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$\mathfrak F$-categories and $\mathfrak F$-functors in Representation Theory IIFeb 06 2018This is a partial derivative of \cite{MR94g:17044}. We give a list of examples/problems that some will find amusing.

On the Universal Central Extension of Hyperelliptic Current AlgebrasMar 11 2015Jul 20 2015Let $p(t)\in\mathbb C[t]$ be a polynomial with distinct roots and nonzero constant term. We describe, using Fa\'a de Bruno's formula and Bell polynomials, the universal central extension in terms of generators and relations for the hyperelliptic current ... More

Certain families of Polynomials arising in the study of hyperelliptic Lie algebrasFeb 03 2016The associative ring $R(P(t))=\mathbb C[t^{\pm1},u \,|\, u^2=P(t)]$, where $P(t)=\sum_{i=0}^na_it^i=\prod_{k=1}^n(t-\alpha_i)$ with $\alpha_i\in\mathbb C$ pairwise distinct, is the coordinate ring of a hyperelliptic curve. The Lie algebra $\mathcal{R}(P(t))=\text{Der}(R(P(t)))$ ... More

Realizations of the three point algebra $\mathfrak{sl}(2, \mathcal R) \oplus\left(Ω_{\mathcal R}/d{\mathcal R}\right)$Mar 27 2013Feb 22 2015We describe the universal central extension of the three point current algebra $\mathfrak{sl}(2,\mathcal R)$ where $\mathcal R=\mathbb C[t,t^{-1},u\,|\,u^2=t^2+4t ]$ and construct realizations of it in terms of sums of partial differential operators.

Free Field Realizations of the Elliptic Affine Lie Algebra sl(2,R)+Ω_R/dRFeb 07 2009May 23 2009In this paper we construct two free field realizations of the elliptic affine Lie algebra sl(2,R) + \Omega_R/dR, where R=C[t,t^{-1},u|u^2=t^3 - 2b t^2 + t]. The first realization gives an analogue of Wakimoto's construction for Affine Kac-Moody algebras, ... More

Imaginary Verma modules and Kashiwara algebras for $U_q(\widehat{\mathfrak{sl}(2)})$Mar 05 2009We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{sl}(2)})$ and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal N_q^-$ of $U_q(\hat{\mathfrak{sl}(2)})$ ... More

Module structure of the center of the universal central extension of a genus zero Krichever-Novikov algebraFeb 20 2016Jun 24 2016We describe how the center of the universal central extension of the genus zero Krichever-Novikov current algebra decomposes as a direct sum of irreducible modules for automorphism group of the coordinate ring of this algebra.

Quantitative photoacoustic tomography using forward and adjoint Monte Carlo models of radianceAug 18 2016Forward and adjoint Monte Carlo (MC) models of radiance are proposed for use in model-based quantitative photoacoustic tomography. A 2D radiance MC model using a harmonic angular basis is introduced and validated against analytic solutions for the radiance ... More

A pseudospectral method for solution of the radiative transport equationJan 19 2018Mar 08 2018The radiative transport equation accurately describes light transport in participating media such as biological tissues, though analytic solutions are known only for simple geometries. We present a pseudospectral technique to efficiently compute numerical ... More

Simple superelliptic Lie algebrasDec 25 2014Let $m\in N$, $P(t)\in C[t]$. Then we have the Riemann surfaces (commutative algebras) $R_m(P)=C[t^{\pm1},u | u^m=P(t)]$ and $S_m(P)=C[t , u| u^m=P(t)].$ The Lie algebras $\mathcal{R}_m(P)=Der(R_m(P))$ and $\mathcal{S}_m(P)=Der(S_m(P))$ are called the ... More

A pseudospectral method for solution of the radiative transport equationJan 19 2018The Boltzmann transport equation accurately describes a number of physical phenomena, though analytic solutions are known only for simple geometries. We present a pseudospectral technique to efficiently compute numerical solutions to the time-domain transport ... More

Reconstruction-classification method for quantitative photoacoustic tomographyAug 04 2015We propose a combined reconstruction-classification method for simultaneously recovering absorption and scattering in turbid media from images of absorbed optical energy. This method exploits knowledge that optical parameters are determined by a limited ... More

