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Dynamical models for Liouville and obstructions to further progress on sign patternsSep 10 2018We define a class of dynamical systems by modifying a construction due to Tao, which includes certain Furstenburg limits arising from the Liouville function. Most recent progress on the Chowla conjectures and sign patterns of the Mobius and Liouville ... More

$\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$Feb 14 2017Mar 18 2019Using etale cohomology, we define a birational invariant for varieties in characteristic $p$ that serves as an obstruction to uniruledness - a variant on an obstruction to unirationality due to Ekedahl. We apply this to $\overline{M}_{1,n}$ and show that ... More

The equidistribution of L-functions of twists by Witt vector Dirichlet characters over function fieldsMay 11 2018Nov 12 2018Katz showed that the L-functions of all Dirichlet characters of F_q(t), with conductor a fixed power of a degree one prime, are equidistributed in the limit as q goes to infinity. We generalize this statement to the L-functions of twists of an arbitrary ... More

Square-root cancellation for sums of factorization functions over short intervals in function fieldsSep 13 2018We present new estimates for sums of the divisor function, and other similar arithmetic functions, in short intervals over function fields. (When the intervals are long, one obtains a good estimate from the Riemann hypothesis.) We obtain an estimate that ... More

$\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$Feb 14 2017Dec 14 2018Using etale cohomology, we define a birational invariant for varieties in characteristic $p$ that serves as an obstruction to uniruledness - a variant on an obstruction to unirationality due to Ekedahl. We apply this to $\overline{M}_{1,n}$ and show that ... More

Bounds for Matchings in Nonabelian GroupsFeb 03 2017We give upper bounds for triples of subsets of a finite group such that the triples of elements that multiply to 1 form a perfect matching. Our bounds are the first to give exponential savings in powers of an arbitrary finite group. Previously, Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans ... More

A representation theory approach to integral moments of L-functions over function fieldsOct 02 2018We propose a new heuristic approach to integral moments of L-functions over function fields, which we demonstrate in the case of Dirichlet characters ramified at one place (the function field analogue of the moments of the Riemann zeta function, where ... More

Correlation of Arithmetic Functions over F_q[T]Nov 12 2018For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we obtain a lower-order ... More

On the Ramanujan conjecture for automorphic forms over function fields I. GeometryMay 30 2018Sep 25 2018Let $G$ be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of $G$, subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of ... More

Free rational points on smooth hypersurfacesJun 20 2019Motivated by a recent question of Peyre, we apply the Hardy-Littlewood circle method to count "sufficiently free" rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the rational numbers.

Free rational curves on low degree hypersurfaces and the circle methodOct 16 2018We use a function field version of the Hardy-Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular ... More

Bounds for the stalks of perverse sheaves in characteristic p and a conjecture of Shende and TsimermanJul 10 2019We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic ... More

South Pointing Chariot: An Invitation to Differential GeometryFeb 24 2015We introduce the south-pointing chariot, an intriguing mechanical device from ancient China. We use its ability to keep track of a global direction as it travels on an arbitrary path as a tool to explore the geometry of curved surfaces. This takes us ... More

Three-Dimensional 2-Framed TQFTs and SurgeryDec 08 1999The notion of 2-framed three-manifolds is defined. The category of 2-framed cobordisms is described, and used to define a 2-framed three-dimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is ... More

Links, Quantum Groups, and TQFT'sJun 02 1995Jul 31 1995The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given. The quantum ... More

Path Integration in Two-Dimensional Toplogical Quantum Field TheoryMay 22 1995A positive, diffeomorphism-invariant generalized measure on the space of metrics of a two-dimensional smooth manifold is constructed. We use the term generalized measure analogously with the generalized measures of Ashtekar and Lewandowski and of Baez. ... More

The Growth Rate of Tri-Colored Sum-Free SetsJun 30 2016Jul 06 2018Let $G$ be an abelian group. A tri-colored sum-free set in $G^n$ is a collection of triples $({\bf a}_i, {\bf b}_i, {\bf c}_i)$ in $G^n$ such that ${\bf a}_i+{\bf b}_j+{\bf c}_k=0$ if and only if $i=j=k$. Fix a prime $q$ and let $C_q$ be the cyclic group ... More

Improved estimates for polynomial Roth type theorems in finite fieldsAug 31 2017Oct 01 2017We prove that, under certain conditions on the function pair $\varphi_1$ and $\varphi_2$, bilinear average $p^{-1}\sum_{y\in \mathbb{F}_p}f_1(x+\varphi_1(y)) f_2(x+\varphi_2(y))$ along curve $(\varphi_1, \varphi_2)$ satisfies certain decay estimate. As ... More

