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Pushing the Limits of Exoplanet Discovery via Direct Imaging with Deep LearningApr 12 2019Further advances in exoplanet detection and characterisation require sampling a diverse population of extrasolar planets. One technique to detect these distant worlds is through the direct detection of their thermal emission. The so-called direct imaging ... More

Dyadic torsion of 2-dimensional hyperelliptic JacobiansOct 29 2014Let $k$ be a field of characteristic $0$, and let $\alpha_{1}$, $\alpha_{2}$, ..., $\alpha_{5}$ be algebraically independent and transcendental over $k$. Let $K$ be the transcendental extension of $k$ obtained by adjoining the elementary symmetric functions ... More

Diffusion in the Mean for an Ergodic Schrödinger Equation Perturbed by a Fluctuating PotentialJun 19 2014May 14 2015Diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates ... More

Distribution of Resonances in Scattering by Thin BarriersApr 14 2014Sep 14 2015We study high energy resonances for the operators $-\Delta +\delta_{\partial\Omega}\otimes V$ and $-\Delta+\delta_{\partial\Omega}'\otimes V\partial_\nu$ where $\Omega$ is strictly convex with smooth boundary, $V:L^2(\partial\Omega)\to L^2(\partial\Omega)$ ... More

Length of the continued logarithm algorithm on rational inputsJun 13 2016Jun 21 2016The continued logarithm algorithm was introduced by Gosper around 1978, and recently studied by Borwein, Calkin, Lindstrom, and Mattingly. In this note I show that the continued logarithm algorithm terminates in at most 2 log_2 p + O(1) steps on input ... More

Extension of valuations in characteristic oneMay 20 2016Aug 22 2016We develop an extension of valuations theorem for suitable extensions of idempotent semirings. As an application, we give a new proof for the classical case of fields. Along the way, we develop characteristic one analogues of some central results in the ... More

The Quantum Sabine Law for Resonances in Transmission ProblemsNov 16 2015We prove a quantum version of the Sabine law from acoustics describing the location of resonances in transmission problems. This work extends the author's previous work to a broader class of systems. Our main applications are to scattering by transparent ... More

Obstruction criteria for modular deformation problemsSep 24 2014For a newform $f=\sum a_n q^n$ of weight $k \geq 3$ and a prime $\lambda$ of $\mathbf{Q}(a_n)$, the deformation problem for its associated mod $\lambda$ Galois representation is unobstructed for all primes outside some finite set. Previous results gave ... More

Boundary Conditions for the Gravitational FieldMar 09 2012Apr 24 2012A review of the treatment of boundaries in general relativity is presented with the emphasis on application to the formulations of Einstein's equations used in numerical relativity. At present, it is known how to treat boundaries in the harmonic formulation ... More

Geometrization of metric boundary data for Einstein's equationsApr 02 2009The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new formulation of ... More

Characteristic Evolution and MatchingOct 10 2008Jan 12 2012I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress in characteristic evolution is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that ... More

A concentration-collapse decomposition for $L^2$ flow singularitiesNov 05 2013We exhibit a concentration-collapse decomposition of singularities of fourth order curvature flows, including the $L^2$ curvature flow and Calabi flow, in dimensions $n \leq 4$. The proof requires the development of several new a priori estimates. First, ... More

Stark conjectures for CM curves over number fieldsAug 30 2001We present an elliptic curve analog of the Stark conjecture for the value of the $L$-function at $s=0$. Although implied by the general Beilinson conjectures, the approach here is very concrete. Several cases are proved.

An interacting particle system with geometric jump rates near a partially reflecting boundaryJan 20 2016This paper constructs a new interacting particle system on a two--dimensional lattice with geometric jumps near a boundary which partially reflects the particles. The projection to each horizontal level is Markov, and on every level the dynamics match ... More

Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank twoApr 27 2015May 22 2015We study two generalizations of the asymmetric simple exclusion process with two types of particles. Particles of type 1 can jump over particles of type 2, while particles of type 2 can only influence the jump rates of particles of type 1. We prove that ... More

On the Relative Gain Array (RGA) with Singular and Rectangular MatricesMay 25 2018Mar 04 2019In this paper we identify a significant deficiency in the literature on the application of the Relative Gain Array (RGA) formalism in the case of singular matrices. Specifically, we show that the conventional use of the Moore-Penrose pseudoinverse is ... More

