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Recent ucscmath items
https://escholarship.org/uc/ucscmath/rss
Recent eScholarship items from Department of Mathematics
Fri, 12 Aug 2022 18:02:39 +0000

The compactness locus of a geometric functor and the formal construction of the Adams isomorphism
https://escholarship.org/uc/item/9bc0s9fx
The compactness locus of a geometric functor and the formal construction of the Adams isomorphism
https://escholarship.org/uc/item/9bc0s9fx
Tue, 19 May 2020 00:00:00 +0000

Chains in CR geometry as geodesics of a Kropina metric
https://escholarship.org/uc/item/3c41x28c
With the help of a generalization of the Fermat principle in general
relativity, we show that chains in CR geometry are geodesics of a certain
Kropina metric constructed from the CR structure. We study the projective
equivalence of Kropina metrics and show that if the kernel distributions of the
corresponding 1forms are nonintegrable then two projectively equivalent
metrics are trivially projectively equivalent. As an application, we show that
sufficiently many chains determine the CR structure up to conjugacy,
generalizing and reproving the main result of [J.H. Cheng, 1988]. The
correspondence between geodesics of the Kropina metric and chains allows us to
use the methods of metric geometry and the calculus of variations to study
chains. We use these methods to reprove the result of [H. Jacobowitz, 1985]
that locally any two points of a strictly pseudoconvex CR manifolds can be
joined by a chain. Finally, we generalize this result to the global setting by
showing that any...
https://escholarship.org/uc/item/3c41x28c
Sat, 27 Jul 2019 00:00:00 +0000

On nonnegatively curved hypersurfaces in H<sup>n</sup><sup>+</sup><sup>1</sup>
https://escholarship.org/uc/item/9q5945xn
In this paper we prove a conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.
https://escholarship.org/uc/item/9q5945xn
Tue, 22 Jan 2019 00:00:00 +0000

On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3manifolds
https://escholarship.org/uc/item/9nj5n7m9
In this note we study constant mean curvature surfaces in asymptotically flat
3manifolds. We prove that, in an asymptotically flat 3manifold with positive
mass, stable spheres of given constant mean curvature outside a fixed compact
subset are unique. Therefore we are able to conclude that there is a unique
foliation of stable spheres of constant mean curvature in an asymptotically
flat 3manifold with positive mass.
https://escholarship.org/uc/item/9nj5n7m9
Tue, 22 Jan 2019 00:00:00 +0000

On $n$superharmonic functions and some geometric applications
https://escholarship.org/uc/item/9fm2d9s6
In this paper we study asymptotic behavior of $n$superharmonic functions at
isolated singularity using the Wolff potential and $n$capacity estimates in
nonlinear potential theory. Our results are inspired by and extend those of
ArsoveHuber and Taliaferro in 2 dimensions. To study $n$superharmonic
functions we use a new notion of $n$thinness by $n$capacity motivated by a
type of Wiener criterion in ArsoveHuber's paper. To extend Taliaferro's work,
we employ the AdamsMoserTrudinger inequality for the Wolff potential, which
is inspired by the one used by BrezisMerle. For geometric applications, we
study the asymptotic end behavior of complete conformally flat manifolds as
well as complete properly embedded hypersurfaces in hyperbolic space. In both
geometric applications the strong $n$capacity lower bound estimate of Gehring
in 1961 is brilliantly used. These geometric applications seem to elevate the
importance of $n$Laplace equations and make a closer tie to the classic
analysis...
https://escholarship.org/uc/item/9fm2d9s6
Tue, 22 Jan 2019 00:00:00 +0000

Norm inflation for incompressible magnetohydrodynamic system ]{Norm inflation for incompressible magnetohydrodynamic system in $\dot{B}_{\infty}^{1,\infty}$
https://escholarship.org/uc/item/9714r4zn
We demonstrate that the solutions to the Cauchy problem for the three
dimensional incompressible magnetohydrodynamics (MHD) system can develop
diferent types of norm inflations in $\dot{B}_{\infty}^{1, \infty}$.
Particularly the magnetic field can develop norm inflation in short time even
when the velocity remains small and vice verse. Efforts are made to present a
very expository development of the inginious construction of Bourgain and
Pavlovi\'{c} for NavierStokes equation.
https://escholarship.org/uc/item/9714r4zn
Tue, 22 Jan 2019 00:00:00 +0000

Weakly horospherically convex hypersurfaces in hyperbolic space
https://escholarship.org/uc/item/8df3j0tn
In [2], the authors develop a global correspondence between immersed weakly
horospherically convex hypersurfaces $\phi:M^n \to \mathbb{H}^{n+1}$ and a
class of conformal metrics on domains of the round sphere $\mathbb{S}^n$. Some
of the key aspects of the correspondence and its consequences have dimensional
restrictions $n\geq3$ due to the reliance on an analytic proposition from [5]
concerning the asymptotic behavior of conformal factors of conformal metrics on
domains of $\mathbb{S}^n$. In this paper, we prove a new lemma about the
asymptotic behavior of a functional combining the gradient of the conformal
factor and itself, which allows us to extend the global correspondence and
embeddedness theorems of [2] to all dimensions $n\geq2$ in a unified way. In
the case of a single point boundary $\partial_{\infty}\phi(M)=\{x\} \subset
\mathbb{S}^n$, we improve these results in one direction. As an immediate
consequence of this improvement and the work on elliptic problems in [2],...
https://escholarship.org/uc/item/8df3j0tn
Tue, 22 Jan 2019 00:00:00 +0000

On nonnegatively curved hypersurfaces in hyperbolic space
https://escholarship.org/uc/item/5zw633z8
In this paper we prove the conjecture of Alexander and Currier that states,
except for covering maps of equidistant surfaces in hyperbolic 3space, a
complete, nonnegatively curved immersed hypersurface in hyperbolic space is
necessarily properly embedded.
https://escholarship.org/uc/item/5zw633z8
Tue, 22 Jan 2019 00:00:00 +0000

Compactness of conformally compact Einstein 4manifolds II
https://escholarship.org/uc/item/5x6733b6
In this paper, we establish compactness results of some class of conformally
compact Einstein 4manifolds. In the first part of the paper, we improve the
earlier results obtained by ChangGe. In the second part of the paper, as
applications, we derive some compactness results under perturbation conditions
when the L^2norm of the Weyl curvature is small. We also derive the global
uniqueness of conformally compact Einstein metrics on the 4Ball constructed in
the earlier work of GrahamLee.
https://escholarship.org/uc/item/5x6733b6
Tue, 22 Jan 2019 00:00:00 +0000

Hypersurfaces in Hyperbolic Poincaré Manifolds and Conformally Invariant PDEs
https://escholarship.org/uc/item/49t4v63w
We derive a relationship between the eigenvalues of the WeylSchouten tensor
of a conformal representative of the conformal infinity of a hyperbolic
Poincar\'e manifold and the principal curvatures on the level sets of its
uniquely associated defining function with calculations based on [9] [10]. This
relationship generalizes the result for hypersurfaces in ${\H}^{n+1}$ and their
connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives
a correspondence between Weingarten hypersurfaces in hyperbolic Poincar\'e
manifolds and conformally invariant equations on the conformal infinity. In
particular, we generalize an equivalence exhibited in [7] between
Christoffeltype problems for hypersurfaces in ${\H}^{n+1}$ and scalar
curvature problems on the conformal infinity ${\SS}^n$ to hyperbolic Poincar\'e
manifolds.
https://escholarship.org/uc/item/49t4v63w
Tue, 22 Jan 2019 00:00:00 +0000