Jones-Witten Invariants for Nonsimply-Connected Lie Groups and The Geometry of the Weyl AlcoveMay 03 1999Oct 18 1999The quotient process of M\"uger and Brugui\`eres is used to construct modular categories and TQFTs out of closed subsets of the Weyl alcove of a simple Lie algebra. In particular it is determined at which levels closed subsets associated to nonsimply-connected ... More

A geometric version of the circle methodNov 28 2017Jun 19 2019We develop a geometric version of the circle method and use it to compute the compactly supported cohomology of the space of rational curves through a point on a smooth affine hypersurface of sufficiently low degree.

Periodic twists of $GL_3$-modular formsMay 13 2019We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding ... More

Lectures on Applied $\ell$-adic CohomologyDec 08 2017Dec 12 2018We describe how a systematic use the deep methods from $\ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an ... More

Monodromy of elliptic curve convolution, seven-point sheaves of $G_2$-type and motives of Beauville typeMar 15 2018Dec 07 2018We study the Tannakian properties of the category of perverse sheaves on elliptic curves endowed with the convolution product. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka group is isomorphic ... More

Chebotarev Density Theorem in Short Intervals for Extensions of $\mathbb{F}_q(T)$Oct 15 2018An old open problem in number theory is whether Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension $E$ of $\mathbb{Q}$ with Galois group $G$, a conjugacy class $C$ in $G$ and an $1\geq \varepsilon>0$, one wants ... More

Lectures on Applied $\ell$-adic CohomologyDec 08 2017Apr 16 2019We describe how a systematic use the deep methods from $\ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an ... More

Closed subsets of the Weyl alcove and TQFTsMar 29 2005For an arbitrary simple Lie algebra $\g$ and an arbitrary root of unity $q,$ the closed subsets of the Weyl alcove of the quantum group $U_q(\g)$ are classified. Here a closed subset is a set such that if any two weights in the Weyl alcove are in the ... More

Path integrals, SUSY QM and the Atiyah-Singer index theorem for twisted DiracMay 23 2016Jan 05 2017Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator ... More

A Rigorous Path Integral for N=1 Supersymmetic Quantum Mechanics on a Riemannian ManifoldJul 11 2012Mar 28 2013Following Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a short-time approximation to the propagator for imaginary time, N=1 supersymmetric quantum mechanics on a compact, even-dimensional ... More

Positivity of GIT heights of zero-cycles and hyperplane arrangementsJul 09 2015In 1996 as part of the development of arithmetic intersection theory and Arakelov theory, Zhang defined a "GIT height function" for semi-stable algebraic cycles in projective space. In the same work, Zhang conjectured that this height function was positive. ... More

Ramanujan Coverings of GraphsJun 08 2015Dec 03 2017Let $G$ be a finite connected graph, and let $\rho$ be the spectral radius of its universal cover. For example, if $G$ is $k$-regular then $\rho=2\sqrt{k-1}$. We show that for every $r$, there is an $r$-covering (a.k.a. an $r$-lift) of $G$ where all the ... More

Diffeomorphism-Invariant Spin Network StatesAug 04 1997Aug 08 1997We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose that G is a compact connected semisimple Lie group and P -> M is a smooth principal G-bundle. A `cylinder function' on ... More

Functional Integration on Spaces of ConnectionsJul 20 1995Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal $G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections on $P$ be a continuous function of the holonomies of $A$ along finitely many piecewise smoothly immersed ... More

Irreducibility of polynomials with a large gapMar 28 2018We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > \operatorname{deg}(c) + \operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is irreducible over $\mathbb ... More

The second moment theory of families of L-functionsApr 04 2018For a fairly general family of L-functions, we survey the known consequences of the existence of asymptotic formulas with power-sawing error term for the (twisted) first and second moments of the central values in the family. We then consider in detail ... More

On Finiteness of Certain Vassiliev InvariantsJul 31 1995The best known examples of Vassiliev invariants are the coefficients of a Jones-type polynomial expanded after exponential substitution. We show that for a given knot, the first $N$ Vassiliev invariants in this family determine the rest for some integer ... More

Shift-preserving maps on $ω^*$May 04 2016The shift map $\sigma$ on $\omega^*$ is the continuous self-map of $\omega^*$ induced by the function $n \mapsto n+1$ on $\omega$. Given a compact Hausdorff space $X$ and a continuous function $f: X \rightarrow X$, we say that $(X,f)$ is a quotient of ... More