Sylow 0-unipotent subgroups in groups of finite Morley rankNov 26 2007One of the central tools in the classification of simple algebraic groups is the distinction between semisimple subgroups and unipotent subgroups. It is not a priori clear how to make this distinction for torsion-free subgroups of a group of finite Morley ... More

A signalizer functor theorem for groups of finite Morley rankAug 06 2003There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The "trichotomy" here is a case ... More

Does Ten Have a Friend?Jun 05 2008Jun 06 2008Any positive integer $n$ other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent ... More

The K-level crossings of a random algebraic polynomial with dependent coefficientsDec 21 2009For a random polynomial with standard normal coefficients, two cases of the K-level crossings have been considered by Farahmand. When the coefficients are independent, Farahmand was able to derive an asymptotic value for the expected number of level crossings, ... More

The real zeros of a random algebraic polynomial with dependent coefficientsJun 10 2009Jul 18 2010Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the order of (2/pi)log(n), ... More

Semistable models of elliptic curves over residue characteristic 2May 14 2019Given an elliptic curve $E$ in Legendre form $y^2 = x(x - 1)(x - \lambda)$ over the fraction field of a Henselian ring $R$ of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of $E$ over $R$ which depends only ... More

Distinguished Cuspidal Representations over p-adic and Finite FieldsMar 26 2017May 12 2017The author's work with Murnaghan on distinguished tame supercuspidal representations is re-examined using a simplified treatment of Jiu-Kang Yu's construction of tame supercuspidal representations of $p$-adic reductive groups. This leads to a unification ... More

Stability of Critical p-Improper Interval GraphsMar 15 2019A $p$-improper interval graph is an interval graph that has an interval representation in which no interval contains more than $p$ other intervals. A critical $p$-improper interval graph is $p-1$ improper when any vertex is removed. In this paper we investigate ... More

Gromov-- Witten Invariants of Toric FibrationsJan 09 2009We prove a conjecture of Artur Elezi in a generalized form suggested by Givental. Namely, our main result relates genus-0 Gromov--Witten invariants of a bundle space with such invariants of the base, provided that the fiber is a toric manifold. When the ... More

Elementary Deuring-Heilbronn PhenomenonJan 03 2012Jul 02 2012Adapting a technique of Pintz, we give an elementary demonstration of the Deuring phenomenon: a zero of \zeta(s) off the critical line gives a lower bound on L(1,\chi). The necessary tools are Dirichlet's 'method of the hyperbola', Euler summation, summation ... More

The stable mapping class group of simply connected 4-manifoldsOct 27 2005Apr 10 2007We consider mapping class groups \Gamma(M) = pi_0 Diff(M fix \partial M) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3-spheres. We show that if M contains CP^2 (with either orientation) as a connected summand then ... More

Ideal triangulations and geometric transitionsMay 22 2013May 13 2014Thurston introduced a technique for finding and deforming three-dimensional hyperbolic structures by gluing together ideal tetrahedra. We generalize this technique to study families of geometric structures that transition from hyperbolic to anti de Sitter ... More

An abelian subfield of the dyadic division field of a hyperelliptic JacobianFeb 28 2018Feb 13 2019Given a field $k$ of characteristic different from $2$ and an integer $d \geq 3$, let $J$ be the Jacobian of the "generic" hyperelliptic curve given by $y^2 = \prod_{i = 1}^d (x - \alpha_i)$, where the $\alpha_i$'s are transcendental and independent over ... More

A Geometric Reverse To The Plus Construction And Some Examples Of Pseudo-Collars On High-Dimensional ManifoldsAug 14 2015In this paper, we develop a geometric procedure for producing a reverse to Quillen's plus construction, a construction called a 1-sided h-cobordism or semi-h-cobordism. We then use this reverse to the plus construction to produce uncountably many distinct ... More

Unit Consistency, Generalized Inverses, and Effective System Design MethodsApr 24 2016This paper examines the potential role of unit consistency as a system design principle. Unit-consistent generalized matrix inverses and unit-invariant matrix decompositions are derived in support of this principle. Applications of the methods described ... More

A Quantitative Vainberg Method for Black Box ScatteringNov 18 2015Mar 04 2016We give a quantitative version of Vainberg's method relating pole free regions to propagation of singularities for black box scatterers. In particular, we show that there is a logarithmic resonance free region near the real axis of size $\tau$ with polynomial ... More

Worldtube conservation laws for the null-timelike evolution problemMay 17 2011I treat the worldtube constraints which arise in the null-timelike initial-boundary value problem for the Bondi-Sachs formulation of Einstein's equations. Boundary data on a worldtube and initial data on an outgoing null hypersurface determine the exterior ... More

