Pure Math Seminar, UNSW Sydney

2023 Presentations


Title: Make Like a Graph and Split

Abstract: In symbolic dynamics, shift spaces are used to approximate topological (or smooth) dynamical systems by using an alphabet of symbols to represent states of the system. In the 1970s, Williams showed that two shifts of finite type are conjugate if and only if we can get from one to the other using a series of "in-splits" and "out-splits" of their associated directed graphs. In this talk I aim to give a fresh perspective on splittings for directed graphs. I'll also show how this perspective lends itself to splittings of more general dynamical systems, including topological graphs and those arising in the noncommutative world of C*-dynamics.


This talk is based on recent work with Kevin Brix and Adam Rennie.


Title: Algebraic Geometry and Analytic Geometry

Abstract: A scheme locally of finite type over the complex numbers C is a scheme which is locally the spectrum of a finitely generated C-algebra. The classical complex topology can be imposed onto the Zariski-closed points of such a scheme, forming a complex analytic space; this is called the ``analytification'' of the scheme. A generalisation of Jean-Pierre Serre's 1956 paper Géométrie algébrique et géometrie analytique (abbreviated to GAGA) establishes a deep connection between the Čech cohomology of these schemes and their analytifications. This talk will give, in the language of category theory, an overview of GAGA and some of its applications to vector bundles and Chow's theorem. An emphasis will be made to understand manifolds and vector bundles as concrete examples of the more abstract constructions.


Title: X+X = X

Abstract: The equation X+X = X looks pretty easy to solve. In this talk, we explain how, if we make it much more complicated than it needs to be, we can obtain all sorts of interesting mathematical objects including: Cantor space, a machine for manipulating the input stream of a computer program, and various combinatorial objects familiar from topological dynamics.

Title: Irreducible Pythagorean representations of Thompson's groups

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. Surprisingly, these discrete groups were recently discovered by Vaughn Jones while working on the very continuous structures of conformal nets and subfactors.

We will explain how using the novel technology of Jones, a so-called Pythagorean unitary representation of Thompson’s groups can be constructed given any isometry from a Hilbert space to its square. These representations are particularly amenable to a diagrammatic calculus which we use to develop powerful techniques to study their properties. For a large class of Pythagorean representations we introduce necessary and sufficient conditions for irreducibility and pair-wise equivalence. This provides a new rich class of irreducible representations of F that extends many previously known classes of irreducible representations of F.

Abstract: TBA

Title: Nijenhuis Geometry

Abstract: Nijenhuis Geometry is a recently initiated  research programme; it studies   Nijenhuis operators, i.e., a field of endomorphisms with vanishing Nijenhuis torsion. Vanishing of the Nijenhuis torsion is  the simplest differential-geometric condition on a field of endomorphisms; for this reason it appears independently and many times in geometry, analysis and mathematical physics, in particular in the theory of projectively equivalent metrics and in the theory of finite- and infinite-dimensional  integrable systems.

The talk is based on a series of papers  (most  are co authored with A. Bolsinov and A. Konyaev) aiming at re-directing the research agenda in this area to "next level topics":

• Singular points: what does it mean for a point to be generic or singular in the context of Nijenhuis geometry? What singularities are non-degenerate? What singularities are stable? How do Nijenhuis operators behave near non-degenerate and stable singular points?

• Global properties: what restrictions on a Nijenhuis operator are imposed by compactness of the manifold? And conversely, what are topological obstructions on a manifold carrying a Nijenhuis operator with specific properties (e.g.with no singular points)?

My talk will  demonstrate that this research programme is realistic by proving  a series of new nontrivial  results, and will touch  applications in differential geometry and mathematical physics.

Title: A journey to the Devil's Domain

Abstract: Max Noether said that algebraic curves were created by God and algebraic surfaces by the Devil. In this talk I will show some modern evidence supporting this statement. A moduli space is a parameter space for isomorphism classes of certain types of objects. Moduli spaces of curves have received a lot of attention for more than a century now. Discoveries about their properties have led to many applications. Interestingly, several of their properties were established before their existence was known. Mumford's Geometric Invariant Theory (GIT) brought an extremely powerful approach to moduli theory. In particular, using GIT, Mumford was able to prove the existence of those moduli spaces. In fact, together with Deligne, he also proved that there exist natural compactifications of these moduli spaces and that these compact moduli spaces are actually projective algebraic varieties. All of this was done in the 1960s and the expectation at the time was that this powerful theory would work for constructing and compactifying moduli spaces of higher dimensional varieties. It turns out that there are new difficulties in higher dimensions that GIT is not equipped to work with and eventually it took more than 5 decades of additional work to achieve the above goal.