Hypersurfaces with nonnegative Ricci curvature in hyperbolic space
https://escholarship.org/uc/item/2pz5z60q
Based on properties of nsubharmonic functions we show that a complete,
noncompact, properly embedded hypersurface with nonnegative Ricci curvature in
hyperbolic space has an asymptotic boundary at infinity of at most two points.
Moreover, the presence of two points in the asymptotic boundary is a rigidity
condition that forces the hypersurface to be an equidistant hypersurface about
a geodesic line in hyperbolic space. This gives an affirmative answer to the
question raised by Alexander and Currier in 1990.
https://escholarship.org/uc/item/2pz5z60q
Tue, 22 Jan 2019 00:00:00 +0000

On the topology of conformally compact Einstein 4manifolds
https://escholarship.org/uc/item/1rs6r4kk
In this paper we study the topology of conformally compact Einstein
4manifolds. When the conformal infinity has positive Yamabe invariant and the
renormalized volume is also positive we show that the conformally compact
Einstein 4manifold will have at most finite fundamental group. Under the
further assumption that the renormalized volume is relatively large, we
conclude that the conformally compact Einstein 4manifold is diffeomorphic to
$B^4$ and its conformal infinity is diffeomorphic to $S^3$.
https://escholarship.org/uc/item/1rs6r4kk
Tue, 22 Jan 2019 00:00:00 +0000

Conformal Ricci flow on asymptotically hyperbolic manifolds
https://escholarship.org/uc/item/01n253rd
In this article we study the shorttime existence of conformal Ricci flow on
asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature
derivative estimate for conformal Ricci flow.
https://escholarship.org/uc/item/01n253rd
Tue, 22 Jan 2019 00:00:00 +0000

A note on triangulated monads and categories of module spectra
https://escholarship.org/uc/item/0g52252t
A note on triangulated monads and categories of module spectra
https://escholarship.org/uc/item/0g52252t
Wed, 24 Oct 2018 00:00:00 +0000

Higher comparison maps for the spectrum of a tensor triangulated category
https://escholarship.org/uc/item/62p078v7
Higher comparison maps for the spectrum of a tensor triangulated category
https://escholarship.org/uc/item/62p078v7
Wed, 1 Aug 2018 00:00:00 +0000

The spectrum of the equivariant stable homotopy category of a finite group
https://escholarship.org/uc/item/4n97047b
The spectrum of the equivariant stable homotopy category of a finite group
https://escholarship.org/uc/item/4n97047b
Wed, 1 Aug 2018 00:00:00 +0000

Restriction to finiteindex subgroups as étale extensions in topology, KK–theory and geometry
https://escholarship.org/uc/item/4kj8h126
For equivariant stable homotopy theory, equivariant KK–theory and equivariant derived categories, we show how restriction to a subgroup of finite index yields a finite commutative separable extension, analogous to finite étale extensions in algebraic geometry.
https://escholarship.org/uc/item/4kj8h126
Wed, 1 Aug 2018 00:00:00 +0000

GrothendieckNeeman duality and the Wirthmiiller isomorphism
https://escholarship.org/uc/item/3fk4x3fm
We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensortriangulated categories, which leads to a surprising trichotomy: there exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of socalled relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin–Matlis duality à la Dwyer–Greenless–Iyengar in the theory of ring spectra, and of Brown–Comenetz duality à la Neeman in stable homotopy theory.
https://escholarship.org/uc/item/3fk4x3fm
Wed, 1 Aug 2018 00:00:00 +0000

On Picard groups of blocks of finite groups
https://escholarship.org/uc/item/9512b72c
We show that the subgroup of the Picard group of a $p$block of a finite
group given by bimodules with endopermutation sources modulo the automorphism
group of a source algebra is determined locally in terms of the fusion system
on a defect group. We show that the Picard group of a block over the a complete
discrete valuation ring ${\mathcal O}$ of characteristic zero with an algebraic
closure $k$ of ${\mathbb F}_p$ as residue field is a colimit of finite Picard
groups of blocks over $p$adic subrings of ${\mathcal O}$. We apply the results
to blocks with an abelian defect group and Frobenius inertial quotient, and
specialise this further to blocks with cyclic or Klein four defect groups.
https://escholarship.org/uc/item/9512b72c
Tue, 19 Jun 2018 00:00:00 +0000

Fibered biset functors
https://escholarship.org/uc/item/8b48v9p1
The theory of biset functors, introduced by Serge Bouc, gives a unified
treatment of operations in representation theory that are induced by
permutation bimodules. In this paper, by considering fibered bisets, we
introduce and describe the basic theory of fibered biset functors which is a
natural framework for operations induced by monomial bimodules. The main result
of this paper is the classification of simple fibered biset functors.
https://escholarship.org/uc/item/8b48v9p1
Tue, 19 Jun 2018 00:00:00 +0000

The (+) and (+) constructions for biset functors
https://escholarship.org/uc/item/0dm6k0qx
In this article we define the $_+$construction and the $^+$construction,
that was crucial in the theory of canonical induction formulas (see
\cite{Boltje1998b}), in the setting of biset functors, thus providing the
necessary framework to define and construct canonical induction formulas for
representation rings that are most naturally viewed as biset functors.
Additionally, this provides a unified approach to the study of a class of
functors including the Burnside ring, the monomial Burnside ring and global
representation ring.
https://escholarship.org/uc/item/0dm6k0qx
Tue, 19 Jun 2018 00:00:00 +0000

Blocks in the asymmetric simple exclusion process
https://escholarship.org/uc/item/44c3g3xm
In earlier work, the authors obtained formulas for the probability in the asymmetric simple exclusion process that the mth particle from the left is at site x at time t. They were expressed in general as sums of multiple integrals and, for the case of step initial condition, as an integral involving a Fredholm determinant. In the present work, these results are generalized to the case where the mth particle is the leftmost one in a contiguous block of L particles. The earlier work depended in a crucial way on two combinatorial identities, and the present work begins with a generalization of these identities to general L.
https://escholarship.org/uc/item/44c3g3xm
Mon, 14 May 2018 00:00:00 +0000

Airy Kernel and Painleve II
https://escholarship.org/uc/item/9v6729zz
We prove that the distribution function of the largest eigenvalue in the Gaussian
Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlev\'e II.
Our goal is to concentrate on this important example of the connection between random
matrix theory and integrable systems, and in so doing to introduce the newcomer to the
subject as a whole. We also give sketches of the results for the limiting distribution of
the largest eigenvalue in the Gaussian Orthogonal Ensemble (GOE) and the Gaussian
Symplectic Ensemble (GSE). This work we did some years ago in a more general setting. These
notes, therefore, are not meant for experts in the field.
https://escholarship.org/uc/item/9v6729zz
Thu, 22 Feb 2018 00:00:00 +0000

