Past

https://arxiv.org/pdf/2004.06929.pdf

Motivation for the study of fusion categories is twofold: Fusion categories arise in wide array of mathematical subjects, and provide the necessary input for some fascinating topological constructions. We will carefully define what fusion categories are, and give representation theoretic examples. Then, we will explain how fusion categories are inherently finite combinatorial objects. We proceed to construct an example that does not come from group theory. Time permitting, we will go some way towards introducing so-called modular tensor categories.

Let $X_1, \ldots$ be i.i.d. copies of some real random variable $X$. For any $\varepsilon_2, \varepsilon_3, \ldots$ in $\{0,1\}$, a basic algorithm introduced by H.A. Simon yields a reinforced sequence $\hat{X}_1, \hat{X}_2, \ldots$ as follows. If $\varepsilon_n=0$, then $\hat{X}_n$ is a uniform random sample from $\hat{X}_1, …, \hat{X}_{n-1}$; otherwise $\hat{X}_n$ is a new independent copy of $X$. The purpose of this talk is to compare the scaling exponent of the usual random walk $S(n)=X_1 +\ldots + X_n$ with that of its step reinforced version $\hat{S}(n)=\hat{X}_1+\ldots + \hat{X}_n$. Depending on the tail of $X$ and on asymptotic behavior of the sequence $\varepsilon_j$, we show that step reinforcement may speed up the walk, or at the contrary slow it down, or also does not affect the scaling exponent at all. Our motivation partly stems from the study of random walks with memory, notably the so-called elephant random walk and its variations.

Fyodorov-Hiary-Keating established a series of conjectures concerning the large values of the Riemann zeta function in a random short interval. After reviewing the origins of these predictions through the random matrix analogy, I will explain recent work with Louis-Pierre Arguin and Maksym Radziwill, which proves a strong form of the upper bound for the maximum.

We say that a graph $G$ is $H$-free if $G$ does not contain $H$ as a (not necessarily induced) subgraph. For a positive integer $n$, denote by $\text{ex}(n,H)$ the largest number of edges in an $H$-free graph with $n$ vertices (the Turán number of $H$). The classical theorem of Erdős, Kleitman, and Rothschild states that, for every $r\geq3$, there are $2^{\text{ex}(n,H)+o(n2)}$ many $K_r$-free graphs with vertex set $\{1,…, n\}$. There exist (at least) three different derivations of this estimate in the literature: an inductive argument based on the Kővári-Sós-Turán theorem (and its generalisation to hypergraphs due to Erdős), a proof based on Szemerédi's regularity lemma, and an argument based on the hypergraph container theorems. In this talk, we present yet another proof of this bound that exploits connections between entropy and independence. This argument is an adaptation of a method developed in a joint work with Gady Kozma, Tom Meyerovitch, and Ron Peled that studied random metric spaces.

We study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the Functional Ito calculus to our more general framework. In particular, unlike in the Functional Ito calculus, where one needs only to consider stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths.  We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.

Joint work with JianFeng Zhang (USC).

In some ways the theory of mapping class groups of 4-manifolds is in 2020 at the same place where the theory of mapping class groups of 2-manifolds was in 1973, before Thurston changed everything.  In this talk I will describe some first steps in an ongoing joint project with Eduard Looijenga where we are trying to understand mapping class groups of certain algebraic surfaces (e.g. rational elliptic surfaces, and also K3 surfaces) from the Thurstonian point of view.

TBA

Given a log Calabi-Yau surface Y with maximal boundary D, I'll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I'll explain how to relate this to the total space of the SYZ fibration predicted by Gross--Hacking--Keel, and, time permitting, explain ties with earlier work of Auroux--Katzarkov--Orlov and Abouzaid. Joint work with Paul Hacking.

I will revisit the question of what is the holographic dual of N=6 supersymmetric Chern-Simons theories.

Eigenvalue-eigenvector pairs of combinatorial graph Laplacians are extensively used in graph theory and network analysis. It is well known that the spectrum of the Laplacian L of a given graph G encodes aspects of the geometry of G  - the multiplicity of the eigenvalue 0 counts the number of connected components while the second smallest eigenvalue (called the Fiedler eigenvalue) quantifies the well-connectedness of G . In network analysis, one uses Laplacian eigenvectors associated with small eigenvalues to perform spectral clustering. In graph signal processing, graph Fourier transforms are defined in terms of an orthonormal eigenbasis of L. Eigenvectors of L also play a central role in graph neural networks.