In this talk I will explain one of the simplest issues we face in higher dimensions which forces us to reconsider how we approach the problem of moduli. I will also mention recent results that have finally completed the quest for the existence and projective compactification of moduli spaces of higher dimensional varieties.

Title: The Type (B) Problem for Well-bounded Operators

Abstract: A scalar-type spectral operator T is an operator acting on a Banach space that can be represented as an integral over σ(T) with respect to a spectral measure. When the Banach space is reflexive, it had been shown by Dunford that T being scalar-type spectral is equivalent to T having a C(σ(T)) functional calculus. In 1994, Doust and deLaubenfels showed that this equivalence holds precisely on Banach spaces that do not have a subspace isomorphic to c0. A well-bounded operator T on a Banach space is an operator that has an AC[a, b] functional calculus, and it is said to be of type (B) if it can be written as an integral with respect to a spectral family of projections. On a reflexive Banach space, all well-bounded operators are of type (B). However, classifying the Banach spaces on which all well-bounded operators are of type (B) is still an open problem, and it has been conjectured by Doust and deLaubenfels that these are precisely the reflexive Banach spaces. In this talk, we will discuss the current standing of this problem and some recent progress.

Title: Divergence in Coxeter groups

Abstract: The divergence of a pair of geodesic rays in a metric space is a measure of how fast they spread apart. For example, in the Euclidean plane the divergence of any pair of geodesic rays is linear, while in the hyperbolic plane it is exponential.  In the 1980s Gersten used this idea to formulate a large-scale invariant of groups, also called divergence, which has been investigated for many important families.  We study divergence in Coxeter groups, which are groups generated by sets of "reflections".  We use the geometry of spaces on which Coxeter groups act, the algebraic structure of certain families of their subgroups, and the combinatorics of their reduced words.  This is joint work with Pallavi Dani, Yusra Naqvi and Ignat Soroko.

Title: Is F automatic?

Abstract: Thompson's group F is a much-studied and interesting example of a finitely presented group. The definition of an automatic group goes back to Thurston and others in the 1980's, which is about efficient computation for finitely generated groups.

I will explain what automatic is, what F is, and give some failed attempts at proving that F is or is not automatic.

Title: Introducing inverse Hamiltonian reduction

Abstract: In this talk, I want to give an introduction to and overview of a new idea in mathematical physics and representation theory motivated by conformal field theory (CFT).

The problem at hand is classifying certain families of weight modules for affine Lie algebras, namely, those of interest in the mathematical study of CFT. For example, the well-studied Wess-Zumino-Witten (WZW) models are built from the nicest irreducible highest-weight modules over the corresponding affine algebra. However, in many models of interest in CFT, such as the so-called fractional-level WZW modules, these modules are insufficient to explain the physics completely; something more is required.

Related to this picture are the so-called W-algebras, which are constructed from affine Lie algebras by a procedure known as hamiltonian reduction. The reduction procedure also gives a way to construct W-algebra modules from affine Lie algebra modules. This is useful because the W-algebra modules of interest in CFT are easier to understand than the corresponding affine Lie algebra modules.

Recent work has shown that in certain cases, it is possible to invert the reduction procedure. This means one can reconstruct the messy representation theory of the affine algebras from their reduced W-algebra counterparts. Generalising this construction (which should be possible) would rise to a constructive and holistic approach to studying the weight modules over W-algebras and affine Lie algebras of interest in CFT. Inverse reduction involves many parts and is still very much a work in progress by myself and many others. Therefore, my goal for the talk is to try and impart some of the flavour of the problem along with some of the key ideas using simple examples.

Title: Poincaré duality and resonance

Abstract: Every graded, graded-commutative algebra (such as the cohomology ring of a space) determines a family of cochain complexes, parametrized by the elements in degree one of the algebra. The resonance varieties are the loci where the cohomology of these cochain complexes jumps. In this talk, I will discuss some of the constraints imposed by Poincaré duality on the geometry of these resonance varieties.