Probability distributions of multi-species q-TAZRP and ASEP as double cosets of parabolic subgroupsJan 08 2018We write explicit contour integral formulas for probability distributions of the multi-species q-TAZRP and the multi-species ASEP starting with q-exchangeable initial conditions. The formulas are equal to the corresponding explicit contour integral formulas ... More

Three-dimensional Gaussian fluctuations of non-commutative random surfaces along time-like pathsJan 23 2014Jul 31 2014We construct a continuous-time non-commutative random walk on $U(\mathfrak{gl}_N)$ with dilation maps $U(\mathfrak{gl}_N)\rightarrow L^2(U(N))^{\otimes\infty}$. This is an analog of a continuous-time non-commutative random walk on the group von Neumann ... More

A Multi-species ASEP(q,j) and q-TAZRP with Stochastic DualityMay 02 2016This paper introduces a multi-species version of a process called ASEP(q,j). In this process, up to 2j particles are allowed to occupy a lattice site, the particles drift to the right with asymmetry 0<q^{2j}<1, and there are n-1 species of particles in ... More

Finite Extensions of $\mathbb{Z}_\mathrm{max}$Jun 04 2014Aug 22 2016We classify the semifields and division semirings containing the max-plus semifield $\mathbb{Z}_\mathrm{max}$, which are finitely generated as $\mathbb{Z}_\mathrm{max}$-semimodules.

Deforming Representations of SL(2,R)Jan 20 2017The spherical principal series representations $\pi(\nu)$ of SL(2,$\mathbb R$) is a family of infinite dimensional representations parametrized by $\nu\in\mathbb C$. The representation $\pi(\nu)$ is irreducible unless $\nu$ is an odd integer, in which ... More

Categorified Algebra and Quantum MechanicsJan 19 2006Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial ... More

Lifting images of standard representations of symmetric groupsMar 14 2019May 02 2019We investigate closed subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose modulo-$2$ images coincide with the image $\mathfrak{S}_{2g + 1} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 1}$ or the image $\mathfrak{S}_{2g + 2} \subseteq ... More

Some Results on Pseudo-Collar Structures on High-Dimensional ManifoldsFeb 15 2015In this paper, we recall Quillen's plus construction for high-dimensional smooth manifolds and the solution to the group extension problem. We then develop a geometric procedure due for producing a "reverse" to the plus construction, a one-sided s-cobordism ... More

Cremona transformations, surface automorphisms and plane cubicsNov 19 2008Jul 26 2010We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the group structure ... More

The framed little 2-discs operad and diffeomorphisms of handlebodiesAug 19 2010Aug 21 2010The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular ... More

A hybrid OpenMP and MPI implementation of a conservative spectral method for the Boltzmann equationJan 17 2013We demonstrate the implementation of a hybrid OpenMP and MPI parallelization of a conservative spectral method for the Boltzmann equation originally developed by Gamba and Tharkabhushaman. We perform a scaling analysis to demonstrate that the problem ... More

The diffeomorphism group of a K3 surface and Nielsen realizationMay 31 2007Nov 12 2017The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough ... More

Small Seifert fibered surgery on hyperbolic pretzel knotsOct 29 2012Jul 23 2013We complete the classification of hyperbolic pretzel knots admitting Seifert fibered surgeries. This is the final step in understanding all exceptional surgeries on hyperbolic pretzel knots. We also present results toward similar classifications for non-pretzel ... More

N derivatives are necessary for order N+1 convergence in quadrature: a converse resultJan 28 2014Results on the error bounds of quadrature methods are well known - most state that if the method has degree N, and the integrand has N derivatives, then the error is order N+1. We prove here a converse: that if the integrand fails to have N derivatives, ... More

An equivalence between two approaches to limits of local fieldsNov 18 2015Nov 20 2015Marc Krasner proposed a theory of limits of local fields in which one relates the extensions of a local field to the extensions of a sequence of related local fields. The key ingredient in his approach was the notion of valued hyperfields, which occur ... More

More Examples of Pseudo-Collars on High-Dimensional ManifoldsApr 07 2018In a previous paper, we developed general techniques for constructing a variety of pseudo-collars, as defined by Guilbault and Tinsley, with roots in earlier work by Chapman and Siebenmann. As an application of our techniques, we exhibited an uncountable ... More