Motivated by this we study eigenvalue-eigenvector pairs of Laplacians of random graphs and their potential use in TDA. I will present simulation results on what persistent homology barcodes of Bernoulli random graphs G(n, p) look like when we use Laplacian eigenvectors as filter functions. Also, I will discuss the conjectures made from the simulations as well as the challenges that arise when trying to prove them. This is work in progress.
 

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints.  We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we prove that LASSO leads to parameter shrinkage, propose measures to quantify robustness of neural networks to adversarial examples and compute sensitivities of optimised certainty equivalents in finance. We also propose extensions of this framework to a multiperiod setting. This talk is based on joint work with Daniel Bartl, Samuel Drapeau and Jan Obloj.

The many facets of community detection on networks 

Community detection, the decomposition of a graph into essential building blocks, has been a core research topic in network science over the past years. Since a precise notion of what consti- tutes a community has remained evasive, community detection algorithms have often been com- pared on benchmark graphs with a particular form of assortative community structure and classified based on the mathematical techniques they employ. However, this comparison can be misleading because apparent similarities in their mathematical machinery can disguise different goals and rea- sons for why we want to employ community detection in the first place. Here we provide a focused review of these different motivations that underpin community detection. This problem-driven classification is useful in applied network science, where it is important to select an appropriate algorithm for the given purpose. Moreover, highlighting the different facets of community detection also delineates the many lines of research and points out open directions and avenues for future research.

Given a curve , what is the surface  that has smallest area among all surfaces spanning ? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable manifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the interior regularity of these minimizers. Much less is known about the boundary regularity, in the case of codimension greater than 1. I will speak about some recent progress in this direction.

We discuss two recent extensions of work on Reynolds-robust preconditioners for the Navier-Stokes equations, to non-Newtonian fluids and to the equations of magnetohydrodynamics.  We model non-Newtonian fluids by means of an implicit constitutive relation between stress and strain. This framework is broadly applicable and allows for proofs of convergence under quite general assumptions. Since the stress cannot in general be solved for in terms of the strain, a three-field stress-velocity-pressure formulation is adopted. By combining the augmented Lagrangian approach with a kernel-capturing space decomposition, we derive a preconditioner that is observed to be robust to variations in rheological parameters in both two and three dimensions.  In the case of magnetohydrodynamics, we consider the stationary incompressible resistive Newtonian equations, and solve a four-field formulation for the velocity, pressure, magnetic field and electric field. A structure-preserving discretisation is employed that enforces both div(u) = 0 and div(B) = 0 pointwise. The basic idea of the solver is to split the fluid and electromagnetic parts and to employ our existing Navier-Stokes solver in the Schur complement. We present results in two dimensions that exhibit robustness with respect to both the fluids and magnetic Reynolds numbers, and describe ongoing work to extend the solver to three dimensions.

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Recently in a joint work with J. Dobrowolski and N. Ramsey it is shown that in any NSOP1 theory with existence,
Kim-independence satisfies all the basic axioms over sets (except base monotonicity) that hold in simple theories with forking-independence. This is an extension of the earlier work by I. Kaplan and N. Ramsey that such hold over models in any NSOP1 theory. All simple theories; unbounded PAC fields; vector spaces over ACF with bilinear maps; the model companion of the empty theory in any language are typical NSOP1 examples.

   An important issue now is to know the existence of canonical bases. In stable and simple theories well-behaving notion of canonical bases for types over models exists, which is used in almost all the advanced studies. But there are a couple of crucial obstacles in finding canonical bases in NSOP1 theories. In this talk I will report a partial success/limit of the project. Namely, a type of a certain Morley sequence over a model has the weak canonical base. In my talk I will try to explain all the related notions.

https://arxiv.org/abs/2003.02844

Leibniz’s principle of identity of indiscernibles at first sight appears completely unrelated to set theory, but Mycielski (1995) formulated a set-theoretic axiom nowadays referred to as LM (for Leibniz-Mycielski) which captures the spirit of Leibniz’s dictum in the following sense:  LM holds in a model M of ZF iff M is elementarily equivalent to a model M* in which there is no pair of indiscernibles.  LM was further investigated in a 2004  paper of mine, which includes a proof that LM is equivalent to the global form of the Kinna-Wagner selection principle in set theory.  On the other hand, one can formulate a strong negation of Leibniz’s principle by first adding a unary predicate I(x) to the usual language of set theory, and then augmenting ZF with a scheme that ensures that I(x) describes a proper class of indiscernibles, thus giving rise to an extension ZFI of ZF that I showed (2005) to be intimately related to Mahlo cardinals of finite order. In this talk I will give an expository account of the above and related results that attest to a lively interaction between set theory and Leibniz’s principle of identity of indiscernibles.