Title: Perturbation theory of commuting self-adjoint operators

Abstract: In this talk, I give an overview of the perturbation theory of commuting self-adjoint operators developed by Voiculescu, which generalizes the classical Weyl-von Neumann theorem and Kato-Rosenblum theorem. 

Quasicentral modulus of a tuple of operators with respect to a normed ideal plays an essential role in the theory, and we present an example of its computation in the case where the joint spectrum of the tuple is contained in a self-similar set. 

2021 Presentations


Title: The metric geometry of subsets of the Hamming cube


Abstract: 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).


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!


This is joint work with Gavin Robertson, Alan Stoneham, Tony Weston and Reihard Wolf. 

Title: From Thompson's groups to holographic toy models.

Abstract: 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. 

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.

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.

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.

I will assume zero knowledge of any of the physics involved – everything will be explained and illustrated.

Title: Groups acting on trees with prescribed local actions

Abstract: 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: https://arxiv.org/abs/2002.11766.  

Title: Polynomial Link Invariants from Markov Traces on Braid Group Representations

Abstract:  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. 

Title: Chern-Weil theory for singular foliations

Abstract: 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". 

Title: Recent advances for integrable long-range spin chains 

Abstract: I will introduce the landscape of quantum-integrable long-range spin chains and the associated (quantum-)algebraic structures, and describe recent advances and open problems in the field.

Since their origin as quantum-mechanical models for magnetism, spin systems have made their way into various areas of mathematics, including operator algebras, probability theory and representation theory. I will focus on spin chains that are quantum integrable: their spectrum admits an exact characterisation thanks to an underlying quantum-algebraic structure. Unlike for the traditionally studied nearest-neighbour Heisenberg spin chains, the quantum integrability of long-range spin chains exploits connections to a different kind of integrable models: quantum-many body systems of Calogero–Sutherland (or Ruijsenaars–Macdonald) type.

For the so-called Haldane–Shastry spin chains this can be understood in terms of affine Hecke algebras, which yield Yangian (or quantum-loop) invariance as well as an explicit description of the spectrum via Jack (or Macdonald) polynomials.

The Inozemtsev spin chain interpolates between Heisenberg and Haldane–Shastry while admitting an exact description of its spectrum throughout, this time in terms of eigenfunctions of the elliptic Calogero–Sutherland model; here the underlying quantum-algebraic structure is not understood yet.

My talk is based on joint work with R. Klabbers (Nordita), with V. Pasquier and D. Serban (IPhT CEA/Saclay), and work in progress. 

Title: Thermodynamic Formalism for Random Interval Maps

Abstract: 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. 

Title: Powers in Orbits of Rational Dynamical Systems

Abstract: 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.

Title: Representation theory of a Quantum SU(2)

Abstract: To avoid paradoxes, the geometry of physics at the very smallest scales must be non-commutative. The symmetries of such a geometry constitute a structure called a Quantum Group. In this talk we build one instance of a quantum group - quantum SU(2), and study its representation theory. This theory leads us to a beautiful generalisation of Pontryagin duality - Tannaka-Krein duality for compact quantum groups, which says that any C*-tensor category with a fibre functor can be realised as the representation category of some compact quantum group up to a natural equivalence. 

Title: Projective families of varieties through Birational Geometry and Hodge Theory

Abstract:  In the 1920s, building on Fermat's Last Theorem, Mordell conjectured that the set of rational points of any smooth projective curve of genus at least two, over any number field, is finite. In the 1960s, Shafarevich turned this into a purely algebro-geometric conjecture involving families of smooth projective curves. Parshin, Arakelov and Faltings settled this conjecture by showing that the base spaces of such families are in some sense hyperbolic, as long as there is some variation in the algebraic structure of the fibers. Inspired by recent advances in Birational Geometry, Kebekus and Kovács conjectured that these hyperbolicity-type properties should hold for a vast class of projective families, with fibers of arbitrary dimension. In this talk I will discuss this conjecture and my solution to it. I will also talk about more recent advances in this area, based on a joint work with Kovács (University of Washington).