The Real Chevalley InvolutionMar 08 2012Feb 13 2014We consider the Chevalley involution in the context of real reductive groups. We show that if G(R) is the real points of a connected reductive group, there is an involution, unique up to conjugacy by G(R), taking any semisimple element to a conjugate ... More

Semistable models of elliptic curves over residue characteristic 2May 14 2019May 21 2019Given an elliptic curve $E$ in Legendre form $y^2 = x(x - 1)(x - \lambda)$ over the fraction field of a Henselian ring $R$ of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of $E$ over $R$ which depends only ... More

Time dilation relation for a Dirac Hamiltonian generator on a qubit arrayDec 08 2015Dec 13 2015Dirac particle dynamics is encoded as a unitary path summation rule and implemented on a qubit array, where the qubit array represents both spacetime and the fermions contained therein. The unitary path summation rule gives a quantum algorithm to model ... More

Images of 2-adic representations associated to hyperelliptic JacobiansOct 10 2014Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$-power roots of unity, and let $K = k(\alpha_{1}, \alpha_{2}, ... , \alpha_{2g + 1})$, where the $\alpha_{i}$'s are independent and transcendental over $k$, and $g$ is a positive integer. We ... More

Global viscosity solutions of generalized Kahler-Ricci flowOct 06 2016We apply ideas from viscosity theory to establish the existence of a unique global weak solution to the generalized Kahler-Ricci flow in the setting of commuting complex structures. Our results are restricted to the case of a smooth manifold with smooth ... More

The long time behavior of fourth-order curvature flowsMar 21 2011Nov 10 2011We show precompactness results for solutions to parabolic fourth order geometric evolution equations. As part of the proof we obtain smoothing estimates for these flows in the presence of a curvature bound, an improvement on prior results which also require ... More

Asymptotic Curvature Decay and Removal of Singularities of Bach-Flat MetricsAug 07 2007We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special ... More

Characteristic Evolution and MatchingAug 23 2005Dec 08 2005I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations ... More

A Geometric transition from hyperbolic to anti de Sitter geometryFeb 22 2013We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe ... More

Quantum Correlations and the Measurement ProblemOct 23 2012Jan 14 2013The transition from classical to quantum mechanics rests on the recognition that the structure of information is not what we thought it was: there are operational, i.e., phenomenal, probabilistic correlations that lie outside the polytope of local correlations. ... More

Trapping planar Brownian motion in a non circular trapOct 30 2016Nov 01 2016Brownian motion in the plane in the presence of a "trap" at which motion is stopped is shown. If the trap $T$ is a connected compact set, it is shown that the probability for planar Brownian motion to hit this set before a given time $t$ is well approximated ... More

Text Data Mining: Theory and MethodsJul 16 2008This paper provides the reader with a very brief introduction to some of the theory and methods of text data mining. The intent of this article is to introduce the reader to some of the current methodologies that are employed within this discipline area ... More

Two constructions of Markov chains on the dual of U(n)Dec 12 2012Apr 09 2017We provide two new constructions of Markov chains which had previously arisen from the representation theory of the infinite-dimensional unitary group. The first construction uses the combinatorial rule for the Littlewood-Richardson coefficients, which ... More

Lifting images of standard representations of symmetric groupsMar 14 2019We investigate closed subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose modulo-$2$ images coincide with the image $\mathfrak{S}_{2g + 1} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 1}$ or the image $\mathfrak{S}_{2g + 2} \subseteq ... More

A Scale-Consistent Approach for Recommender SystemsApr 30 2019In this paper we propose and develop a relatively simple and efficient approach for estimating unknown elements of a user-rating matrix in the context of a recommender system (RS). The critical theoretical property of the method is its consistency with ... More

On the shock wave spectrum for isentropic gas dynamics with capillarityJun 15 2008We consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. ... More

The Bender method in groups of finite Morley rankNov 27 2007Jaligot's Lemma states that the Fitting subgroups of distinct Borel subgroups do not intersect in a tame minimal simple groups of finite Morley. Such a strong result appears hopeless without tameness. Here we use the 0-unipotence theory to build a toolkit ... More

Constructing tame supercuspidal representationsJan 26 2017Nov 28 2017A new approach to Jiu-Kang Yu's construction of tame supercuspidal representations of $p$-adic reductive groups is presented. Connections with the theory of cuspidal Deligne-Lusztig representations of finite groups of Lie type are also discussed.