The C*-algebra of a Single Invertible ElementDec 31 2010Oct 01 2011This paper characterizes the unital C*-algebra generated by a single invertible element as the unital free product of C[0,1] and C(T). To do this, I develop techniques to split and merge presentations of C*-algebras using free products in tandem with ... More

Möbius PolynomialsDec 13 2013We introduce the M\"obius polynomial $ M_n(x) = \sum_{d|n} \mu\left( \frac nd \right) x^d $, which gives the number of aperiodic bracelets of length $n$ with $x$ possible types of gems, and therefore satisfies $M_n(x) \equiv 0$ (mod $n$) for all $x \in ... More

Nakayama automorphisms of Frobenius algebrasFeb 19 2014We show that the Nakayama automorphism of a Frobenius algebra $R$ over a field $k$ is independent of the field (Theorem 4). Consequently, the $k$-dual functor on left $R$-modules and the bimodule isomorphism type of the $k$-dual of $R$, and hence the ... More

Bilinear Forms on Frobenius AlgebrasJan 25 2014We analyze the homothety types of associative bilinear forms that can occur on a Hopf algebra or on a local Frobenius \(k\)-algebra \(R\) with residue field \(k\). If \(R\) is symmetric, then there exists a unique form on \(R\) up to homothety iff \(R\) ... More

Universal Properties of Some QuiversJun 06 2011Jun 03 2017In this paper, I characterize four particular classes of directed multigraphs, or quivers, as images under left and right adjoints to the natural vertex and edge functors. In particular, the following notions coincide: (1) independent sets of vertices ... More

Scaled-free objects IIMay 28 2014Feb 26 2015This work creates two categories of "array-weighted sets" for the purposes of constructing universal matrix-normed spaces and algebras. These universal objects have the analogous universal property to the free vector space, lifting maps completely bounded ... More

Matricial Banach spacesMay 23 2014Feb 08 2015This work performs a study of the category of complete matrix-normed spaces, called matricial Banach spaces. Many of the usual constructions of Banach spaces extend in a natural way to matricial Banach spaces, including products, direct sums, and completions. ... More

Birthday Inequalities, Repulsion, and Hard SpheresJun 03 2015Nov 26 2015We study a birthday inequality in random geometric graphs: the probability of the empty graph is upper bounded by the product of the probabilities that each edge is absent. We show the birthday inequality holds at low densities, but does not hold in general. ... More

anesthetic: nested sampling visualisationMay 12 2019anesthetic is a Python package for processing nested sampling runs, and will be useful for any scientist or statistician who uses nested sampling software. anesthetic unifies many existing tools and techniques in an extensible framework that is intuitive ... More

Isoperimetric functions for subdirect products and Bestvina-Brady groupsOct 22 2008In this thesis we investigate the Dehn functions of two different classes of groups: subdirect products, in particular subdirect products of limit groups; and Bestvina-Brady groups. Let D = \Gamma_1 \times ... \times \Gamma_n be a direct product of n ... More

Generalised Sobolev Stable Flux ReconstructionApr 04 2018Mar 08 2019A new set of symmetric correction functions is presented for high-order flux reconstruction, that expands upon, while incorporating, all previous correction function sets and opens the possibility for improved performance. By considering FR applied to ... More

No Maximal Models from Looking DownNov 03 2015In [Sh893], Shelah proves that (on a stationary set of cardinals) an AEC has not too many models or every model has extensions of arbitrary cardinality. We show that, if we assume limited amalgamation, then the second condition holds for a larger class ... More

The Γ-Ultraproduct and Averageable ClassesNov 03 2015Sep 03 2016We consider an ultraproduct that is designed to omit a fixed set of unary types $\Gamma$, called the $\Gamma$-ultraproduct. The $\Gamma$-ultraproduct is not always well-behaved, but we discuss several general conditions under which it is and several examples. ... More

Ideals and idempotents in the uniform ultrafiltersMay 08 2015If $S$ is a discrete semigroup, then $\beta S$ has a natural, left-topological semigroup structure extending $S$. Under some very mild conditions, $U(S)$, the set of uniform ultrafilters on $S$, is a two-sided ideal of $\beta S$, and therefore contains ... More

On the universality of the countable random graph: a density theoremSep 02 2016It is proved that for almost every graph on $\mathbb{N}$, every $A \subseteq \mathbb{N}$ of positive upper density contains every countable graph as an induced subgraph. This can be viewed as a stronger, Ramsey-theoretic version of the well-known embedding ... More