Asymptotics of a Class of Solutions to the Cylindrical Toda Equations
https://escholarship.org/uc/item/9j9670wn
The small t asymptotics of a class of solutions to the 2D cylindrical Toda
equations is computed. The solutions, q_k(t), have the representation q_k(t) = log
det(Ilambda K_k)  log det(Ilambda K_{k1}) where K_k are integral operators. This class
includes the nperiodic cylindrical Toda equations. For n=2 our results reduce to the
previously computed asymptotics of the 2D radial sinhGordon equation and for n=3 (and with
an additional symmetry contraint) they reduce to earlier results for the radial
BulloughDodd equation.
https://escholarship.org/uc/item/9j9670wn
Wed, 21 Feb 2018 00:00:00 +0000

On ASEP with Step Bernoulli Initial Condition
https://escholarship.org/uc/item/9gf9d47c
This paper extends results of earlier work on ASEP to the case of step Bernoulli
initial condition. The main results are a representation in terms of a Fredholm determinant
for the probability distribution of a fixed particle, and asymptotic results which in
particular establish KPZ universality for this probability in one regime. (And, as a
corollary, for the current fluctuations.)
https://escholarship.org/uc/item/9gf9d47c
Wed, 21 Feb 2018 00:00:00 +0000

Fluctuations in the composite regime of a disordered growth model
https://escholarship.org/uc/item/9fq5t704
We continue to study a model of disordered interface growth in two dimensions. The
interface is given by a height function on the sites of the onedimensional integer
lattice and grows in discrete time: (1) the height above the site $x$ adopts the height
above the site to its left if the latter height is larger, (2) otherwise, the height above
$x$ increases by 1 with probability $p_x$. We assume that $p_x$ are chosen independently at
random with a common distribution $F$, and that the initial state is such that the origin
is far above the other sites. Provided that the tails of the distribution $F$ at its right
edge are sufficiently thin, there exists a nontrivial composite regime in which the
fluctuations of this interface are governed by extremal statistics of $p_x$. In the
quenched case, the said fluctuations are asymptotically normal, while in the annealed case
they satisfy the appropriate...
https://escholarship.org/uc/item/9fq5t704
Wed, 21 Feb 2018 00:00:00 +0000

Asymptotics in ASEP with Step Initial Condition
https://escholarship.org/uc/item/9899n7n7
In previous work the authors considered the asymmetric simple exclusion process on
the integer lattice in the case of step initial condition, particles beginning at the
positive integers. There it was shown that the probability distribution for the position of
an individual particle is given by an integral whose integrand involves a Fredholm
determinant. Here we use this formula to obtain three asymptotic results for the positions
of these particles. In one an apparently new distribution function arises and in another
the distribuion function F_2 arises. The latter extends a result of Johansson on TASEP to
ASEP.
https://escholarship.org/uc/item/9899n7n7
Tue, 20 Feb 2018 00:00:00 +0000

Formulas for Joint Probabilities for the Asymmetric Simple Exclusion Process
https://escholarship.org/uc/item/93g97253
In earlier work the authors obtained integral formulas for probabilities for a
single particle in the asymmetric simple exclusion process. Here formulas are obtained for
joint probabilities for several particles. In the case of a single particle the derivation
here is simpler than the one in the earlier work for one of its main results.
https://escholarship.org/uc/item/93g97253
Tue, 20 Feb 2018 00:00:00 +0000

On the ground state energy of the deltafunction Bose gas
https://escholarship.org/uc/item/92n455ww
The weak coupling asymptotics, to order $(c/\rho)^2$, of the ground state energy of
the deltafunction Bose gasmis derived. Here $2c\ge 0$ is the deltafunction potential
amplitude and $\rho$ the density of the gas in the thermodynamic limit. The analysis uses
the electrostatic interpretation of the LiebLiniger integral equation.
https://escholarship.org/uc/item/92n455ww
Tue, 20 Feb 2018 00:00:00 +0000

The asymmetric simple exclusion process with an open boundary
https://escholarship.org/uc/item/8qc9f8kw
We consider the asymmetric simple exclusion process confined to the nonnegative
integers with an open boundary at 0. The point 0 is connected to a reservoir where
particles are injected and ejected at prescribed rates subject to the exclusion rule. We
derive formulas for the transition probability as a function of time from states where
initially there are m particles to states where there are n particles.
https://escholarship.org/uc/item/8qc9f8kw
Tue, 20 Feb 2018 00:00:00 +0000

On ASEP with Periodic Step Bernoulli Initial Condition
https://escholarship.org/uc/item/8m31k9wr
We consider the asymmetric simple exclusion process (ASEP) on the integers in which
the initial density at a site (the probability that it is occupied) is given by a periodic
function on the positive integers. (When the function is constant this is the step
Bernoulli initial condition.) Starting with a result in earlier work we find a formula for
the probability distribution for a given particle at a given time which is a sum over
positive integers k of integrals of order k.
https://escholarship.org/uc/item/8m31k9wr
Tue, 20 Feb 2018 00:00:00 +0000

Differential Equations for Dyson Processes
https://escholarship.org/uc/item/8jd537h0
We call "Dyson process" any process on ensembles of matrices in which the entries
undergo diffusion. We are interested in the distribution of the eigenvalues (or singular
values) of such matrices. In the original Dyson process it was the ensemble of n by n
Hermitian matrices, and the eigenvalues describe n curves. Given sets X_1,...,X_m the
probability that for each k no curve passes through X_k at time \tau_k is given by the
Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this
reason we call this Dyson process the Hermite process. Similarly, when the entries of a
complex matrix undergo diffusion we call the evolution of its singular values the Laguerre
process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite
process at the edge leads to the Airy process and in the bulk to the sine process; scaling
the Laguerre process at...
https://escholarship.org/uc/item/8jd537h0
Tue, 20 Feb 2018 00:00:00 +0000

Correlation Functions, Cluster Functions and Spacing Distributions for Random
Matrices
https://escholarship.org/uc/item/8h04j0tx
The usual formulas for the correlation functions in orthogonal and symplectic
matrix models express them as quaternion determinants. From this representation one can
deduce formulas for spacing probabilities in terms of Fredholm determinants of
matrixvalued kernels. The derivations of the various formulas are somewhat involved. In
this article we present a direct approach which leads immediately to scalar kernels for
unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles, and the
representations of the correlation functions, cluster functions and spacing distributions
in terms of them.
https://escholarship.org/uc/item/8h04j0tx
Tue, 20 Feb 2018 00:00:00 +0000

Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth
Models
https://escholarship.org/uc/item/8f51w562
We introduce a class of onedimensional discrete spacediscrete time stochastic
growth models described by a height function $h_t(x)$ with corner initialization. We prove,
with one exception, that the limiting distribution function of $h_t(x)$ (suitably centered
and normalized) equals a Fredholm determinant previously encountered in random matrix
theory. In particular, in the universal regime of large $x$ and large $t$ the limiting
distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called
the critical regime, the limiting distribution seems not to have previously occurred. The
proofs use the dual RSK algorithm, Gessel's theorem, the BorodinOkounkov identity and a
novel, rigorous saddle point analysis. In the fixed $x$, large $t$ regime, we find a
Brownian motion representation. This model is equivalent to the Sepp\"al\"ainenJohansson
model. Hence some of...
https://escholarship.org/uc/item/8f51w562
Tue, 20 Feb 2018 00:00:00 +0000