DJKM algebras I: Their Universal Central ExtensionSep 05 2010The purpose of this paper is to explicitly describe in terms of generators and relations the universal central extension of the infinite dimensional Lie algebra, $\mathfrak g\otimes \mathbb C[t,t^{-1},u|u^2=(t^2-b^2)(t^2-c^2)]$, appearing in the work ... More

Intermediate Wakimoto modules for Affine sl(n+1)Aug 13 2003We construct certain boson type realizations of affine sl(n+1) that depend on a parameter r. When r=0 we get a Fock space realization of Imaginary Verma modules appearing in the work of the first author and when r=n they are the Wakimoto modules described ... More

On the Adjoint Operator in Photoacoustic TomographyFeb 05 2016Aug 01 2016Photoacoustic Tomography (PAT) is an emerging biomedical "imaging from coupled physics" technique, in which the image contrast is due to optical absorption, but the information is carried to the surface of the tissue as ultrasound pulses. Many algorithms ... More

Model based learning for accelerated, limited-view 3D photoacoustic tomographyAug 31 2017Recent advances in deep learning for tomographic reconstructions have shown great potential to create accurate and high quality images with a considerable speed-up. In this work we present a deep neural network that is specifically designed to provide ... More

Acoustic Wave Field Reconstruction from Compressed Measurements with Application in Photoacoustic TomographySep 09 2016We present a method for the recovery of compressively sensed acoustic fields using patterned, instead of point-by-point, detection. From a limited number of such compressed measurements, we propose to reconstruct the field on the sensor plane in each ... More

Families of orthogonal Laurent polynomials, hyperelliptic Lie algebras and elliptic integralsAug 03 2015We describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties are listed in ... More

Structure of Intermediate Wakimoto ModulesJan 19 2006We show that our construction of boson type realizations of affine $\mathfrak{sl}(n+1)$ in terms of intermediate Wakimoto modules gives representations that are generically isomorphic to certain Verma type modules. We then use this identification to obtain ... More

DJKM algebras and non-classical orthogonal polynomialsOct 08 2012Sep 07 2014We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa in their work on the Landau-Lifshitz equations. Two of the families of polynomials we show satisfy ... More

An imaginary PBW basis for quantum affine algebras of type 1Jul 10 2013Mar 28 2014Let $\hat{\mathfrak g}$ be an affine Lie algebra of type 1. We give a PBW basis for the quantum affine algebra $U_q(\hat{\mathfrak g})$ with respect to the triangular decomposition of $\hat{\mathfrak g}$ associated with the imaginary positive root system. ... More

Virasoro action on Imaginary Verma modules and the operator form of the KZ-equationDec 22 2011We define the Virasoro algebra action on imaginary Verma modules for affine sl(2) and construct the analogs of Knizhnik-Zamolodchikov equation in the operator form. Both these results are based on a free field realization of imaginary Verma modules.

Imaginary Verma Modules for $U_q(\widehat{\mathfrak{sl}(2)})$ and Crystal-like basesSep 02 2015We consider imaginary Verma modules for quantum affine algebraU_q(\widehat{\mathfrak{sl}(2)}) and define a crystal-like base which we call an imaginary crystal basis using the Kashiwara algebra K_q constructed in earlier work of the authors. In particular, ... More

Free field realizations of the Date-Jimbo-Kashiwara-Miwa algebraSep 27 2013We use the description of the universal central extension of the DJKM algebra $\mathfrak{sl}(2, R)$ where $ R=\mathbb C[t,t^{-1},u\,|\,u^2=t^4-2ct^2+1 ]$ given in earlier work to construct realizations of the DJKM algebra in terms of sums of partial differential ... More

Representations of $a_{\infty}$ and $d_{\infty}$ with central charge 1 on the single neutral fermion Fock space $\mathit{F^{\otimes \frac{1}{2}}}$Sep 28 2013Nov 29 2013We construct a new representation of the infinite rank Lie algebra $a_{\infty}$ with central charge $c=1$ on the Fock space $\mathit{F^{\otimes \frac{1}{2}}}$ of a single neutral fermion. We show that $\mathit{F^{\otimes \frac{1}{2}}}$ is a direct sum ... More

The 3-point Virasoro algebra and its action on a Fock spaceFeb 13 2015Jul 20 2015We define a 3-point Virasoro algebra, and construct a representation of it on a previously defined Fock space for the 3-point affine algebra $\mathfrak{sl}(2, \mathcal R) \oplus\left( \Omega_{\mathcal R}/d{\mathcal R}\right)$.