Title: Equations for Schubert varieties, affine Grassmanians, and nilpotent orbits.

Abstract: A common goal in algebraic geometry is to understand a geometric object as a moduli space. One fundamental difficulty is determining whether the proposed moduli space is reduced, i.e. that the corresponding defining ideals are radical ideals. This talk will be about problems of this sort that arise in the study of affine Grassmannians, their Schubert varieties, and nilpotent orbit closures. 

Specifically, I will discuss work on a conjectural moduli description of Schubert varieties in the affine Grassmannian and proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on equations defining type A affine Grassmannians. As an application of our ideas, we prove a conjecture of Pappas and Rapoport about nilpotent orbit closures. This involves work with Joel Kamnitzer, Alex Weekes, and Oded Yacobi.

Title: The Algebra of Boole

Abstract: George Boole introduced Boolean algebra in 1847 and since then it has played a central role in various areas such as logic circuit design, complexity theory and propositional logic. The conventional Boolean Algebra has three basic operations with them being, conjunction, disjunction and negation. In this talk, we will explore a different framework, which we call the Algebra of Boole that utilises only two operations, the antivalence and the conjunction operation over the Boolean ring. We will see that the Algebra of Boole leads to a simpler framework to represent Boolean functions. Furthermore, we will also see that there is a connection between the Mobius function from the theory of posets with the Algebra of Boole.

Title: A tour via examples of Beilinson-Berstein localisation

Abstract: This is a talk about my favourite theorem. The Beilinson—Bernstein localisation theorem provides a concrete link between two seemingly disparate mathematical worlds: the world of Lie theory (the study of continuous symmetry groups), and the world of algebraic D-modules (the study of differential equations on algebraic varieties). This theorem was introduced in 1981 to prove the most important open problem in Lie theory of the time, the Kazhdan—Lusztig conjecture, and has remained at the heart of geometric representation theory for the past 40 years. In this talk, we’ll explore this powerful theorem through the lens of examples, both classical and modern, and I’ll explain how it shapes my personal mathematical landscape.   

Title: Compactifications of cluster varieties and convexity

Abstract: Cluster varieties are log Calabi-Yau varieties which are unions of algebraic tori glued by birational  "mutation" maps. They can be seen as a generalization of the toric varieties. In toric geometry, projective toric varieties can be described by polytopes. We will see how to generalize the polytope construction to cluster convexity which satisfies piecewise linear structure. As an application, we will see the non-integral vertex in the Newton Okounkov body of Grassmannian comes from broken line convexity. We will also see links to the symplectic geometry and application to mirror symmetry. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vianna.

Title: Automorphism tower of groups of homeomorphism of Cantor space

Abstract: The class of full and sufficiently transitive (i.e. flexible) groups of homeomorphisms of Cantor space contains many groups of interest including  generalisations of the Higman-Thompson groups G_{n,r}, and the Rational group R_2 of Grigorchuk, Nekrashevych, and Suchanskiı̆. We will introduce this class of groups and go on to describe recent results on the automorphism tower of such groups. For the groups Gn,r our result can be seen as extending results of Brin and Guzmán for Thompson’s group T and generalisations of Thompson’s group F.


Title: Pseudoholomorphic curves in symplectic geometry

Abstract: Pseudoholomorphic curves were introduced by Gromov as a fundamental tool for the study of symplectic manifolds. I will review some applications of pseudoholomorphic curves in symplectic geometry, which often crucially rely (among other things) on a version of Gromov's compactness result for pseudoholomorphic curves. Then I will discuss some basic facts at the core of Gromov's compactness result, with the goal of conveying why and in what settings such a result can be expected.

Title: Mixing via shearing for parabolic flows

Abstract: Parabolic flows are smooth flows for which nearby points diverge slowly (namely, at polynomial speed) and include, for example, smooth area-preserving flows on surfaces, horocycle flows and nilflows. Apart from a few exceptions, not much is known for general smooth parabolic flows and there is no classification of the characteristic phenomena for this kind of behavior.

We will survey some recent progress in the study of the chaotic features of non-homogeneous parabolic flows. We will explain how a common geometric shearing mechanism plays a key role in understanding their mixing properties. 

Some of the results discussed are joint work with Artur Avila, Giovanni Forni and Corinna Ulcigrai, and with Adam Kanigowski.