Quivers and Three-Dimensional Lie AlgebrasSep 23 2014We study a family of three-dimensional Lie algebras $L_\mu$ that depend on a continuous parameter $\mu$. We introduce certain quivers, which we denote by $Q_{m,n}$ $(m,n \in \mathbb{Z})$ and $Q_{\infty \times \infty}$, and prove that idempotented versions ... More

On Frattini arguments in L-groups of finite Morley rankApr 20 2009We modify the Frecon-Jaligot construction of Carter subgroups to show that a degenerate type group has a Carter subgroup invariant under the Sylow 2-subgroup of a group of automorphisms; thus reducing the need to know that Carter subgroups are conjugate ... More

Galois Cohomology of Real GroupsOct 29 2013Jul 01 2014Real forms of a complex reductive group are classified in terms of Galois cohomology $H^1(\Gamma,G_{ad})$ where $G_{ad}$ is the adjoint group. Alternatively, the theory of the Cartan involution gives a description in terms of cohomology with respect to ... More

Notes on $\log(ζ(s))^{\prime\prime}$Nov 21 2013Dec 22 2014Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann zeta function, in particular the zeros of this function. Theorem 1 gives a zero-free region. Theorem 2 gives ... More

Distinguished Regular Supercuspidal Representations and Inductive Constructions of RepresentationsAug 12 2018This paper develops the theory of distinguished regular supercuspidal representations, and it highlights how the correspondence between regular characters and regular supercuspidal representations resembles induction in certain ways.

The computational complexity of the local postage stamp problemDec 22 2001The well-studied local postage stamp problem (LPSP) is the following: given a positive integer k, a set of postive integers 1 = a1 < a2 < ... < ak and an integer h >= 1, what is the smallest positive integer which cannot be represented as a linear combination ... More

A note on 8-division fields of elliptic curvesApr 20 2017Aug 02 2017Let $K$ be a field of characteristic different from $2$ and let $E$ be an elliptic curve over $K$, defined either by an equation of the form $y^{2} = f(x)$ with degree $3$ or as the Jacobian of a curve defined by an equation of the form $y^{2} = f(x)$ ... More

Notes on Low discriminants and the generalized Newman conjectureJan 14 2013Apr 10 2013Generalizing work of Polya, de Bruijn and Newman, we allow the backward heat equation to deform the zeros of quadratic Dirichlet L-functions. There is a real constant \Lambda_Kr (generalizing the de Bruijn-Newman constant \Lambda) such that for time t>=\Lambda_Kr ... More

The quadratic character experimentFeb 28 2008Jun 29 2010A fast new algorithm is used compute the zeros of the quadratic character L-functions for all negative fundamental discriminants with absolute value 10^12<d<10^12+10^7. These are compared to the 1-level density, including various lower order terms. These ... More

Bounds on Accumulation Rates of Eigenvalues on Manifolds with Degenerating MetricsNov 14 2003Jan 09 2004We consider a family of manifolds with a class of degenerating warped product metrics $g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$, with $M$ compact, $\rho$ homogeneous degree one, $a \le -1$ and $b > 0$. We study the Laplace operator ... More

Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-AnosovsJun 05 2019In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number ... More

Error Bounds for Approximations of Geometrically Ergodic Markov ChainsFeb 24 2017May 20 2017A common tool in the practice of Markov Chain Monte Carlo is to use approximating transition kernels to speed up computation when the true kernel is slow to evaluate. A relatively limited set of quantitative tools exist to determine whether the performance ... More

A Lattice-Gas with Long-Range Interactions Coupled to a Heat BathMar 18 1994Introduced is a lattice-gas with long-range 2-body interactions. An effective inter-particle force is mediated by momentum exchanges. There exists the possibility of having both attractive and repulsive interactions using finite impact parameter collisions. ... More

Homology of framed links embedded in thickened surfacesOct 30 2008We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming ... More

Distinguishing topologically and smoothly doubly slice knotsJan 06 2014Nov 03 2014We construct an infinite family of smoothly slice knots that we prove are topologically doubly slice. Using the correction terms coming from Heegaard Floer homology, we show that none of these knots is smoothly doubly slice. We use these knots to show ... More

Generalized geometry, T-duality, and renormalization group flowOct 18 2013We interpret the physical $B$-field renormalization group flow in the language of Courant algebroids, clarifying the sense in which this flow is the natural "Ricci flow" for generalized geometry. Next we show that the $B$-field renormalization group flow ... More

The gradient flow of the $L^2$ curvature energy on surfacesAug 25 2010We investigate the gradient flow of the $L^2$ norm of the Riemannian curvature on surfaces. We show long time existence with arbitrary initial data, and exponential convergence of the volume normalized flow to a constant scalar curvature metric when the ... More