Kashiwara Algebras and Imaginary Verma Modules for $U_q(\hat{\mathfrak{g}})$Aug 08 2013Aug 15 2013We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{g}})$, where $\hat{\mathfrak{g}}$ is of type 1 i.e. of non-twisted type, and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a ... More

$N$-point locality for vertex operators: normal ordered products, operator product expansions, twisted vertex algebrasJul 18 2013In this paper we study fields satisfying $N$-point locality and their properties. We obtain residue formulae for $N$-point local fields in terms of derivatives of delta functions and Bell polynomials. We introduce the notion of the space of descendants ... More

Representations of quantum groups defined over commutative rings IIJan 19 2006In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.

A Wakimoto type realization of toroidal $\mathfrak{sl}_{n+1}$Jul 07 2010The authors construct a Wakimoto type realization of toroidal $\mathfrak{sl}_{n+1}$ The representation constructed in this paper utilizes non-commuting differential operators acting on the tensor product of two polynomial rings in many commuting variables. ... More

A note on the $R_\infty$ property for groups $\mathrm{FAlt}(X)\leqslant G\leqslant \mathrm{Sym}(X)$Feb 08 2016Aug 22 2018Given a set $X$, the group $\mathrm{Sym}(X)$ consists of all bijections from $X$ to $X$, and $\mathrm{FSym}(X)$ is the subgroup of maps with finite support i.e. those that move only finitely many points in $X$. We describe the automorphism structure of ... More

Molecular simulations of heterogeneous ice nucleation. II. Peeling back the layersJan 08 2015May 29 2015Coarse grained molecular dynamics simulations are presented in which the sensitivity of the ice nucleation rate to the hydrophilicity of a graphene nanoflake is investigated. We find that an optimal interaction strength for promoting ice nucleation exists, ... More

Molecular simulations of heterogeneous ice nucleation. I. Controlling ice nucleation through surface hydrophilicityJan 08 2015May 29 2015Ice formation is one of the most common and important processes on earth and almost always occurs at the surface of a material. A basic understanding of how the physicochemical properties of a material's surface affect its ability to form ice has remained ... More

$N$-point Virasoro Algebras and Their Modules of DensitiesAug 30 2013In this paper we introduce and study $n$-point Virasoro algebras, $\tilde{\W_a}$, which are natural generalizations of the classical Virasoro algebra and have as quotients multipoint genus zero Krichever-Novikov type algebras. We determine necessary and ... More

A viscous froth model adapted to wet foamsMar 30 2017We describe the extension of a "viscous froth" model to the dynamics of a wet foam in a Hele-Shaw cell. The two-dimensional model includes the friction experienced by the soap films as they are dragged along the cell walls, while retaining accurate geometrical ... More

A perturbation result for quasi-linear stochastic differential equations in UMD Banach spacesMar 07 2012We consider the effect of perturbations to a quasi-linear parabolic stochastic differential equation set in a UMD Banach space $X$. To be precise, we consider perturbations of the linear part, i.e. the term concerning a linear operator $A$ generating ... More

Skorokhod embeddings, minimality and non-centred target distributionsOct 27 2003Mar 24 2005In this paper we consider the Skorokhod embedding problem for target distributions with non-zero mean. In the zero-mean case, uniform integrability provides a natural restriction on the class of embeddings, but this is no longer suitable when the target ... More

Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noiseDec 18 2018We provide convergence rates for space approximations of semi-linear stochastic differential equations with multiplicative noise in a Hilbert space. The space approximations we consider are spectral Galerkin and finite elements, and the type of convergence ... More