Fredholm Determinants, Differential Equations and Matrix Models
https://escholarship.org/uc/item/8bk4859w
Orthogonal polynomial random matrix models of NxN hermitian matrices lead to
Fredholm determinants of integral operators with kernel of the form (phi(x) psi(y)  psi(x)
phi(y))/xy. This paper is concerned with the Fredholm determinants of integral operators
having kernel of this form and where the underlying set is a union of open intervals. The
emphasis is on the determinants thought of as functions of the endpoints of these
intervals. We show that these Fredholm determinants with kernels of the general form
described above are expressible in terms of solutions of systems of PDE's as long as phi
and psi satisfy a certain type of differentiation formula. There is also an exponential
variant of this analysis which includes the circular ensembles of NxN unitary matrices.
https://escholarship.org/uc/item/8bk4859w
Tue, 20 Feb 2018 00:00:00 +0000

A Fredholm Determinant Representation in ASEP
https://escholarship.org/uc/item/8b20z95h
In previous work the authors found integral formulas for probabilities in the
asymmetric simple exclusion process (ASEP) on the integer lattice. The dynamics are
uniquely determined once the initial state is specified. In this note we restrict our
attention to the case of step initial condition with particles at the positive integers,
and consider the distribution function for the m'th particle from the left. In the previous
work an infinite series of multiple integrals was derived for this distribution. In this
note we show that the series can be summed to give a single integral whose integrand
involves a Fredholm determinant. We use this determinant representation to derive
(nonrigorously, at this writing) a scaling limit.
https://escholarship.org/uc/item/8b20z95h
Tue, 20 Feb 2018 00:00:00 +0000

Universality of the Distribution Functions of Random Matrix Theory. II
https://escholarship.org/uc/item/895083pz
This paper is a brief review of recent developments in random matrix theory. Two
aspects are emphasized: the underlying role of integrable systems and the occurrence of the
distribution functions of random matrix theory in diverse areas of mathematics and physics.
https://escholarship.org/uc/item/895083pz
Tue, 20 Feb 2018 00:00:00 +0000

The Pearcey Process
https://escholarship.org/uc/item/86n8b4v7
The extended Airy kernel describes the spacetime correlation functions for the
Airy process, which is the limiting process for a polynuclear growth model. The Airy
functions themselves are given by integrals in which the exponents have a cubic
singularity, arising from the coalescence of two saddle points in an asymptotic analysis.
Pearcey functions are given by integrals in which the exponents have a quartic singularity,
arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears
in a random matrix model and a Brownian motion model for a fixed time. This paper derives
an extended Pearcey kernel by scaling the Brownian motion model at several times, and a
system of partial differential equations whose solution determines associated distribution
functions. We expect there to be a limiting nonstationary process consisting of infinitely
many paths, which we call...
https://escholarship.org/uc/item/86n8b4v7
Tue, 20 Feb 2018 00:00:00 +0000

Universality of the distribution functions of random matrix theory
https://escholarship.org/uc/item/82n709wn
This paper first surveys the connection of integrable systems of the Painleve type
to various distribution functions appearing in WignerDyson random matrix theory. A short
discussion is then given of the appearance of these same distributions in other areas of
mathematics.
https://escholarship.org/uc/item/82n709wn
Tue, 20 Feb 2018 00:00:00 +0000

A Limit Theorem for Shifted Schur Measures
https://escholarship.org/uc/item/7gs562vg
To each partition $\lambda$ with distinct parts we assign the probability
$Q_\lambda(x) P_\lambda(y)/Z$ where $Q_\lambda$ and $P_\lambda$ are the Schur $Q$functions
and $Z$ is a normalization constant. This measure, which we call the shifted Schur measure,
is analogous to the muchstudied Schur measure. For the specialization of the first $m$
coordinates of $x$ and the first $n$ coordinates of $y$ equal to $\alpha$
($0<\alpha<1$) and the rest equal to zero, we derive a limit law for $\lambda_1$ as
$m,n\ra\infty$ with $\tau=m/n$ fixed. For the Schur measure the $\alpha$specialization
limit law was derived by Johansson. Our main result implies that the two limit laws are
identical.
https://escholarship.org/uc/item/7gs562vg
Tue, 20 Feb 2018 00:00:00 +0000

On Orthogonal and Symplectic Matrix Ensembles
https://escholarship.org/uc/item/7dc8c391
The focus of this paper is on the probability, $E_\beta(0;J)$, that a set $J$
consisting of a finite union of intervals contains no eigenvalues for the finite $N$
Gaussian Orthogonal ($\beta=1$) and Gaussian Symplectic ($\beta=4$) Ensembles and their
respective scaling limits both in the bulk and at the edge of the spectrum. We show how
these probabilities can be expressed in terms of quantities arising in the corresponding
unitary ($\beta=2$) ensembles. Our most explicit new results concern the distribution of
the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that
these largest eigenvalue distributions are given in terms of a particular Painlev\'e II
function.
https://escholarship.org/uc/item/7dc8c391
Tue, 20 Feb 2018 00:00:00 +0000

On the diagonal susceptibility of the 2D Ising model
https://escholarship.org/uc/item/7bz6f4n3
We consider the diagonal susceptibility of the isotropic 2D Ising model for
temperatures below the critical temperature. For a parameter k related to temperature and
the interaction constant, we extend the diagonal susceptibility to complex k inside the
unit disc, and prove the conjecture that the unit circle is a natural boundary.
https://escholarship.org/uc/item/7bz6f4n3
Tue, 20 Feb 2018 00:00:00 +0000

On the ground state energy of the deltafunction fermi gas II: Further asymptotics
https://escholarship.org/uc/item/14t316mb
Building on previous work of the authors, we here derive the weak coupling asymptotics to order γ2 of the ground state energy of the deltafunction Fermi gas. We use a method that can be applied to a large class of finite convolution operators.
https://escholarship.org/uc/item/14t316mb
Tue, 20 Feb 2018 00:00:00 +0000

Formulas for ASEP with TwoSided Bernoulli Initial Condition
https://escholarship.org/uc/item/77k0b40f
For the asymmetric simple exclusion process on the integer lattice with twosided
Bernoulli initial condition, we derive exact formulas for the following quantities: (1) the
probability that site x is occupied at time t; (2) a correlation function, the probability
that site 0 is occupied at time 0 and site x is occupied at time t; (3) the distribution
function for the total flux across 0 at time t and its exponential generating function.
https://escholarship.org/uc/item/77k0b40f
Fri, 16 Feb 2018 00:00:00 +0000

Hamiltonian Structure of Equations Appearing in Random Matrices
https://escholarship.org/uc/item/7405c948
The level spacing distributions in the Gaussian Unitary Ensemble, both in the
``bulk of the spectrum,'' given by the Fredholm determinant of the operator with the sine
kernel ${\sin \pi(xy) \over \pi(xy)}$ and on the ``edge of the spectrum,'' given by the
Airy kernel ${\rm{Ai}(x) \rm{Ai}'(y)  \rm{Ai}(y) \rm{Ai}'(x) \over (xy)}$, are determined
by compatible systems of nonautonomous Hamiltonian equations. These may be viewed as
special cases of isomonodromic deformation equations for first order $ 2\times 2 $ matrix
differential operators with regular singularities at finite points and irregular ones of
Riemann index 1 or 2 at $\infty$. Their Hamiltonian structure is explained within the
classical Rmatrix framework as the equations induced by spectral invariants on the loop
algebra ${\tilde{sl}(2)}$, restricted to a Poisson subspace of its dual space
${\tilde{sl}^*_R(2)}$, consisting...
https://escholarship.org/uc/item/7405c948
Fri, 16 Feb 2018 00:00:00 +0000