Title: Demazure-Lusztig opertors and Metaplectic Whittaker functions

Abstract: The study of objects from number theory such as metaplectic Whittaker functions has led to surprising applications of combinatorial representation theory. Classical Whittaker functions can be expressed in terms of symmetric polynomials, such as Schur polynomials via the Casselmann-Shalika formula. Tokuyama's theorem is an identity that links Schur polynomials to highest-weight crystals, a symmetric structure that has interesting combinatorial parameterisations. In this talk, we will discuss constructions of metaplectic Iwahori-Whittaker functions inspired by Tokuyama's theorem. These can be related and better understood using representations of the Iwahori-Hecke algebra. This theory carries over to the infinite-dimensional setting, and connects with work on double affine Hecke algebras, where several intriguing open questions remain. 

Title: Interactions between invariant theory and complexity theory

Abstract: The study of symmetries in the setting of group actions via "invariant" polynomials is called invariant theory. Computation has played a key role in the evolution of invariant theory from its very beginnings with "masters of computation" such as Gordan, Sylvester, Cayley, etc in the mid 19th century. Over the last two decades, new directions in invariant theory have emerged from connections to computational complexity, the subject dedicated to understanding rigorously the notion of computational efficiency. In this talk, I will discuss how fundamental problems in complexity such as graph isomorphism, identity testing and even the celebrated P vs NP problem arise in the context of invariant theory. I will explain how developments in invariant theory can inform and make progress in complexity theory and in particular illustrate this in the case of identity testing. Towards the end, I will try to give a glimpse of the various directions in which this rapidly expanding field is headed. Based on joint works with Harm Derksen, Ankit Garg, Christian Ikenmeyer, Rafael Oliveira, Michael Walter, and Avi Wigderson. 

Title: Groups acting on trees: constructions, computations and classifications

Abstract: Following a motivation of groups acting on trees by situating them within the broader theory of all (locally compact) groups, I demonstrate that the ’global’ structure of such a group is innately connected to the ’local’ actions that vertex stabilisers induce on balls around the fixed vertex. This is illustrated by a particularly accessible class of groups acting on trees with ’prescribed’ local actions, following Burger-Mozes. Being defined solely in terms of finite permutation groups, these groups allow us to introduce computational methods to the world of locally compact groups: I will outline the capabilities of a recently developed GAP package that provides methods to create, analyse and find suitable local actions (joint work with Khalil Hannouch). Finally, I will outline a strategy to classify closed, vertex-transitive groups acting on trees.

Title: Quantum Subgroups of Lie Algebras

 

Abstract: Quantum subgroups are module categories, which encode the ``higher representation theory'' of the Lie algebras. They appear naturally in mathematical physics, where they correspond to extensions of the Wess-Zumino-Witten models. The classification of these quantum subgroups has been a long-standing open problem. The main issue at hand being the possible existence of exceptional examples. Despite considerable attention from both physicists and mathematicians, full results are only known for sl_2 and sl_3.

 

In this talk I will describe recent progress in the quantum subgroup classification program for sl_n. Our results finish off the classification for n = 5,6,7, and pave the way for higher ranks. In particular we discover several new exceptionals.

 

Despite being an algebraic question at heart, our techniques draw heavily from the theory of operator algebras. In particular Vaughan Jones planar algebras, and the Cuntz algebras make key appearances.


2020 Presentations


Title: Mixing times of Markov chains and the cutoff phenomenon

Abstract: Understanding the rate of convergence to stationarity, or mixing time, of a Markov chain is of interest for problems from shuffling cards to providing rigorous bounds for the runtime of Monte Carlo algorithms. In this talk, we will explain how the probabilistic technique of coupling can be used to bound mixing times. We will also discuss the cutoff phenomenon, which describes how mixing occurs very abruptly at, and not before, a precise point for certain Markov chains.

Title: Ultrapower techniques in classical and noncommutative probability theory

 Abstract: In 1979 János Komlós proved a generalised result of the strong law of large numbers by removing the i.i.d condition, commonly known as the Komlós Theorem, which guarantees almost everywhere convergence of a subsequence of Cesàro averages for any L^1-bounded sequence of random variables. Recently, Junge, Scheckter, and Sukochev proved that a generalisation of the Komlós Theorem holds in noncommutative probability theory, where certain von Neumann algebras take the role of L^∞ spaces.