On decoupling in Banach spacesMay 31 2018We consider decoupling inequalities for random variables taking values in a Banach space $X$. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a ... More

Homomorphisms and Higher Extensions for Schur algebras and symmetric groupsAug 26 2005This paper surveys, and in some cases generalises, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the ... More

Homomorphisms between Weyl modules for SL_3(k)Aug 26 2005We classify all homomorphisms between Weyl modules for SL_3(k) when k is an algebraically closed field of characteristic at least three, and show that the Hom-spaces are all at most one-dimensional. As a corollary we obtain all homomorphisms between Specht ... More

Bezoutians and Tate ResolutionsJun 19 2006This paper gives an explicit construction of the Tate resolution of sheaves arising from the d-fold Veronese embedding of P^n. Our description involves the Bezoutian of n+1 homogenous forms of degree d in n+1 variables. We give applications to duality ... More

A presentation for the Chow ring of \bar{M}_{0,2}(P^1,2)Apr 28 2005May 10 2005We give a presentation for the Chow ring of the moduli space of degree two stable maps from two-pointed rational curves to P^1. Also, integrals of of all degree four monomials in the hyperplane pullbacks and boundary divisors of this ring are computed ... More

Namba forcing, weak approximation, and guessingOct 02 2016We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle $\textsf{IGMP}$, $\textsf{GMP}$ together with $2^\omega \le \omega_2$ is consistent with the existence of a nontrivial $\omega_1$-distributive ... More

Tate Resolutions for Segre EmbeddingsNov 04 2007Jun 07 2008We give an explicit description of the terms and differentials of the Tate resolution of sheaves arising from Segre embeddings of $\P^a\times\P^b$. We prove that the maps in this Tate resolution are either coming from Sylvester-type maps, or from Bezout-type ... More

A presentation for the Chow ring A^*(\bar{M}_{0,2}(P^1,2))May 06 2005The purpose of this dissertation is to study the intersection theory of the moduli spaces of stable maps of degree two from two-pointed, genus zero nodal curves to arbitrary-dimensional projective space. Toward this end, first the Betti numbers of \bar{M}_{0,2}(P^r,2) ... More

An Additive Basis for the Chow Ring of \bar{M}_{0,2}(P^r,2)Jan 20 2005Aug 31 2007We begin a study of the intersection theory of the moduli spaces of degree two stable maps from two-pointed rational curves to arbitrary-dimensional projective space. First we compute the Betti numbers of these spaces using Serre polynomial and equivariant ... More

Strongly proper forcing and some problems of ForemanDec 05 2016We provide solutions to several problems of Foreman about ideals, several of which are closely related to Mitchell's notion of \emph{strongly proper} forcing. We prove: 1) Presaturation of a normal ideal implies projective antichain catching, enabling ... More

Regularity and Segre-Veronese embeddingsMay 29 2008This paper studies the regularity of certain coherent sheaves that arise naturally from Segre-Veronese embeddings of a product of projective spaces. We give an explicit formula for the regularity of these sheaves and show that their regularity is subadditive. ... More

Namba forcing, weak approximation, and guessingOct 02 2016May 25 2018We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle $\textsf{IGMP}$, $\textsf{GMP}$ together with $2^\omega \le \omega_2$ is consistent with the existence of an $\omega_1$-distributive ... More

Quotients of Strongly Proper Forcings and Guessing ModelsJun 12 2014Jun 05 2015We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain ... More

Prevalence of Generic Laver DiamondMay 12 2014Viale \cite{Viale_GuessingModel} introduced the notion of Generic Laver Diamond at $\kappa$---which we denote $\Diamond_{\text{Lav}}(\kappa)$---asserting the existence of a single function from $\kappa \to H_\kappa$ that behaves much like a supercompact ... More

On a fractional version of Haemers' boundFeb 01 2018Dec 13 2018In this note, we present a fractional version of Haemers' bound on the Shannon capacity of a graph, which is originally due to Blasiak. This bound is a common strengthening of both Haemers' bound and the fractional chromatic number of a graph. We show ... More

