11pt; font-family: Arial; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; vertical-align: baseline; white-space: pre-wrap;">. </span></p></div></div></section><section class="field field-name-taxonomy-vocabulary-4 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">School Seminar Series: </h2><ul class="field-items"><li class="field-item even"><a href="/category/school-seminar-series/pure-maths-seminar">Pure Maths Seminar</a></li></ul></section>Sat, 21 Aug 2021 23:32:02 +0000z35245655732 at https://www.maths.unsw.edu.auPolynomial Link Invariants from Markov Traces on Braid Group Representations
https://www.maths.unsw.edu.au/seminars/2021-07/polynomial-link-invariants-markov-traces-braid-group-representations
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Jiayi Li</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">UNSW Sydney</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Tue, 27/07/2021 - 12:00pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/88532217896</div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><p><span style="font-size: 11pt; font-family: Arial; color: #000000; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; vertical-align: baseline; white-space: pre-wrap;">In 1984 Vaughan Jones discovered a polynomial knot invariant that is now called the Jones polynomial. This talk will explore the process of obtaining polynomial knot invariants from Markov traces on braid group representations. In particular, we will focus on the Jones polynomial, and we will see two different constructions of the polynomial as a Markov trace on the Temperley-Lieb algebra. We will explore Jones' original algebraic construction as well as a diagrammatic construction due to the work of Kauffman. Finally, we will briefly address two generalisations of the Jones polynomial: the HOMFLY polynomial which is a Markov trace on the Hecke algebra and the Kauffman polynomial which is a Markov trace on the BMW algebra. </span></p></div></div></section><div class="field field-name-field-seminar-latex field-type-text-long field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><div class="field field-name-field-seminar-url field-type-link-field field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><section class="field field-name-taxonomy-vocabulary-4 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">School Seminar Series: </h2><ul class="field-items"><li class="field-item even"><a href="/category/school-seminar-series/pure-maths-seminar">Pure Maths Seminar</a></li></ul></section>Tue, 20 Jul 2021 04:08:28 +0000z35245655714 at https://www.maths.unsw.edu.auChern-Weil theory for singular foliations
https://www.maths.unsw.edu.au/seminars/2021-06/chern-weil-theory-singular-foliations
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Lachlan MacDonald</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">University of Adelaide</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Tue, 22/06/2021 - 12:00pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/81740603495 </div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><p><span style="box-sizing: border-box; font-size: 14.6667px; font-variant-ligatures: none; white-space: pre-wrap; color: #000000; font-family: Arial;">Chern-Weil theory describes a procedure for constructing the characteristic classes of a smooth manifold from geometric data (such as a Riemannian metric). In the 1970s and 1980s, Chern-Weil theory was successfully adapted by R. Bott to describe the characteristic classes of the leaf space of any regular foliation, including the so-called secondary classes such as the Godbillon-Vey invariant. The extension of Chern-Weil theory to singular foliations has, however, remained elusive. In this talk, I will describe recent, joint work with Benjamin McMillan which gives a Chern-Weil homomorphism for a family of singular foliations whose singularities are not ``too big".</span></p></div></div></section><div class="field field-name-field-seminar-latex field-type-text-long field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><div class="field field-name-field-seminar-url field-type-link-field field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><section class="field field-name-taxonomy-vocabulary-4 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">School Seminar Series: </h2><ul class="field-items"><li class="field-item even"><a href="/category/school-seminar-series/pure-maths-seminar">Pure Maths Seminar</a></li></ul></section>Tue, 15 Jun 2021 01:41:36 +0000z35245655685 at https://www.maths.unsw.edu.auThermodynamic Formalism for Random Interval Maps
https://www.maths.unsw.edu.au/seminars/2021-06/termodynamic-formalism-random-interval-maps
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Jason Atnip</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">UNSW Sydney</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Tue, 08/06/2021 - 12:00pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/86263973939</div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><p><span style="font-size: 11pt; font-family: Arial; color: #201f1e; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; vertical-align: baseline; white-space: pre-wrap;">In this talk we will consider a collection of piecewise monotone interval maps, which we iterate randomly, together with a collection of holes placed randomly throughout phase space. Birkhoff’s Ergodic Theorem implies that the trajectory of almost every point will eventually land in one of these holes. We prove the existence of an absolutely continuous conditionally invariant measure, conditioned according to survival from the infinite past. Absolute continuity is with respect to a conformal measure on the closed systems without holes. Furthermore, we prove that the rate at which mass escapes from phase space is equal to the difference in the expected pressures of the closed and open systems. Finally, we prove a formula for the Hausdorff dimension of the fractal set of points whose trajectories never land in a hole in terms of the expected pressure function.</span></p></div></div></section><section class="field field-name-taxonomy-vocabulary-4 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">School Seminar Series: </h2><ul class="field-items"><li class="field-item even"><a href="/category/school-seminar-series/pure-maths-seminar">Pure Maths Seminar</a></li></ul></section>Mon, 31 May 2021 04:55:57 +0000z35245655679 at https://www.maths.unsw.edu.auPowers in Orbits of Rational Dynamical Systems
https://www.maths.unsw.edu.au/seminars/2021-04/powers-orbits-rational-dynamical-systems
<section class="field field-name-field-seminar-speaker field-type-text field-label-above view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Conrad Martin</div></div></section><section class="field field-name-field-seminar-affiliation field-type-text field-label-above view-mode-rss"><h2 class="field-label">Affiliation: </h2><div class="field-items"><div class="field-item even">UNSW Sydney</div></div></section><section class="field field-name-field-seminar-date field-type-date field-label-above view-mode-rss"><h2 class="field-label">Date: </h2><div class="field-items"><div class="field-item even"><span class="date-display-single">Tue, 20/04/2021 - 12:00pm</span></div></div></section><section class="field field-name-field-seminar-venue field-type-text field-label-above view-mode-rss"><h2 class="field-label">Venue: </h2><div class="field-items"><div class="field-item even">Zoom link: https://unsw.zoom.us/j/84758330445</div></div></section><section class="field field-name-field-seminar-abstract field-type-text-long field-label-above view-mode-rss"><h2 class="field-label">Abstract: </h2><div class="field-items"><div class="field-item even"><p><span style="font-size: 11pt; font-family: Lato; color: #201f1e; background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; vertical-align: baseline; white-space: pre-wrap;">In the field of arithmetic dynamics, we are interested in classifying points in a number field K depending on their orbit, that is, how they behave under repeated application of a given rational function $f$. Recently, Ostafe, Pottymeyer and Shparlinski have given a classification of points whose orbits generated by a given polynomial contain perfect powers. This talk will provide an effective background to understand these results and will then give a generalisation for this work which focuses on powers in the orbits of rational functions.</span></p></div></div></section><div class="field field-name-field-seminar-latex field-type-text-long field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><div class="field field-name-field-seminar-url field-type-link-field field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"></div></div></div><section class="field field-name-taxonomy-vocabulary-4 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">School Seminar Series: </h2><ul class="field-items"><li class="field-item even"><a href="/category/school-seminar-series/pure-maths-seminar">Pure Maths Seminar</a></li></ul></section>Fri, 09 Apr 2021 04:06:49 +0000z35245655649 at https://www.maths.unsw.edu.au

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