The proof relies on ultrafilters and ultrapowers, which originated in model theory but have found a wide range of usages throughout mathematics.

In this talk, we discuss the relationship between the classical and noncommutative probability theories as well as the application of ultrapower techniques in the framework of a proof of the classical Komlós Theorem, including an original result on the ultrapower of L^∞ spaces.

Title: The Feuerbach theorem in universal geometry

Abstract: Geometric proofs are often the subject of revisiting and streamlining as new perspectives and tools emerge in the world of mathematics. Tangential to this exercise is the ever-growing web that is the study of triangle centres, and at the point of tangency we can find the Feuerbach theorem and the Feuerbach point. This thesis will revisit this theorem once more, this time from the perspective of a universal geometry founded upon linear algebra, working without the assumptions that accompanied previous methods. Such an approach results in some nice expressions which leads to a deeper understanding of the Feuerbach theorem. And of course, because this is exploration is geometric in nature, we also stumble across some tangential discoveries concerning the Feuerbach point's relation to another triangle centre.

Title: Attacks on the Polynomial Learning with Errors problem 

Abstract: The Polynomial Learning with Errors problem (PLWE) is a promising contender for lattice based, post quantum cryptography, with applications revolving around the security and implementation of Public Key Encryption schemes, and Fully Homomorphic Encryption schemes.  This talk will cover an introduction to the PLWE problem, including parameter choices and the underlying ring structure. We will then review specific choices that expose the PLWE to attacks, as well as hardness reductions that exist, relating the difficulty of PLWE to other known difficult problems.   

Title: Convolution of Harmonic Polynomials

Abstract: In this talk we will present an original result connecting the convolution of harmonic polynomials (those satisfying Laplace’s equation) to a certain algebra of symmetric tensors. Although the result itself is simple, we will discuss some beautiful results from the representation theory of compact groups that are utilized in the proof. 

Title: A Proof of Quantumness

Abstract: As quantum computers continue to increase in size, an efficient proof of quantumness is required to certify that they are performing computations as expected. In this talk, we examine a cryptographic protocol designed to solve this problem, which draws on connections between quantum computation and binary linear codes. Recently, this protocol was shown to be vulnerable to a classical attack, allowing for quantumness to be forged. We show how this attack can be generalised and suggest modifications to the original protocol such that it is no longer amenable to such an attack. 

Title: On Applied Mathematics, Formal Systems, and Structuralism 

Abstract:  In 1960 Eugene Wigner wrote The Unreasonable Effectiveness of Mathematics in the Natural Sciences, an essay questioning the role of mathematics in the physical sciences, and especially questioning the extremely accurate results mathematical theories provide in quantum physics. In this talk we lay out a philosophical framework for the application of mathematics in the physical sciences which draws on Aristotelian Realism, Strong Formalism, and an Empiricist schema. We then apply this framework to the application of Euclidean geometry as formalised by Euclid and Hilbert. Finally we consider in light of this case study the legitimacy of Wigner's questions. 

Title: Gelfand Duality and its Applications 

Abstract: The Gelfand Duality is an equivalence of categories between the category of commutative unital C*-algebras and the category of compact Hausdorff spaces. The talk will cover this equivalence of categories, some of the language needed to describe it (categories, C*-algebras), some generalisations and specialisations, and some interesting applications in harmonic analysis and group theory.

Title: How can we understand an abelian category?

Abstract: Abelian categories are a generalisation of a module categories and play an important role in areas such as algebraic topology and algebraic geometry. A well-known result of Morita states that any abelian category with a progenerator is equivalent to a module category. However, not all abelian categories are modules categories.

In this talk we will introduce the notion of a quotient of an abelian category and present a Morita-type result, giving necessary and sufficient conditions for an abelian category to be equivalent to a quotient of the category of graded right modules over some graded algebra.

Title: The group of automorphisms of the shift dynamical system and the Higman-Thompson groups

Abstract: We give a survey of recent results exploring connections between the Higman-Thompson groups and their automorphism groups and the group of automorphisms of the shift dynamical system. Our survey takes us from dynamical systems to group theory via groups of homeomorphisms with a segue through combinatorics, in particular, de Bruijn graphs.  Joint work with Collin Bleak and Peter Cameron.