Much progress in the study of 3-manifolds has been made by considering the geometric structures they admit. This is nowhere more true than for 3-manifolds which admit a hyperbolic structure. However, in the land of algorithms a more combinatorial approach is necessary, replacing our charts and isometries with finite simplicial complexes that are defined by a finite amount of data. 

In this talk we'll have a look at how in fact one can combine the two approaches, using the geometry of hyperbolic 3-manifolds to assist in this more combinatorial approach. To do so we'll combine tools from Hyperbolic Geometry, Triangulations, and perhaps suprisingly Polynomial Algebra to find explicit bounds on the runtime of an algorithm for comparing Hyperbolic manifolds.

We study the following question at the intersection of extremal and structural graph theory. Given a fixed graph $H$ that embeds in a fixed surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph that embeds in $\Sigma$? Exact answers to this question are known for specific graphs $H$ when $\Sigma$ is the sphere. We aim for more general, albeit less precise, results. We show that the answer to the above question is $\Theta(nf(H))$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers. This is joint work with Tony Huynh and Gwenaël Joret [https://arxiv.org/abs/2003.13777]

The small subgraph conditioning method is an analysis of variance technique which was introduced by Robinson and Wormald in 1992, in their proof that almost all cubic graphs are Hamiltonian. The method has been used to prove many structural results about random regular graphs, mostly to show that a certain substructure is present with high probability. I will discuss some applications of the small subgraph conditioning method to hypergraphs, and describe a subtle issue which is absent in the graph setting.

Abstract: 

In this talk I will discuss regularization by noise from a pathwise perspective using non-linear Young integration, and discuss the relations with occupation measures and local times. This methodology of pathwise regularization by noise was originally proposed by Gubinelli and Catellier (2016), who use the concept of averaging operators and non-linear Young integration to give meaning to certain ill posed SDEs. 

In a recent work together with   Nicolas Perkowski we show that there exists a class of paths with exceptional regularizing effects on ODEs, using the framework of Gubinelli and Catellier. In particular we prove existence and uniqueness of ODEs perturbed by such a path, even when the drift is given as a Scwartz distribution. Moreover, the flow associated to such ODEs are proven to be infinitely differentiable. Our analysis can be seen as purely pathwise, and is only depending on the existence of a sufficiently regular occupation measure associated to the path added to the ODE. 

As an example, we show that a certain type of Gaussian processes has infinitely differentiable local times, whose paths then can be used to obtain the infinitely regularizing effect on ODEs. This gives insight into the powerful effect that noise may have on certain equations. I will also discuss an ongoing extension of these results towards regularization of certain PDE/SPDEs by noise.​

I will discuss some ongoing work on three-dimensional supersymmetric gauge theories and their relationship to (equivariant) quantum K-theory. I will emphasise the interplay between the physical and mathematical motivations and approaches, and attempt to build a dictionary between the two.  As an interesting example, I will discuss the quantum K-theory of flag manifolds. The QK ring will be related to the vacuum structure of a gauge theory with Chern-Simons interactions, and the (genus-0) K-theoretic invariants will be computed in terms of explicit residue formulas that can be derived from the relevant supersymmetric path integrals.

We discuss hyperkahler implosion spaces, their relevance to group actions and why they should fit into the symplectic duality picture. For certain groups we present candidates for the symplectic duals of the associated implosion spaces and provide computational evidence. This is joint work with Amihay Hanany and Frances Kirwan.
 

Modern cellular radio systems such as 4G and 5G use antennas with multiple elements, a technique known as MIMO, and the intention is to increase the capacity of the radio channel.  5G allows even more possibilities, such as massive MIMO, where there can be hundreds of elements in the transmit antenna, and beam-forming (or beam-steering), where the phase of the signals fed to the antenna elements is adjusted to focus the signal energy in the direction of the receivers.  However, this technology poses some difficult optimization problems, and here mathematicians can contribute.   In this talk I will explain the background, and then look at questions such as: what is an appropriate objective function?; what constraints are there?; are any problems of this type convex (or quasi-convex, or difference-of-convex)?; and, can big problems of this type be solved in real time?