Fredholm determinants and the mKdV/sinhGordon hierarchies
https://escholarship.org/uc/item/70r004pw
For a particular class of integral operators $K$ we show that the quantity
\[\ph:=\log \det (I+K)\log \det (IK)\] satisfies both the integrated mKdV hierarchy and
the sinhGordon hierarchy. This proves a conjecture of Zamolodchikov.
https://escholarship.org/uc/item/70r004pw
Fri, 16 Feb 2018 00:00:00 +0000

On Exact Solutions to the Cylindrical PoissonBoltzmann Equation with Applications to
Polyelectrolytes
https://escholarship.org/uc/item/6v8769hg
Using exact results from the theory of completely integrable systems of the
Painleve/Toda type, we examine the consequences for the theory of polyelectrolytes in the
(nonlinear) PoissonBoltzmann approximation.
https://escholarship.org/uc/item/6v8769hg
Thu, 15 Feb 2018 00:00:00 +0000

On the limit of some Toeplitzlike determinants
https://escholarship.org/uc/item/6tm18254
In this article we derive, using standard methods of Toeplitz theory, an asymptotic
formula for certain large minors of Toeplitz matrices. D. Bump and P. Diaconis obtained the
same asymptotics using representation theory, with an answer having a different form.
https://escholarship.org/uc/item/6tm18254
Thu, 15 Feb 2018 00:00:00 +0000

Distribution functions for largest eigenvalues and their applications
https://escholarship.org/uc/item/6sd3b9mp
It is now believed that the limiting distribution function of the largest
eigenvalue in the three classic random matrix models GOE, GUE and GSE describe new
universal limit laws for a wide variety of processes arising in mathematical physics and
interacting particle systems. These distribution functions, expressed in terms of a certain
Painlev\'e II function, are described and their occurences surveyed.
https://escholarship.org/uc/item/6sd3b9mp
Thu, 15 Feb 2018 00:00:00 +0000

LevelSpacing Distributions and the Bessel Kernel
https://escholarship.org/uc/item/6q472228
The level spacing distributions which arise when one rescales the Laguerre or
Jacobi ensembles of hermitian matrices is studied. These distributions are expressible in
terms of a Fredholm determinant of an integral operator whose kernel is expressible in
terms of Bessel functions of order $\alpha$. We derive a system of partial differential
equations associated with the logarithmic derivative of this Fredholm determinant when the
underlying domain is a union of intervals. In the case of a single interval this Fredholm
determinant is a Painleve tau function.
https://escholarship.org/uc/item/6q472228
Thu, 15 Feb 2018 00:00:00 +0000

LevelSpacing Distributions and the Airy Kernel
https://escholarship.org/uc/item/5s7414dw
Scaling levelspacing distribution functions in the ``bulk of the spectrum'' in
random matrix models of $N\times N$ hermitian matrices and then going to the limit
$N\to\infty$, leads to the Fredholm determinant of the sine kernel $\sin\pi(xy)/\pi
(xy)$. Similarly a double scaling limit at the ``edge of the spectrum'' leads to the Airy
kernel $[{\rm Ai}(x) {\rm Ai}'(y) {\rm Ai}'(x) {\rm Ai}(y)]/(xy)$. We announce analogies
for this Airy kernel of the following properties of the sine kernel: the completely
integrable system of P.D.E.'s found by Jimbo, Miwa, M{\^o}ri and Sato; the expression, in
the case of a single interval, of the Fredholm determinant in terms of a Painlev{\'e}
transcendent; the existence of a commuting differential operator; and the fact that this
operator can be used in the derivation of asymptotics, for general $n$, of the probability
that an interval contains precisely...
https://escholarship.org/uc/item/5s7414dw
Thu, 15 Feb 2018 00:00:00 +0000

The Distribution of the Largest Eigenvalue in the Gaussian Ensembles
https://escholarship.org/uc/item/5738x9r6
The focus of this survey paper is on the distribution function for the largest
eigenvalue in the finite N Gaussian ensembles (GOE,GUE,GSE) in the edge scaling limit of
N>infinity. These limiting distribution functions are expressible in terms of a
particular Painleve II function. Comparisons are made with finite N simulations as well as
a discussion of the universality of these distribution functions.
https://escholarship.org/uc/item/5738x9r6
Thu, 15 Feb 2018 00:00:00 +0000

Random Unitary Matrices, Permutations and Painleve
https://escholarship.org/uc/item/3xp1c95c
This paper is concerned with certain connections between the ensemble of n x n
unitary matrices  specifically the characteristic function of the random variable tr(U)
 and combinatorics  specifically Ulam's problem concerning the distribution of the
length of the longest increasing subsequence in permutation groups  and the appearance of
Painleve functions in the answers to apparently unrelated questions. Among the results is a
representation in terms of a Painleve V function for the characteristic function of tr(U)
and (using recent results of Baik, Deift and Johansson) an expression in terms of a
Painleve II function for the limiting distribution of the length of the longest increasing
subsequence in the hyperoctahedral group.
https://escholarship.org/uc/item/3xp1c95c
Wed, 14 Feb 2018 00:00:00 +0000

Introduction to Random Matrices
https://escholarship.org/uc/item/3w1317n7
These notes provide an introduction to the theory of random matrices. The central
quantity studied is $\tau(a)= det(1K)$ where $K$ is the integral operator with kernel
$1/\pi} {\sin\pi(xy)\over xy} \chi_I(y)$. Here $I=\bigcup_j(a_{2j1},a_{2j})$ and
$\chi_I(y)$ is the characteristic function of the set $I$. In the Gaussian Unitary Ensemble
(GUE) the probability that no eigenvalues lie in $I$ is equal to $\tau(a)$. Also $\tau(a)$
is a taufunction and we present a new simplified derivation of the system of nonlinear
completely integrable equations (the $a_j$'s are the independent variables) that were first
derived by Jimbo, Miwa, M{\^o}ri, and Sato in 1980. In the case of a single interval these
equations are reducible to a Painlev{\'e} V equation. For large $s$ we give an asymptotic
formula for $E_2(n;s)$, which is the probability in the GUE that exactly $n$ eigenvalues
lie in an interval...
https://escholarship.org/uc/item/3w1317n7
Wed, 14 Feb 2018 00:00:00 +0000

Formulas and Asymptotics for the Asymmetric Simple Exclusion Process
https://escholarship.org/uc/item/3v68t7s6
This is an expanded version of a series of lectures delivered by the second author
in June, 2009. It describes the results of three of the authors' papers on ASEP, from the
derivation of exact formulas for configuration probabilities, through Fredholm determinant
representation, to asymptotics for ASEP with step initial condition establishing KPZ
universality. Although complete proofs are in general not given, at least the main elements
of them are.
https://escholarship.org/uc/item/3v68t7s6
Wed, 14 Feb 2018 00:00:00 +0000

Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz
https://escholarship.org/uc/item/3j3369bq
We prove that the solution to a pair of nonlinear integral equations arising in the
thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear
integral operator with kernel exp(u(theta)u(theta'))/cosh[(1/2)(thetatheta')]
https://escholarship.org/uc/item/3j3369bq
Mon, 5 Feb 2018 00:00:00 +0000

The Dynamics of the OneDimensional DeltaFunction Bose Gas
https://escholarship.org/uc/item/39b6f3h8
We give a method to solve the timedependent Schroedinger equation for a system of
onedimensional bosons interacting via a repulsive delta function potential. The method
uses the ideas of Bethe Ansatz but does not use the spectral theory of the associated
Hamiltonian.
https://escholarship.org/uc/item/39b6f3h8
Mon, 5 Feb 2018 00:00:00 +0000

Nonintersecting Brownian excursions
https://escholarship.org/uc/item/2w73b31w
We consider the process of $n$ Brownian excursions conditioned to be
nonintersecting. We show the distribution functions for the top curve and the bottom curve
are equal to Fredholm determinants whose kernel we give explicitly. In the simplest case,
these determinants are expressible in terms of Painlev\'{e} V functions. We prove that as
$n\to \infty$, the distributional limit of the bottom curve is the Bessel process with
parameter 1/2. (This is the Bessel process associated with Dyson's Brownian motion.) We
apply these results to study the expected area under the bottom and top curves.
https://escholarship.org/uc/item/2w73b31w
Thu, 1 Feb 2018 00:00:00 +0000

Painlev\'e Functions in Statistical Physics
https://escholarship.org/uc/item/2v86f7jn
We review recent progress in limit laws for the onedimensional asymmetric simple
exclusion process (ASEP) on the integer lattice. The limit laws are expressed in terms of a
certain Painlev\'e II function. Furthermore, we take this opportunity to give a brief
survey of the appearance of Painlev\'e functions in statistical physics.
https://escholarship.org/uc/item/2v86f7jn
Thu, 1 Feb 2018 00:00:00 +0000

A System of Differential Equations for the Airy Process
https://escholarship.org/uc/item/2g30f47p
The Airy process is characterized by its finitedimensional distribution functions.
We show that each finitedimensional distribution function is expressible in terms of a
solution to a system of differential equations.
https://escholarship.org/uc/item/2g30f47p
Thu, 1 Feb 2018 00:00:00 +0000

LevelSpacing Distributions and the Airy Kernel
https://escholarship.org/uc/item/27f342vd
Scaling levelspacing distribution functions in the ``bulk of the spectrum'' in
random matrix models of $N\times N$ hermitian matrices and then going to the limit
$N\to\infty$, leads to the Fredholm determinant of the sine kernel $\sin\pi(xy)/\pi
(xy)$. Similarly a scaling limit at the ``edge of the spectrum'' leads to the Airy kernel
$[{\rm Ai}(x) {\rm Ai}'(y) {\rm Ai}'(x) {\rm Ai}(y)]/(xy)$. In this paper we derive
analogues for this Airy kernel of the following properties of the sine kernel: the
completely integrable system of P.D.E.'s found by Jimbo, Miwa, M{\^o}ri and Sato; the
expression, in the case of a single interval, of the Fredholm determinant in terms of a
Painlev{\'e} transcendent; the existence of a commuting differential operator; and the fact
that this operator can be used in the derivation of asymptotics, for general $n$, of the
probability that an interval contains...
https://escholarship.org/uc/item/27f342vd
Thu, 1 Feb 2018 00:00:00 +0000

On the Distributions of the Lengths of the Longest Monotone Subsequences in Random
Words
https://escholarship.org/uc/item/27185530
We consider the distributions of the lengths of the longest weakly increasing and
strongly decreasing subsequences in words of length N from an alphabet of k letters. We
find Toeplitz determinant representations for the exponential generating functions (on N)
of these distribution functions and show that they are expressible in terms of solutions of
Painlev\'e V equations. We show further that in the weakly increasing case the generating
function gives the distribution of the smallest eigenvalue in the k x k Laguerre random
matrix ensemble and that the distribution itself has, after centering and normalizing, an N
> infinity limit which is equal to the distribution function for the largest eigenvalue
in the Gaussian Unitary Ensemble of k x k hermitian matrices of trace zero.
https://escholarship.org/uc/item/27185530
Thu, 1 Feb 2018 00:00:00 +0000

The Bose Gas and Asymmetric Simple Exclusion Process on the HalfLine
https://escholarship.org/uc/item/26g1d7xf
In this paper we find explicit formulas for: (1) Green's function for a system of
onedimensional bosons interacting via a deltafunction potential with particles confined
to the positive halfline; and (2) the transition probability for the onedimensional
asymmetric simple exclusion process (ASEP) with particles confined to the nonnegative
integers. These are both for systems with a finite number of particles. The formulas are
analogous to ones obtained earlier for the Bose gas and ASEP on the line and integers,
respectively. We use coordinate Bethe Ansatz appropriately modified to account for
confinement of the particles to the halfline. As in the earlier work, the proof for the
ASEP is less straightforward than for the Bose gas.
https://escholarship.org/uc/item/26g1d7xf
Thu, 1 Feb 2018 00:00:00 +0000

A Distribution Function Arising in Computational Biology
https://escholarship.org/uc/item/1pv2k3xf
Karlin and Altschul in their statistical analysis for multiple highscoring
segments in molecular sequences introduced a distribution function which gives the
probability there are at least r distinct and consistently ordered segment pairs all with
score at least x. For long sequences this distribution can be expressed in terms of the
distribution of the length of the longest increasing subsequence in a random permutation.
Within the past few years, this last quantity has been extensively studied in the
mathematics literature. The purpose of these notes is to summarize these new mathematical
developments in a form suitable for use in computational biology.
https://escholarship.org/uc/item/1pv2k3xf
Wed, 31 Jan 2018 00:00:00 +0000

On the Distribution of a Second Class Particle in the Asymmetric Simple Exclusion
Process
https://escholarship.org/uc/item/1jx76452
We give an exact expression for the distribution of the position X(t) of a single
second class particle in the asymmetric simple exclusion process (ASEP) where initially the
second class particle is located at the origin and the first class particles occupy the
sites {1,2,...}.
https://escholarship.org/uc/item/1jx76452
Wed, 31 Jan 2018 00:00:00 +0000

Random Words, Toeplitz Determinants and Integrable Systems. I
https://escholarship.org/uc/item/1gq736d2
It is proved that the limiting distribution of the length of the longest weakly
increasing subsequence in an inhomogeneous random word is related to the distribution
function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject
to an overall constraint that the eigenvalues lie in a hyperplane.
https://escholarship.org/uc/item/1gq736d2
Wed, 31 Jan 2018 00:00:00 +0000

Total Current Fluctuations in ASEP
https://escholarship.org/uc/item/18z3q1k9
A limit theorem for the total current in the asymmetric simple exclusion process
(ASEP) with step initial condition is proved. This extends the result of Johansson on TASEP
to ASEP.
https://escholarship.org/uc/item/18z3q1k9
Wed, 31 Jan 2018 00:00:00 +0000