Title: An introduction to groupoids and their C*-algebras

Abstract: C*-algebras were first introduced in order to model physical observables in quantum mechanics, but are now studied more abstractly in pure mathematics. Much of the current research of C*-algebraists involves constructing interesting classes of C*-algebras from various mathematical objects—such as groups, group actions, and graphs—and studying their properties. In this talk, I will define topological groupoids and examine Renault's 1980 construction of a groupoid C*-algebra, which provides a unifying model for C*-algebras associated to groups, group actions, and graphs. 

Title: Higher categories and comparisons of different models

Abstract: Ordinary category theory provides a formal language to explain similar phenomena that happen in various contexts, enabling the transfer of ideas from one area of study to another. For many relevant categorical structures that occur naturally (e.g. most flavors of cobordism categories), the relations expected to hold in a category, such as associativity of composition, are not satisfied exactly but rather up to a "higher isomorphism". To accommodate these examples, it is appropriate to relax the conditions that define a category and work with a "higher category" instead. The defining principles of a higher category leave a lot of freedom of interpretation and can be implemented by many models, each with their own advantages and disadvantages depending on the purpose. I will give an overview of how one should think about a higher categorical structure, mentioning the role they play in different situations, and I will discuss some results and work in progress in the context of comparing different models of higher categories. 

Title: Locally conformal Kähler structures on Lie groups

Abstract: During this talk I will recall some basic definitions involved in the notion of locally conformal Kähler (LCK) manifolds (complex structure, symplectic form, Kähler metric, etc.). I'll focus on LCK structures on solvmanifolds, compact quotients $\Gamma\backslash G$, where G is a solvable Lie group, and $\Gamma$ a co-compact discrete subgroup.

We will see some examples, recent results and open problems in this topic. This is based on joint works with Adrián Andrada (UNC-CONICET, Argentina).

Title: When the smallest is greates

Abstract: At the intersection of graph theory, extremal set theory and optimisation theory lies a collection of mathematical results often known as ‘min-max theorems’ and include celebrated combinatorial results such as Hall’s Theorem and Dilworth’s Theorem. These results are not only beautiful and fascinating but also have great theoretical and practical value. This talk will present some of these theorems along with their applications. 

Title: A surprising connection between symplectic geometry and geometric invariant theory 

Abstract: In the 1980s, a relationship was found between the two fields of symplectic geometry and geometric invariant theory (GIT) via the Kempf-Ness theorem. Symplectic geometry generalises the notion of the phase space in classical mechanics, while GIT studies quotients by group actions in algebraic geometry.

In this talk, we give a brief overview of both symplectic geometry and GIT with the intent to discuss a corollary of the Kempf-Ness theorem. The corollary is the following: Given a smooth complex projective variety X and a complex reductive Lie group G, the GIT quotient of X by G is homeomorphic to the symplectic reduction of X by a maximal compact subgroup of G.

Title: Random graph models and their applications to social networks

Abstract: A social network can be represented as a collection of nodes and edges. The nodes represent the people in the network and the edges (or links) between nodes signify a relationship, such as friendship or professional acquaintance. In mathematical terms, such a structure is often called a graph. This talk will introduce the notion of a random graph model, which is essentially a non-deterministic algorithm for producing a graph. It will investigate various random graph models and provide an overview of their strengths and limitations in capturing some essential features of social networks.

Title:  Elliptic curves over the complex field: lattice - elliptic curve correspondence  

Abstract: An elliptic curve over a field k is a smooth projective curve of genus 1 with a distinguished point O. With Riemann-Roch theorem and assuming that char(K) $\neq$ 2,3 ,  every elliptic curve can be represented by $y^2 = x^3 + Ax + B,$ for $A , B \in k$. In this talk,  we give an overview of the uniformisation theorem, which states that any elliptic curve defined over $\mathbb{C}$ is isomorphic to $\mathbb{C}/L$ for some lattice $L$. We start by looking at maps that carry $\mathbb{C}/L$ to $E(\mathbb{C})$ isomorphically.  Then we look at two proofs of the inversion problem. The first proof uses the fact that j-invariant classifies elliptic curves over C up to isomorphism. The second proof is more concrete and is achieved via construction of a Riemann surface. 