Efficient and affordable catadioptric spectrograph designs for 4MOST and HectorJun 30 2014Spectrograph costs have become the limiting factor in multiplexed fiber-based spectroscopic instruments, because tens of millions of resolution elements (spectral x spatial) are now required. Catadioptric (Schmidt-like) designs allow faster cameras and ... More

A fast new catadioptric design for fiber-fed spectrographsJul 30 2012Jul 31 2012The next generation of massively multiplexed multi-object spectrographs (DESpec, SUMIRE, BigBOSS, 4MOST, HECTOR) demand fast, efficient and affordable spectrographs, with higher resolutions (R = 3000-5000) than current designs. Beam-size is a (relatively) ... More

A Functorial Link between Quivers and HypergraphsJul 30 2016May 22 2018This paper discusses some issues arising from the category $\mathfrak{H}$ of hypergraphs, the category $\mathfrak{M}$ of (undirected) multigraphs, and the topos $\mathfrak{Q}$ of quivers. First, the natural inclusion of $\mathfrak{M}$ into $\mathfrak{H}$ ... More

Direct Sum Decompositions and Indecomposable TQFT'sMay 23 1995The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT's in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space ... More

Tameness from Large Cardinal AxiomsMar 03 2013May 13 2014We show that Shelah's Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with $LS(K)$ below ... More

Definable Coherent Ultrapowers and Elementary ExtensionsSep 09 2016We develop the notion of coherent ultrafilters (extenders without normality or well-foundedness). We then use definable coherent ultraproducts to characterize any extension of a model $M$ in any fragment of $\mathbb{L}_{\infty, \omega}$ that defines Skolem ... More

Scaled-Free ObjectsNov 02 2010Apr 17 2012In this work, I address a primary issue with adapting categorical and algebraic concepts to functional analytic settings, the lack of free objects. Using a "normed set" and associated categories, I describe constructions of normed objects, which build ... More

Universal flows and automorphisms of $\mathcal P(ω)/\mathrm{fin}$Feb 06 2018We prove that for every countable discrete group $G$, there is a $G$-flow on $\omega^*$ that has every $G$-flow of weight $\leq\! \aleph_1$ as a quotient. It follows that, under the Continuum Hypothesis, there is a universal $G$-flow of weight $\leq\!\mathfrak{c}$. ... More

From Theory to Systems: A Grounded Approach to Programming Language EducationApr 14 2019I present a new approach to teaching a graduate-level programming languages course focused on using systems programming ideas and languages like WebAssembly and Rust to motivate PL theory. Drawing on students' prior experience with low-level languages, ... More

Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot GameJul 25 2011We begin by reviewing and proving the basic facts of combinatorial game theory. We then consider scoring games (also known as Milnor games or positional games), focusing on the "fixed-length" games for which all sequences of play terminate after the same ... More

A pathological o-minimal quotientApr 11 2014We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an o-minimal structure ... More

Injective Envelopes and Projective Covers of QuiversNov 09 2011Jun 04 2012This paper characterizes the injective and projective objects in the category of directed multigraphs, or quivers. Further, the injective envelope and projective cover of any quiver in this category is constructed.

Very fast transmissive spectrograph designs for highly multiplexed fiber spectroscopyJul 26 2016Very fast (f/1.2 and f/1.35) transmissive spectrograph designs are presented for Hector and MSE. The designs have 61mm x 61mm detectors, 4 or 5 camera lenses of aperture less than 228mm, with just 6 air/glass surfaces, and rely on extreme aspheres for ... More

A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative GeometryMay 17 2019The Accardi-Boukas quantum Black-Scholes equation can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck ... More

An Isoperimetric Function for Bestvina-Brady GroupsMay 29 2007Dec 17 2008Given a right-angled Artin group A, the associated Bestvina-Brady group is defined to be the kernel of the homomorphism A \to \mathbb{Z} that maps each generator in the standard presentation of A to a fixed generator of \mathbb{Z}. We prove that the Dehn ... More

A Subgroup of a Direct Product of Free Groups whose Dehn Function has a Cubic Lower BoundAug 06 2007Jan 07 2009We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of 3 free groups.