A growth model in a random environment
https://escholarship.org/uc/item/0zq7g7f3
We consider a model of interface growth in two dimensions, given by a height
function on the sites of the onedimensional integer lattice. According to the discrete
time update rule, the height above the site $x$ increases to the height above $x1$, if the
latter height is larger; otherwise the height above $x$ increases by 1 with probability
$p_x$. We assume that $p_x$ are chosen independently at random with a common distribution
$F$, and that the initial state is such that the origin is far above the other sites. We
explicitly identify the asymptotic shape and prove that, in the pure regime, the
fluctuations about that shape, normalized by the square root of time, are asymptotically
normal. This contrasts with the quenched version: conditioned on the environment, and
normalized by the cube root of time, the fluctuations almost surely approach a distribution
known from random matrix theory.
https://escholarship.org/uc/item/0zq7g7f3
Fri, 26 Jan 2018 00:00:00 +0000

Integral Formulas for the Asymmetric Simple Exclusion Process
https://escholarship.org/uc/item/0t31f8mv
In this paper we obtain general integral formulas for probabilities in the
asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor
hopping rates p to the right and q=1p to the left. For the most part we consider an
Nparticle system but for certain of these formulas we can take the limit as N goes to
infinity. First we obtain, for the Nparticle system, a formula for the probability of a
configuration at time t, given the initial configuration. For this we use Bethe Ansatz
ideas to solve the master equation, extending a result of Schuetz for the case N=2. The
main results of the paper, derived from this, are integral formulas for the probability,
for given initial configuration, that the m'th leftmost particle is at x at time t. In one
of these formulas we can take the limit as N goes to infinity, and it gives the probability
for an infinite system where...
https://escholarship.org/uc/item/0t31f8mv
Fri, 26 Jan 2018 00:00:00 +0000

Random Words, Toeplitz Determinants and Integrable Systems. II
https://escholarship.org/uc/item/06g660f8
This paper, a continuation of math.CO/9909169, connects the analysis of the length
of the longest weakly increasing subsequence of inhomogeneous random words to a
RiemannHilbert problem and an associated system of integrable PDEs. In particular, we show
that the Poissonization of the distribution function of this length can be identified as
the JimboMiwaUeno tau function.
https://escholarship.org/uc/item/06g660f8
Fri, 26 Jan 2018 00:00:00 +0000

On the Determinant of a Certain WienerHopf + Hankel Operator
https://escholarship.org/uc/item/9454v124
We establish an asymptotic formula for determinants of truncated WienerHopf+Hankel operators with symbol equal to the exponential of a constant times the characteristic function of an interval. This is done by reducing it to the corresponding (known) asymptotics for truncated Toeplitz+Hankel operators. The determinants in question arise in random matrix theory in determining the limiting distribution for the number of eigenvalues in an interval for a scaled Laguerre ensemble of positive Hermitian matrices.
https://escholarship.org/uc/item/9454v124
Tue, 19 Dec 2017 00:00:00 +0000

Stability of the surface area preserving mean curvature flow in Euclidean space
https://escholarship.org/uc/item/5x99t347
The surface area preserving mean curvature flow is a mean curvaturetype flow with a global forcing term to keep the hypersurface areafixed. By iteration techniques, we show that the surface area preservingmean curvature flow in Euclidean space exists for all time and convergesexponentially to a round sphere, if initially the L2norm of the traceless second fundamental form is small (but the initial hypersurface is notnecessarily convex).
https://escholarship.org/uc/item/5x99t347
Thu, 6 Jul 2017 00:00:00 +0000

Padic Lfunctions and Euler systems: A tale in two trilogies
https://escholarship.org/uc/item/5nt3z8v1
This chapter surveys six different special value formulae for padic Lfunctions, stressing their common features and their eventual arithmetic applications via Kolyvagin’s theory of “Euler systems”, in the spirit of CoatesWiles and KatoPerrinRiou.
https://escholarship.org/uc/item/5nt3z8v1
Tue, 10 May 2016 00:00:00 +0000

Scalar invariants of surfaces in the conformal 3sphere via Minkowski spacetime
https://escholarship.org/uc/item/8rf4t5w4
For a surface in 3sphere, by identifying the conformal round 3sphere as the
projectivized positive light cone in Minkowski 5spacetime, we use the
conformal Gauss map and the conformal transform to construct the associate
homogeneous 4surface in Minkowski 5spacetime. We then derive the local
fundamental theorem for a surface in conformal round 3sphere from that of the
associate 4surface in Minkowski 5spacetime. More importantly, following the
idea of Fefferman and Graham, we construct local scalar invariants for a
surface in conformal round 3sphere. One distinct feature of our construction
is to link the classic work of Blaschke to the works of Bryan and
FeffermanGraham.
https://escholarship.org/uc/item/8rf4t5w4
Tue, 15 Mar 2016 00:00:00 +0000

Estimates for the energy density of critical points of a class of conformally invariant variational problems
https://escholarship.org/uc/item/86j2v038
Estimates for the energy density of critical points of a class of conformally invariant variational problems
https://escholarship.org/uc/item/86j2v038
Tue, 10 Nov 2015 00:00:00 +0000

Modified mean curvature flow of starshaped hypersurfaces in hyperbolic space
https://escholarship.org/uc/item/5rm5w502
Modified mean curvature flow of starshaped hypersurfaces in hyperbolic space
https://escholarship.org/uc/item/5rm5w502
Tue, 10 Nov 2015 00:00:00 +0000

Existence of good sweepouts on closed manifolds
https://escholarship.org/uc/item/3qh4j8qb
In this note we establish estimates for the harmonic map heat flow from $S^1$
into a closed manifold, and use it to construct sweepouts with the following
good property: each curve in the tightened sweepout, whose energy is close to
the maximal energy of curves in the sweepout, is itself close to a closed
geodesic.
https://escholarship.org/uc/item/3qh4j8qb
Tue, 10 Nov 2015 00:00:00 +0000

Uniformity of harmonic map heat flow at infinite time
https://escholarship.org/uc/item/29r2j8r8
Uniformity of harmonic map heat flow at infinite time
https://escholarship.org/uc/item/29r2j8r8
Tue, 10 Nov 2015 00:00:00 +0000

Closed Geodesics in Alexandrov Spaces of Curvature Bounded from Above
https://escholarship.org/uc/item/1226b2xb
Closed Geodesics in Alexandrov Spaces of Curvature Bounded from Above
https://escholarship.org/uc/item/1226b2xb
Tue, 10 Nov 2015 00:00:00 +0000

A note on static spaces and related problems
https://escholarship.org/uc/item/8gn049c3
In this paper we study static spaces introduced in Hawking and Ellis (1975) [1], Fischer and Marsden (1975) [3] and Riemannian manifolds possessing solutions to the critical point equation introduced in Besse (1987) [11], Hwang (2000) [12]. In both cases, on the manifolds there is a function satisfying a particular RicciHessian type equation (1.6). With an idea similar to that used in Cao et al. (2012) [15,16], we have made progress in solving the problem raised in Fischer and Marsden (1975) [3] of classifying vacuum static spaces and in proving the conjecture proposed in Besse (1987) [11] concerning manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3. © 2013 Elsevier B.V.
https://escholarship.org/uc/item/8gn049c3
Sun, 1 Nov 2015 00:00:00 +0000