We will introduce the Baum-Connes conjecture without coefficients, in the setting of discrete groups, and try to explain why it is interesting for operator algebraists. We will give some idea of the LHS and the RHS of the conjecture, without being too formal, and rather than trying to define the assembly map, we will explain what it does for finite groups, for the integers, for free groups, and finally for wreath products of a finite group with the integers (the latter result is joint work with R. Flores and S. Pooya; it raises a few open questions about classifying the corresponding group C*-algebras up to isomorphism).

In January of this year, a solution to Connes' Embedding Problem was announced on arXiv. The paper itself deals firmly in the realm of information theory and relies on a vast network of implications built by many hands over many years to get from an efficient reduction of the so-called Halting problem back to the existence of finite von Neumann algebras that lack nice finite-dimensional approximations. The seminal link in this chain was forged by astonishing results of Kirchberg which showed that Connes' Embedding Problem is equivalent to what is now known as Kirchberg's QWEP Conjecture. In this talk, I aim to introduce Kirchberg's conjecture and to touch on some of the many deep insights in the theory surrounding it.

We study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. We will see that the frontier of the LOB (e.g., the best ask price) is the value function of a mean-field stochastic control problem, as the limiting version of a Bertrand-type competition among the liquidity providers.
With a detailed analysis on the N-seller static Bertrand game, we formulate a continuous time limiting mean-field control problem of the representative seller.
We then validate the dynamic programming principle (DPP) and show that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation.
We argue that the value function can be used to obtain the equilibrium density function of the LOB. (Joint work with Jin Ma)

In 2007, Andrew Mayo and Thanos Antoulas proposed a rational interpolation algorithm to solve a basic problem in control theory: given samples of the transfer function of a dynamical system, construct a linear time-invariant system that realizes these samples.  The resulting theory enables a wide range of data-driven modeling, and has seen diverse applications and extensions.  We will introduce these ideas from a numerical analyst's perspective, show how the selection of interpolation points can be guided by a Sylvester equation and pseudospectra of matrix pencils, and mention an application of these ideas to a contour algorithm for the nonlinear eigenvalue problem. (This talk involves collaborations with Michael Brennan (MIT), Serkan Gugercin (Virginia Tech), and Cosmin Ionita (MathWorks).)

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 I will report on my recent work on dimension, definable groups, and definable fields in vector spaces over algebraically closed [real closed] fields equipped with a non-degenerate alternating bilinear form or a non-degenerate [positive-definite] symmetric bilinear form. After a brief overview of the background, I will discuss a notion of dimension and some other ingredients of the proof of the main result, which states that, in the above context, every definable group is (algebraic-by-abelian)-by-algebraic [(semialgebraic-by-abelian)-by-semialgebraic]. It follows from this result that every definable field is definable in the field of scalars, hence either finite or definably isomorphic to it [finite or algebraically closed or real closed].
 

https://arxiv.org/abs/2003.10466

https://arxiv.org/abs/2004.00015

https://arxiv.org/abs/2004.06115

Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above.  Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1}, \in \rangle$ and $\langle H_{\omega_2}, \in \rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

In this talk, I would like to advertise the strict equality between two objects from very different areas of mathematical physics:

- Kahane's Gaussian Multiplicative Chaos (GMC), which uses a log-correlated field as input and plays an important role in certain conformal field theories.

- A reference model in random matrices called the Circular Beta Ensemble (CBE).

The goal is to give a precise theorem whose loose form is GMC = CBE. Although it was known that random matrices exhibit log-correlated features, such an exact correspondence is quite a surprise. 

Dobrushin (1972) showed that, at low enough temperatures, the interface of the 3D Ising model - the random surface separating the plus and minus phases above and below the $xy$-plane - is localized: it has $O(1)$ height fluctuations above a fixed point, and its maximum height $M_n$ on a box of side length $n$ is $O_P(\log n)$. We study this interface and derive a shape theorem for its "pillars" conditionally on reaching an atypically large height. We use this to analyze the maximum height $M_n$ of the interface, and prove that at low temperature $M_n/\log n$ converges to $c\beta$ in probability. Furthermore, the sequence $(M_n - E[M_n])_{n\geq 1}$ is tight, and even though this sequence does not converge, its subsequential limits satisfy uniform Gumbel tails bounds.
Joint work with Reza Gheissari.

How long does it take for a pandemic to stop spreading? When modelling an infection process, especially these days, this is one of the main questions that comes to mind. In this talk, we consider this question in the bootstrap percolation setting.

Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollobás in 1968. In this process, we start with initial "infected" set of edges $E_0$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E_t \subseteq E(Kn)$, we infect a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n] , E_t \cup \{e\})$. A question raised by Bollobás asks for the maximum time the process can run before it stabilizes. Bollobás, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H=K_r$. They answered the question for $r \leq 4$ and gave a non-trivial lower bound for every $r \geq 5$. They also conjectured that the maximal running time is $o(n^2)$ for every integer $r$. We disprove their conjecture for every $r \geq 6$ and we give a better lower bound for the case $r=5$; in the proof we use the Behrend construction. This is a joint work with József Balogh, Alexey Pokrovskiy, and Tibor Szabó.

Ambitwistor strings are a class of holomorphic worldsheet models that directly describe massless quantum field theories, such as supergravity and super Yang-Mills. Their correlators give remarkably compact amplitude representations, known as the CHY formulas: characteristic worldsheet integrals that are fully localized on a set of polynomial constraints known as the scattering equations. Moreover, the ambitwistor string models provide a natural way of extending these formulas to loop level, where the constraints can be used to simplify the formulas (originally on higher genus curves) to 'forward limit-like' constructions on nodal spheres. After reviewing these developments, I will discuss one of the peculiar features of this approach: the worldsheet formulas on nodal spheres result in a non-standard integrand representation that makes it difficult to e.g. apply established integration techniques. While several approaches for addressing this look feasible or have been put forward in the literature, they only work for the simplest toy models. Taking inspiration from these attempts, I want to discuss a novel strategy to overcome this difficulty, and formulate compact worldsheet formulas with standard Feynman propagators.

Let ?={??}?∈ℤ be zero-mean stationary Gaussian sequence of random variables with covariance function ρ satisfying ρ(0)=1. Let φ:R→R be a function such that ?[?(?_0)2]<∞ and assume that φ has Hermite rank d≥1. The celebrated Breuer–Major theorem asserts that, if ∑|?(?)|^?<∞ then
the finite dimensional distributions of the normalized sum of ?(??) converge to those of ?? where W is
a standard Brownian motion and σ is some (explicit) constant. Surprisingly, and despite the fact this theorem has become over the years a prominent tool in a bunch of different areas, a necessary and sufficient condition implying the weak convergence in the
space ?([0,1]) of càdlàg functions endowed with the Skorohod topology is still missing. Our main goal in this paper is to fill this gap. More precisely, by using suitable boundedness properties satisfied by the generator of the Ornstein–Uhlenbeck semigroup,
we show that tightness holds under the sufficient (and almost necessary) natural condition that E[|φ(X0)|p]<∞ for some p>2.

Joint work with D Nualart
 

Just as differential equations often boundary conditions of various types, so too do quantum field theories often admit boundary theories. I will explain these notions and then discuss a theorem proved with Constantin Teleman, essentially characterizing certain 3-dimensional topological field theories which admit nonzero boundary theories. One application is to gapped systems in condensed matter physics.

I will describe the results of two projects on the construction of Calabi-Yau threefolds and certain real Lagrangian submanifolds. The first concerns the construction of a novel dataset of Calabi-Yau threefolds via an application of the Gross-Siebert algorithm to a reducible union of toric varieties obtained by degenerating anti-canonical hypersurfaces in a class of (around 1.5 million) Gorenstein toric Fano fourfolds. Many of these constructions correspond to smoothing such a hypersurface; in contrast to the famous construction of Batyrev-Borisov which exploits crepant resolutions of such hypersurfaces. A central ingredient here is the construction of a certain 'integral affine structure with singularities' on the boundary of a class of polytopes from which one can form a topological model, due to Gross, of the corresponding Calabi-Yau threefold X. In general, such topological models carry a canonical (anti-symplectic) involution i and in the second project, which is joint work with H. Argüz, we describe the fixed point locus of this involution. In particular, we prove that the map i*-1 on graded pieces of a Leray filtration of H^3(X,Z2) can be identified with the map D -> D^2, where D is an element of H^2(X',Z2) and X' is mirror-dual to X. We use this to compute the Z2 cohomology group of the fixed locus, answering a question of Castaño-Bernard--Matessi.

Over the last few years, it has become clear that working in sectors of large global charge leads to significant simplifications when studying strongly coupled CFTs. It allows us in particular to calculate the CFT data as an expansion in inverse powers of the large charge.
In this talk, I will introduce the large-charge expansion via the simple example of the O(2) model and will then apply it to a number of other systems which display a richer structure, such as non-Abelian global symmetry groups.