On The Generators Of Quantum Stochastic Operator CocyclesOct 10 2005Jun 16 2006The stochastic generators of Markov-regular operator cocycles on symmetric Fock space are studied in a variety of cases: positive cocycles, projection cocycles, and partially isometric cocycles. Moreover a class of transformations of positive contraction ... More

Contractions of 3-folds: deformations and invariantsNov 05 2015This note discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper, the author and Michael Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It characterises their geometric ... More

Relationship of the Hennings and Chern-Simons Invariants For Higher Rank Quantum GroupsJan 05 2017Jul 24 2018The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with Jones- Witten-Reshetikhin-Turaev invariant arising from Chern-Simons filed theory for the same Lie algebra and the ... More

Grassmannian twists on the derived category via spherical functorsNov 16 2011We construct new examples of derived autoequivalences for a family of higher-dimensional Calabi-Yau varieties. Specifically, we take the total spaces of certain natural vector bundles over Grassmannians G(r,d) of r-planes in a d-dimensional vector space, ... More

On measuring the absolute scale of baryon acoustic oscillationsMay 03 2012Oct 11 2012The baryon acoustic oscillation (BAO) feature in the distribution of galaxies provides a fundamental standard ruler which is widely used to constrain cosmological parameters. In most analyses, the comoving length of the ruler is inferred from a combination ... More

The Combinatorial Game Theory of Well-Tempered Scoring GamesDec 15 2011We consider the class of "well-tempered" integer-valued scoring games, which have the property that the parity of the length of the game is independent of the line of play. We consider disjunctive sums of these games, and develop a theory for them analogous ... More

Negative Ricci curvature on some non-solvable Lie groups IIOct 20 2017We construct many examples of Lie groups with compact Levi factor admitting a left-invariant metric with negative Ricci curvature. We start with a Lie algebra with Levi factor su(n) or so(n) acting on an abelian nilradical via the representation on the ... More

Random k-SAT and the Power of Two ChoicesSep 24 2012Oct 15 2013We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule ... More

An off-axis, wide-field, diffraction-limited, reflective Schmidt TelescopeAug 06 2010Off-axis telescopes with unobstructed pupils offer great advantages in terms of emissivity, throughput, and diffractionlimited energy concentration. For most telescope designs, implementation of an off-axis configuration imposes enormous penalties in ... More

Partial results on dp-finite fieldsMar 27 2019We prove that NIP valued fields of positive characteristic are henselian. Furthermore, we partially generalize the known results on dp-minimal fields to dp-finite fields. We prove a dichotomy: if K is a sufficiently saturated dp-finite expansion of a ... More

Finite burden in multivalued algebraically closed fieldsMay 13 2019We prove that an expansion of an algebraically closed field by $n$ arbitrary valuation rings is NTP${}_2$, and in fact has finite burden. It fails to be NIP, however, unless the valuation rings form a chain. Moreover, the incomplete theory of algebraically ... More

Improvements and Generalizations of Stochastic Knapsack and Multi-Armed Bandit Approximation Algorithms: Full VersionJun 05 2013Sep 13 2016We study the multi-armed bandit problem with arms which are Markov chains with rewards. In the finite-horizon setting, the celebrated Gittins indices do not apply, and the exact solution is intractable. We provide approximation algorithms for a more general ... More

Generalizing the Converse to Pascal's Theorem via Hyperplane Arrangements and the Cayley-Bacharach TheoremAug 16 2011Using a new point of view inspired by hyperplane arrangements, we generalize the converse to Pascal's Theorem, sometimes called the Braikenridge-Maclaurin Theorem. In particular, we show that if 2k lines meet a given line, colored green, in k triple points ... More

Presentations and Tietze transformations of C*-algebrasDec 06 2010Mar 06 2012In this work, I develop a new view of presentation theory for C*-algebras, both unital and non-unital, heavily grounded in classical notions from algebra. In particular, I introduce Tietze transformations for these presentations, which lead to a transformation ... More

Universal Properties of Some QuiversJun 06 2011In this paper, I characterize four particular classes of directed multigraphs, or quivers, as images under left and right adjoints to the natural vertex and edge functors. In particular, the following notions coincide: (1) independent sets of vertices ... More

On finite derived quotients of 3-manifold groupsMay 17 2014This paper studies the set of finite groups appearing as $\pi_1(M)/\pi_1(M)^{(n)}$, where $M$ is a closed, orientable 3-manifold and $\pi_1(M)^{(n)}$ denotes the $n$-th term of the derived series of $\pi_1(M)$. Our main result is that if $M$ is a closed, ... More

Two Generator groups acting on the complex hyperbolic planeOct 06 2015This is an expository article about groups generated by two isometries of the complex hyperbolic plane.