Closed Form of the Biphoton K-Vector Spectrum for Arbitrary Spatio-Temporal Pump ModesFeb 16 2011Sep 02 2011A closed form solution is derived for the biphoton k-vector spectrum for an arbitrary pump spatial mode. The resulting mode coefficients for the pump input that maximize the probability of biphoton detection in the far field are found. It is thus possible ... More

Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifoldsFeb 09 2015We prove long time existence and convergence results for the pluriclosed flow, which imply geometric and topological classification theorems for generalized K\"ahler structures. Our approach centers on the reduction of pluriclosed flow to a degenerate ... More

Resonances for Thin Barriers on the CircleOct 01 2014Jan 14 2016We study high energy resonances for the operator $-\Delta_{V,\partial\Omega}:=-\Delta+\delta_{\partial\Omega}\otimes V $ when $V$ has strong frequency dependence. The operator $-\Delta_{V,\partial\Omega}$ is a Hamiltonian used to model both quantum corrals ... More

Closed-form analytical continuation of spin density interactions in spin-2 superfluidsSep 08 2016Presented is an unitary operator splitting method for handling the spin-density interaction in spinor Bose-Einstein condensates. The zero temperature behavior of a spinor BEC is given by mean field theory, where the Hamiltonian includes a nonlinear hyperfine ... More

Pluriclosed flow on manifolds with globally generated bundlesJun 13 2016We show global existence and convergence results for the pluriclosed flow on manifolds for which certain naturally associated tensor bundles are globally generated.

Patching and Multiplicity $2^k$ for Shimura CurvesFeb 19 2019We use the Taylor--Wiles--Kisin patching method to investigate the multiplicities with which Galois representations occur in the mod $\ell$ cohomology of Shimura curves over totally real number fields. Our method relies on explicit computations of local ... More

Asymptotics of a discrete-time particle system near a reflecting boundaryMar 07 2012Dec 11 2012We examine a discrete-time Markovian particle system on the quarter-plane introduced by M. Defosseux. The vertical boundary acts as a reflecting wall. The particle system lies in the Anisotropic Kardar-Parisi-Zhang with a wall universality class. After ... More

Ramsey theory for monochromatically well-connected subsetsFeb 28 2019We define well-connectedness, an order-theoretic notion of largeness whose associated partition relations $\nu\to_{wc}(\mu)_\lambda^2$ formally weaken those of the classical Ramsey relations $\nu\to(\mu)_\lambda^2$. We show that it is consistent that ... More

Lifting images of standard representations of symmetric groupsMar 14 2019Apr 20 2019We investigate closed subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose modulo-$2$ images coincide with the image $\mathfrak{S}_{2g + 1} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 1}$ or the image $\mathfrak{S}_{2g + 2} \subseteq ... More

Computing rank of finite algebraic structures with limited nondeterminismJun 03 2014Feb 12 2016The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation. In the case of groups, the previous best algorithm for computing rank used ... More

A simple homotopy-theoretical proof of the Sullivan conjectureMay 19 2011We give a new proof, using comparatively simple techniques, of the Sullivan conjecture: the space of pointed maps from the classifying space of the cyclic group of order $p$ to any finite-dimensional CW complex $K$ is contractible.

Row-strict Quasisymmetric Schur Functions, Characterizations of Demazure Atoms, and Permuted Basement Nonsymmetric Macdonald PolynomialsMar 14 2013We give a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. We then discuss a family of polynomials called Demazure ... More

A Littlewood-Richardson Type Rule for Row-Strict Quasisymmetric Schur FunctionsFeb 07 2011We give a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. This expansion follows from several new properties ... More

Ricci Yang-Mills flow on surfacesOct 29 2007Jul 31 2009We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on surfaces. We show that existence for the flow reduces to a bound on the isoperimetric constant. In the presence of such a bound, we show that on $S^2$, if the bundle is nontrivial, ... More

Software Adaptation and Generalization of Physically-Constrained GamesJun 03 2019In this paper we provide a case study of the use of relatively sophisticated mathematics and algorithms to redefine and adapt a simple traditional game/puzzle to exploit the computational power of smart devices. The focus here is not so much on the end ... More

AI Reasoning Systems: PAC and Applied MethodsJul 09 2018Learning and logic are distinct and remarkable approaches to prediction. Machine learning has experienced a surge in popularity because it is robust to noise and achieves high performance; however, ML experiences many issues with knowledge transfer and ... More