GAP PHENOMENA AND CURVATURE ESTIMATES FOR CONFORMALLY COMPACT EINSTEIN MANIFOLDS
https://escholarship.org/uc/item/6tw8n95x
<p>In this paper we obtain first a gap theorem for a class of conformally compact Einstein manifolds with a renormalized volume that is close to its maximum value. We also use a blowup method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The major part of this paper is the study of how a property of the conformal infinity influences the geometry of the interior of a conformally compact Einstein manifold. Specifically we are interested in conformally compact Einstein manifolds with conformal infinity whose Yamabe invariant is close to that of the round sphere. Based on the approach initiated by Dutta and Javaheri we present a complete proof of the relative volume inequality
\[
(
Y
(
∂
X
,
[
g
^
...
https://escholarship.org/uc/item/6tw8n95x
Sun, 1 Nov 2015 00:00:00 +0000

Hypersurfaces in hyperbolic space with support function
https://escholarship.org/uc/item/6mn2x9vw
Based on [19], we develop a global correspondence between immersed hypersurfaces ϕ:Mn→Hn+1 satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e2ρgSn on domains Ω in the boundary Sn at infinity of Hn+1, where ρ is the horospherical support function, ∂∞ϕ(M<sup>n</sup>)=∂Ω, and Ω is the image of the Gauss map G:Mn→Sn. To do so we first establish results on when the Gauss map G:Mn→Sn is injective. We also discuss when an immersed horospherically concave hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles for elliptic problems of both immersed hypersurfaces in Hn+1 and conformal metrics on domains in Sn. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave hypersurface in Hn+1 of constant mean curvature.
https://escholarship.org/uc/item/6mn2x9vw
Sun, 1 Nov 2015 00:00:00 +0000

On scalar curvature rigidity of vacuum static spaces
https://escholarship.org/uc/item/55c6k9ht
In this paper we extend the local scalar curvature rigidity result in [6] to
a small domain on general vacuum static spaces, which confirms the interesting
dichotomy of local surjectivity and local rigidity about the scalar curvature
in general in the light of the paper [10]. We obtain the local scalar curvature
rigidity of bounded domains in hyperbolic spaces. We also obtain the global
scalar curvature rigidity for conformal deformations of metrics in the domains,
where the lapse functions are positive, on vacuum static spaces with positive
scalar curvature, and show such domains are maximal, which generalizes the work
in [15].
https://escholarship.org/uc/item/55c6k9ht
Sun, 1 Nov 2015 00:00:00 +0000

Möbius and Laguerre geometry of Dupin Hypersurfaces
https://escholarship.org/uc/item/4kg558b7
In this paper we show that a Dupin hypersurface with constant M\"{o}bius
curvatures is M\"{o}bius equivalent to either an isoparametric hypersurface in
the sphere or a cone over an isoparametric hypersurface in a sphere. We also
show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre
equivalent to a flat Laguerre isoparametric hypersurface. These results solve
the major issues related to the conjectures of Cecil et al on the
classification of Dupin hypersurfaces.
https://escholarship.org/uc/item/4kg558b7
Sun, 1 Nov 2015 00:00:00 +0000

A note on conformal Ricci flow
https://escholarship.org/uc/item/2m079191
In this note we study the conformal Ricci flow that Arthur Fischer introduced in 2004. We use DeTurck's trick to rewrite the conformal Ricci flow as a strong parabolicelliptic partial differential equation. Then we prove shorttime existence for the conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We show that the Yamabe constant is monotonically increasing along conformal Ricci flow on compact manifolds. We also show that the conformal Ricci flow is the gradient flow for the ADM mass on asymptotically flat manifolds. © 2014 Mathematical Sciences Publishers.
https://escholarship.org/uc/item/2m079191
Sun, 1 Nov 2015 00:00:00 +0000

Fractional conformal laplacians and fractional yamabe problems
https://escholarship.org/uc/item/1pr7j4bj
Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichletto Neumann operators of uniformly degenerate elliptic boundary value problems observed by Chang and González, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopftype maximum principle together with interplay between analysis of weighted trace Sobolev inequalities and conformal structure of the underlying manifolds, which extends the phenomena displayed in the classic Yamabe problem and boundary Yamabe problem. © 2013 Mathematical Sciences Publishers.
https://escholarship.org/uc/item/1pr7j4bj
Sun, 1 Nov 2015 00:00:00 +0000

Integral Eisenstein cocycles on GLn, II: Shintani's method
https://escholarship.org/uc/item/9fz4f7n6
We define a cocycle on GLn(Q) using Shintani's method. This construction is closely related to earlier work of Solomon and Hill, but differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q. As a corollary of our result we obtain a new proof of a theorem of Diaz y Diaz and Friedman on signed fundamental domains, and give a cohomological reformulation of Shintani's proof of the KlingenSiegel rationality theorem on partial zeta functions of totally real fields. Next we relate the Shintani cocycle to the Sczech cocycle by showing that the two differ by the sum of an explicit coboundary and a simple "polar" cocycle. This generalizes a result of Sczech and Solomon in the case n = 2. Finally, we introduce an integral version of our cocycle by smoothing at an auxiliary prime l. This integral refinement has strong arithmetic consequences. We showed in previous work that certain specializations of the smoothed class yield the padic...
https://escholarship.org/uc/item/9fz4f7n6
Thu, 22 Oct 2015 00:00:00 +0000

The padic Lfunctions of evil Eisenstein series
https://escholarship.org/uc/item/7k657713
We compute the padic Lfunctions of evil Eisenstein series, showing that they factor as products of two KubotaLeopoldt padic Lfunctions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.
https://escholarship.org/uc/item/7k657713
Thu, 22 Oct 2015 00:00:00 +0000

Symmetric regularization, reduction and blowup of the planar threebody problem
https://escholarship.org/uc/item/3cq122sb
Symmetric regularization, reduction and blowup of the planar threebody problem
https://escholarship.org/uc/item/3cq122sb
Thu, 2 Jan 2014 00:00:00 +0000

Symmetric regularization, reduction and blowup of the planar threebody problem
https://escholarship.org/uc/item/9wn7j80d
Symmetric regularization, reduction and blowup of the planar threebody problem
https://escholarship.org/uc/item/9wn7j80d
Mon, 16 Dec 2013 00:00:00 +0000

Bubblesandcrashes:Gradientdynamicsinﬁnancial markets
https://escholarship.org/uc/item/3905j8kq
<p>Fund managers respond to the payoff gradient by continuously adjusting leverage in our analytic and simulation models. The base model has a stable equilibrium with classic properties. However, bubbles and crashes occur in extended models incorporating an endogenous market risk premium based on investors' historical losses and constantgain learning. When losses have been small for a long time, asset prices inflate as fund managers increase leverage. Then slight losses can trigger a crash, as a widening risk premium accelerates deleveraging and asset price declines.</p>
https://escholarship.org/uc/item/3905j8kq
Thu, 8 Oct 2009 00:00:00 +0000

Almost Global Solutions of the Eikonal Equation
https://escholarship.org/uc/item/93p4c58m
Almost Global Solutions of the Eikonal Equation
https://escholarship.org/uc/item/93p4c58m
Tue, 16 Nov 2004 00:00:00 +0000