Inverting the Turán ProblemNov 06 2017Nov 07 2017Classical questions in extremal graph theory concern the asymptotics of $\operatorname{ex}(G, \mathcal{H})$ where $\mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a "standard" increasing sequence of host graphs $(G_1, G_2, \dots)$, ... More

The Galois theory of the lemniscateAug 13 2012Aug 21 2012This article studies the Galois groups that arise from division points of the lemniscate. We compute these Galois groups two ways: first, by class field theory, and second, by proving the irreducibility of lemnatomic polynomials, which are analogs of ... More

On closed graphs IIDec 31 2014Apr 22 2015A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we explore various aspects of closed graphs, including the number of closed labelings and clustering ... More

A simple hierarchical Bayesian model for simultaneous inference of tournament graphs and informant errorApr 29 2013Oct 11 2013The paper presents a hierarchical Bayesian model for simultaneous inference of tournament graphs and informant error. From multiple informant reports or measurement instrument outputs, the model estimates the structure of a criterion (i.e., true) tournament ... More

The Calabi-Yau equation on almost-Kahler four-manifoldsApr 18 2006Mar 26 2007Let (M, \omega) be a compact symplectic 4-manifold with a compatible almost complex structure J. The problem of finding a J-compatible symplectic form with prescribed volume form is an almost-K\"ahler analogue of Yau's theorem and is connected to a programme ... More

The configuration basis of a Lie algebra and its dualOct 22 2010We use the Lie coalgebra and configuration pairing framework presented previously by Sinha and Walter to derive a new, left-normed monomial basis for free Lie algebras (built from associative Lyndon-Shirshov words), as well as a dual monomial basis for ... More

Constraints on the Global Mass-to-Light Ratios and Extent of Dark Matter Halos in Globular Clusters and Dwarf SpheroidalsNov 29 1995The detection of stars in the process of being tidally removed from globular clusters and dwarf spheroidals in the Galaxy's halo provides a strong constraint on their mass to light ratios and on the extent of their possible dark matter halos. If a significant ... More

The Kähler-Ricci flow on compact Kähler manifoldsFeb 24 2015These notes are based on a lecture series given at the Park City Math Institute in the summer of 2013. The notes are intended as a leisurely introduction to the K\"ahler-Ricci flow on compact K\"ahler manifolds, aimed at graduate students with some background ... More

Some Optimizations for (Maximal) Multipliers in $L^p$Feb 08 2014We use Oberlin, Nazarov, and Thiele's Multi-Frequency Calder\'{o}n-Zygmund decomposition to lower estimates on maximal multipliers in $L^p$. We also improve on classical multiplier results of Coifman, Rubio de Francia, and Semmes.

Codimension-two Bifurcations Induce Hysteresis Behavior and Multistabilities in Delay-coupled Kuramoto OscillatorsAug 11 2016Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled ... More

Essential dimension and the flats spanned by a point setFeb 25 2016Oct 12 2016Let $P$ be a finite set of points in $\mathbb{R}^d$ or $\mathbb{C}^d$. We answer a question of Purdy on the conditions under which the number of hyperplanes spanned by $P$ is at least the number of $(d-2)$-flats spanned by $P$. In answering this question, ... More

Coxeter groups as Beauville groupsAug 11 2015Apr 21 2016We generalize earlier work of Fuertes and Gonz\'{a}lez-Diez as well as earlier work of Bauer, Catanese and Grunewald to Coxeter groups in general by classifying which of these are strongly real Beauville groups. As a consequence of this we determine which ... More

Recent work on Beauville surfaces, structures and groupsMay 29 2014Beauville surfaces are a class of complex surfaces defined by letting a finite group $G$ act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group $G$. In ... More

Bounded gaps between primesFeb 19 2014Feb 23 2014These are notes on Zhang's work and subsequent developments produced in preparation for 5 hours of talks for a general mathematical audience given in Cambridge, Edinburgh and Auckland over the last year. Being for colloquium-style talks, these notes are ... More

Montreal Lecture Notes on Quadratic Fourier AnalysisApr 04 2006Apr 09 2007These are notes to accompany four lectures that I gave at the School on Additive Combinatorics, held in Montreal, Quebec between March 30th and April 5th 2006. My aim is to introduce ``quadratic fourier analysis'' in so far as we understand it at the ... More