Title: Complex Links and Algebraic Multiplicities

Abstract:  In this talk I will consider two invariants associated to pairs of strata X , Y in an analytic stratification of some ambient complex projective variety W. The main result states that if the closure of X is contained in the closure of Y and both are subvarieties of W with the closure of X irreducible, then the Hilbert-Samuel (or algebraic) multiplicity of the closure of Y along the closure of X equals the Euler characteristic of the space obtained by intersecting Y with the complex link of X in W. I will also discuss the ingredients used to prove this result, and in particular will present a local Lefschetz hyperplane theorem for such complex linking spaces. Applications of this result include a new algebraic formula for MacPherson's local Euler obstruction of W. This talk is based on recent joint work with Vidit Nanda [Oxford] which is available on the arxiv (https://arxiv.org/abs/2006.10452).

Title: On solenoids and their complements 

Abstract: The problem of classifying solenoids, their complements in S^3, and continuous maps on them was posed by Borsuk and Eilenberg in 1936. At the time, it stimulated substantial advances in algebraic topology and cohomology theory, including Eilenberg’s obstruction theory, Steenrod duality, and the Eilenberg—Mac Lane universal coefficient theorem. In this talk, I will give a historical overview of the problem, and touch upon some new developments obtained with methods from descriptive set theory. 

Title: Homogeneous Ricci flows

Abstract: In this talk I will introduce the Ricci flow evolution equation for Riemannian metrics on a smooth manifold, without assuming any previous knowledge on differential geometry. Then I will discuss recent progress on the structure and long-time behavior of the flow on manifolds which are homogeneous, explaining in particular why solutions which are defined for all positive times asymptotically approach equilibrium points of the equation, so called Ricci solitons. This is based on joint work with Christoph Böhm (Münster). 

Title: Stationary actions of higher rank lattices on von Neumann algebras

Abstract: I will talk about a recent joint work with Remi Boutonnet in which we show that for higher rank lattices (e.g. SL(3, Z)), the left regular representation is weakly contained in any weakly mixing unitary representation. This strengthens Margulis’ normal subgroup theorem (1978), Stuck-Zimmer’s stabilizer rigidity result (1992) as well as Peterson’s character rigidity result (2014). We also prove that Uniformly Recurrent Subgroups (URS) of higher rank lattices are finite, answering a question of Glasner-Weiss (2014). The main novelty of our work is a structure theorem for stationary actions of higher rank lattices on von Neumann algebras. 

Title: Playing billiards with Pi

Abstract: Mathematical billiards has been a playground for various topics within geometry and dynamical systems for the last 100 years. In this talk, we give an introduction to mathematical billiards and explore the importance of counting collisions in a billiard system. We show that by simply following the laws of physics we can use a billiard system to produce the first N digits of pi for any natural number N. Further, we will also cover some theorems regarding collisions in a billiard system and extend this technique to linear point billiards.  

Title: Locally compact piecewise full groups 

Abstract: A group G acting faithfully by homeomorphisms of the Cantor set is called piecewise full if any homeomorphism assembled piecewise from elements of G is itself an element of G. I will discuss when such a group admits a non-discrete locally compact second countable group topology and describe a number of examples. This is joint work with Alejandra Garrido and Colin Reid. 

Title: A revisionist History of (Cisinski) Model Category

 Abstract:  We'll begin with a revisionist history of model category theory (we'll imagine that, in place of homological algebra, the notion had its origin in category theory). Using this apocryphal premise, we'll describe Cisinski's theory of accessible localizers as the natural homotopification of the notion of models for a finite limit sketch in presheaf categories, a straightforward notion subsuming Grothendieck toposes, algebraic categories, and essentially algebraic categories.

We'll then use this understanding of model categories to revisit the 1963 work of Kan. While the study of categories, particularly here in Australia, has begotten the study of higher categories, e.g. 2, 3, ... , n, ... , \omega, in strict and various weak guises, little has been made of extending the structure down. We'll present a notion of Z-category, with z-morphisms in all integer dimensions, and show that Kan's 1963 paper can be read as a proof that spectra admit presentation as locally finite pointed Z-groupoids.