The isomorphism class of the shift mapApr 22 2019The \emph{shift map} $\sigma$ is the self-homeomorphism of $\omega^* = \beta\omega \setminus \omega$ induced by the successor function $n \mapsto n+1$ on $\omega$. We prove that the isomorphism classes of $\sigma$ and $\sigma^{-1}$ cannot be separated ... More

On the proof of elimination of imaginaries in algebraically closed valued fieldsJun 13 2014We give a simplified proof of elimination of imaginaries (in the geometric sorts) in ACVF, based on ideas of Hrushovski. This proof manages to avoid many of the technical issues which arose in the original proof by Haskell, Hrushovski, and Macpherson. ... More

Nonlocal Diffusions and The Quantum Black-Scholes Equation: Modelling the Market Fear FactorJun 06 2018Jun 27 2018In this paper, we establish a link between quantum stochastic processes, and nonlocal diffusions. We demonstrate how the non-commutative Black-Scholes equation of Accardi & Boukas (Luigi Accardi, Andreas Boukas, 'The Quantum Black-Scholes Equation', Jun ... More

PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black-Scholes EquationDec 03 2018Jan 18 2019The Accardi-Boukas quantum Black-Scholes framework, provides a means by which one can apply the Hudson-Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via ... More

Bending Fuchsian representations of fundamental groups of cusped surfaces in PU(2,1)Jan 05 2011We describe a family of representations of $\pi_1(\Sigma)$ in PU(2,1), where $\Sigma$ is a hyperbolic Riemann surface with at least one deleted point. This family is obtained by a bending process associated to an ideal triangulation of $\Sigma$. We give ... More

The Growth Rate of Tri-Colored Sum-Free SetsJun 30 2016Let $G$ be an abelian group. A tri-colored sum-free set in $G^n$ is a collection of triples $({\bf a}_i, {\bf b}_i, {\bf c}_i)$ in $G^n$ such that ${\bf a}_i+{\bf b}_j+{\bf c}_k=0$ if and only if $i=j=k$. Fix a prime $q$ and let $C_q$ be the cyclic group ... More

Notes on commutation of limits and colimitsSep 28 2014May 04 2015We show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are intermediate between the class of pseudo-filtered colimits and that of all ... More

Rational cohomology toriSep 03 2015We study normal compact K\"ahler spaces whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give constraints on ... More

Dynamic temperature selection for parallel-tempering in Markov chain Monte Carlo simulationsJan 23 2015Mar 15 2016Modern problems in astronomical Bayesian inference require efficient methods for sampling from complex, high-dimensional, often multi-modal probability distributions. Most popular methods, such as Markov chain Monte Carlo sampling, perform poorly on strongly ... More

On cap sets and the group-theoretic approach to matrix multiplicationMay 21 2016Aug 10 2016In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent $\omega$ of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific ... More

Cosmological constraints from galaxy clusteringJan 24 2006In this manuscript I review the mathematics and physics that underpins recent work using the clustering of galaxies to derive cosmological model constraints. I start by describing the basic concepts, and gradually move on to some of the complexities involved ... More

Markov-chain reconstruction of the 2dF Galaxy Redshift Survey real-space power spectrumOct 26 2004The real-space power spectrum of L* galaxies measured from the 2dF Galaxy Redshift Survey (2dFGRS) is presented. Markov-Chain Monte-Carlo (MCMC) sampling was used to fit radial and angular modes resulting from a Spherical Harmonics decomposition of the ... More

Large Scale Structure ObservationsDec 19 2013Galaxy Surveys are enjoying a renaissance thanks to the advent of multi-object spectrographs on ground-based telescopes. The last 15 years have seen the fruits of this experimental advance, including the 2-degree Field Galaxy Redshift Survey (2dFGRS; ... More

From Haar to Lebesgue via Domain Theory, Revised versionApr 01 2015If ${\mathcal C}\simeq 2^{\mathbb N}$ denotes the Cantor set realized as the infinite product of two-point groups, then a folklore result says the Cantor map from ${\mathcal C}$ into $[0,1]$ sends Haar measure to Lebesgue measure on the interval. In fact, ... More

Chains of saturated models in AECsMar 30 2015May 12 2015We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. Under a natural superstability assumption (which follows from categoricity in a high-enough cardinal), we prove: $\mathbf{Theorem}$ ... More

Forking in Short and Tame Abstract Elementary ClassesJun 27 2013Sep 21 2016We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC $K$ is tame, type-short, and failure of an order-property, we consider {\bf Definition.} Let $M_0 \prec N$ be models ... More