Generalising the Hardy-Littlewood Method for PrimesJan 10 2006The Hardy-Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov's 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van ... More

On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energySep 24 2003Mar 21 2006The J-flow is a parabolic flow on Kahler manifolds. It was defined by Donaldson in the setting of moment maps and by Chen as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. It is shown here that under a certain condition ... More

A uniformly continuous linear extension principle in topological vector spaces with an application to Lebesgue integrationMar 26 2012May 07 2013We prove a uniformly continuous linear extension principle in topological vector spaces from which we derive a very short and canonical construction of the Lebesgue integral of Banach space valued maps on a finite measure space. The Vitali Convergence ... More

Variable Selection for Additive Partial Linear Quantile Regression with Missing CovariatesOct 01 2015Jun 04 2016The standard quantile regression model assumes a linear relationship at the quantile of interest and that all variables are observed. We relax these assumptions by considering a partial linear model while allowing for missing linear covariates. To handle ... More

Garling sequence spacesDec 04 2016By generalizing a construction of Garling, for each $1\leqslant p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related ... More

A proof of the Landsberg-Schaar relation by finite methodsOct 15 2018The Landsberg-Schaar relation is a classical identity between quadratic Gauss sums, normally used as a stepping stone to prove quadratic reciprocity. The Landsberg-Schaar relation itself is usually proved by carefully taking a limit in the functional ... More

The dark matter crisisMar 07 2001Mar 08 2001I explore several possible solutions to the ``missing satellites'' problem that challenges the collisionless cold dark matter model.

CM liftings of Supersingular Elliptic CurvesApr 09 2009Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D<0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $\order_{D}$ to supersingular elliptic curves in ... More

Lie algebra configuration pairingOct 22 2010We give a new description of the configuration pairing of Sinha and Walter by showing that it computes coefficients in the associative, preLie, or graph polynomial of a Lie bracket expression. We also connect the graph complexes of Sinha and Walter with ... More

Higher K-energy functionals and higher Futaki invariantsApr 23 2002May 01 2002This note discusses the higher K-energy functionals which were defined by Bando and Mabuchi, and integrate higher Futaki invariants. Two new formulas for the higher K-energy functionals are given, and the second K-energy is shown to be related to Donaldson's ... More

Spectral gap for the interchange process in a boxMay 05 2008We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to $\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in any graph the spectral ... More

Positively curved surfaces in the three-sphereApr 18 2003In this talk I will discuss an example of the use of fully nonlinear parabolic flows to prove geometric results. I will emphasise the fact that there is a wide variety of geometric parabolic equations to choose from, and to get the best results it can ... More

An infinite family of Legendrian torus knots distinguished by cube numberDec 20 2010For a knot $K$ the cube number is a knot invariant defined to be the smallest $n$ for which there is a cube diagram of size $n$ for $K$. There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will show that the ... More

Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert seriesAug 29 2018The Hilbert series of a polarised algebraic variety $(X,D)$ is a powerful invariant that, while it captures some features of the geometry of $(X,D)$ precisely, often cannot recover much information about its singular locus. This work explores the extent ... More

New Upper Bounds on the Spreads of Some Large Sporadic GroupsJul 12 2009May 03 2011Let G be a group. We say that G has spread r if for any set of distinct elements {x1,..., xr}\subset G there exists an element y\in G with the property that <xi, y>=G for every 0<i<r+1. Few bounds on the spread of finite simple groups are known. In this ... More

A new presentation of the cyclotomic Cherednik algebraSep 18 2016We give an alternate presentation of the cyclotomic rational Cherednik algebra. This presentation has a diagrammatic flavor, and it provides a simple explanation of several surprising facts about this algebra. It allows direct proof of the connection ... More

Big Bang Models in String TheoryMay 19 2006Dec 20 2006These proceedings are based on lectures delivered at the "RTN Winter School on Strings, Supergravity and Gauge Theories", CERN, January 16 - January 20, 2006. The school was mainly aimed at Ph.D. students and young postdocs. The lectures start with a ... More

Dimension-free Maximal Inequalities for Spherical Means in the HypercubeSep 17 2013Dec 08 2014We extend the main result of \cite{HKS} -- the existence of dimension-free $L^2$-bounds for the spherical maximal function in the hypercube -- to all $L^p, p > 1$. Our approach is motivated by the spectral technique developed in \cite{S} and \cite{NS} ... More

Improved mixing time bounds for the Thorp shuffle and L-reversal chainFeb 04 2008We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models. E. Thorp introduced the following card shuffling ... More

Singularity formation in the Yang-Mills flowOct 08 2002May 31 2003This paper studies rapidly forming singularities in the Yang-Mills flow. It is shown that a sequence of blow-ups near the singular point converges, modulo the gauge group, to a homothetically shrinking soliton with non-zero curvature. The proof uses Hamilton's ... More

Renormalization Group and Black Hole Production in Large Extra DimensionsJul 31 2007Aug 22 2007It has been suggested that the existence of a non-Gaussian fixed point in general relativity might cure the ultraviolet problems of this theory. Such a fixed point is connected to an effective running of the gravitational coupling. We calculate the effect ... More

Caustic Rings and Cold Dark MatterMar 06 2001The hierarchical cold dark matter (CDM) model for structure formation is a well defined and testable model. Direct detection is the best technique for confirming the model yet predictions for the energy and density distribution of particles on earth remain ... More

The Nature Of Dark MatterFeb 03 1994Collisionless particles, such as cold dark matter, interact only by gravity and do not have any associated length scale, therefore the dark halos of galaxies should have negligible core radii. This expectation has been supported by numerical experiments ... More

Evolutionary processes in clustersJun 27 2003Are the morphologies of galaxies imprinted during an early and rapid formation epoch or are they due to environmental processes that subsequently transform galaxies between morphological classes? Recent numerical simulations demonstrate that the cluster ... More

Construction of Maximal Hypersurfaces with Boundary ConditionsAug 22 2014Oct 07 2016We construct maximal hypersurfaces with a Neumann boundary condition in Minkowski space via mean curvature flow. In doing this we give general conditions for long time existence of the flow with boundary conditions with assumptions on the curvature of ... More

Polish Models and Sofic EntropyNov 06 2014Dec 31 2015For actions of a sofic group on probability spaces, the entropy has been defined by Bowen, with an extension by Kerr-Li. In particular, when the action is by homeomorphisms of a compact space preserving a given measure, Kerr-Li show one can compute the ... More

The chart based approach to studying the global structure of a spacetime induces a coordinate invariant boundaryJan 07 2014Feb 25 2014I demonstrate that the chart based approach to the study of the global structure of Lorentzian manifolds induces a homeomorphism of the manifold into a topological space as an open dense set. The topological boundary of this homeomorphism is a chart independent ... More

Relative entropy and the Pinsker product formula for sofic groupsMay 05 2016We continue our study of the outer Pinsker factor for probability measure-preserving actions of sofic groups. Using the notion of doubly quenched convergence developed by Austin, we prove that in many cases the outer Pinsker factor of a product action ... More

A compactness result in approach theory with an application to the continuity approach structureApr 28 2015Jun 17 2015We establish a compactness result in approach theory which we apply to obtain a generalization of Prokhorov's Theorem for the continuity approach structure.

Infinite-dimensional $\ell^1$ minimization and function approximation from pointwise dataMar 09 2015Jun 23 2016We consider the problem of approximating a smooth function from finitely-many pointwise samples using $\ell^1$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three advantages of ... More

Independence Tuples and Deninger's ProblemFeb 12 2015Apr 27 2016Motivated by our results in "Polish Models and Sofic Entropy," we define modified version of the independence tuples for sofic entropy developed by Kerr and Li. These modified version essentially require that the independence sequences give rise to representations ... More

Subelliptic Resolvent Estimates for Non-self-adjoint Semiclassical Schrodinger OperatorsSep 02 2016Oct 01 2016In this paper we prove a subelliptic resolvent estimate for a class of semiclassical non-self-adjoint Schr\"odinger operators with purely imaginary potentials when the spectral parameter is in a parabolic neighborhood of the imaginary axis.