Prior knowledge, in the form of linear inequalities that
need to be satisfied over multiple polyhedral sets, is
incorporated into a function approximation generated by a
linear combination of linear or nonlinear kernels. In
addition, the approximation needs to satisfy conventional
conditions such as having given exact or inexact function
values at certain points. Determining such an approximation
leads to a linear programming formulation. By using a
nonlinear kernel and imposing the prior knowledge in the
feature space rather than the input space, the nonlinear
prior knowledge translates into nonlinear inequalities in
the original input space. Through a number of computational
examples, it is shown that prior knowledge can significantly
improve function approximation.

Given a complex simple Lie algebra, Joseph has constructed a
completely prime ideal in the universal enveloping algebra whose
associated variety is the minimal non-zero nilpotent orbit. He also
claimed that J is characterized by this property when the Lie algebra is
not of type A. Unfortunately there is a gap in the proof. We shall
describe a simple proof along different lines. (Joint work with G. Savin).

Joint Seminar with UCI

Let $(X,w)$ be an integral symplectic manifold and let $(L,Delta)$ be a quantum line bundle, with connection, over X having w as curvature. With this data, one can define an induced symplectic manifold Y with $/dim(Y) = dim(X)+2$. This is applied to show that the 5 split exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.

Mathematical models of neurons are formulated as nonlinear
stochastic ordinary or partial differential equations.
Progress in determining the activity of single neurons for
models of varying degrees of mathematical complexity by analytical
and simulation methods will be described. Exact results for the global
activity in a network of such elements will also be
obtained and in particular their approximations by means of
diffusion processes. Analogous models for
viral dynamics and epidemic networks may also be considered.

This talk will be a survey of joint work with Harold Stark on
zeta and L-functions of finite connected graphs, emphasizing parallels
with the zeta and L-functions of number theory.

Geometrical optics and its ingredients, eikonals and amplitudes, have
wide applications, such as optimal control, robotic navigation and
computer vision. We propose a local level set method for constructing
the geometrical optics term in the paraxial formulation for the high
frequency asymptotics of 2-D acoustic wave equations. The geometrical
optics term consists of two multivalued functions: a traveltime function
satisfying the eikonal equation locally and an amplitude function
solving a transport equation locally. The multivalued traveltimes are
obtained by solving a level set equation and a traveltime equation with a
forcing term. The multivalued amplitudes are computed by a new
Eulerian formula based on the gradients of traveltimes and takeoff
angles. As a byproduct the method is also able to capture the caustic
locations. The proposed Eulerian method is second-order accurate and
has complexity of $O(N^2 Log N)$. Several examples including the well
known Marmousi synthetic model illustrate the accuracy and efficiency of
the Eulerian method. We will also discuss the extension of the method
to anisotropic elastic wave equations and other possible future
directions.

The adjoint group of a simple complex Lie algebra Lie(G) has
a unique minimal orbit which we denote by C. We describe for
classical Lie algebras, for any natural number k,
the Zariski closure of the union kC of all spaces spanned
by k points on C. The image of this set in the projective
space {Bbb P}(Lie(G)) is usually called the (k-1)-st
secant variety of {Bbb P}(C), and its dimension and
defect are easily determined from our explicit description.
We give the smallest k for which the closure of kC is
equal to the Lie algebra and compare these results with
the upper bound on secants of general varieties given
in a theorem of F. Zak (eg 1993).

This talk describes recent joint work with Jan Draisma.

I will talk about a stochastic proof of an extension of the strong maximum principle by E. Calabi in the framework of local Dirichlet forms associated with strong Feller diffusions.

Gross has refined the Birch Swinnerton Dyer Conjecture in the case of
an elliptic curve with complex multiplication by a nonmaximal order.
Gross Conjecture has been reformulated in the language of derived
categories and determinants of perfect complexes. Burns and Flach have
realized that this immediately leads to a refinement of Gross
Conjecture. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. This
conjecture is proved by a construction which shows it to follow from
the Explicit Reciprocity Law and Rubin`s Main Conjecture.

Let $v$ be a vector field from ${mathbb R}^2$ to the unit circle
${mathbb S}^1$. We study the operator $$ H_vf(x)= p.v. int_{-1}^{1}f(x-tv(x))frac{dt}{t},.$$ We prove that if the vector field $v$ has $1+epsilon$ derivatives, then $H_v$ extends to a bounded map from $L^2$ into itself.

Contact geometry is a venerable subject that arose out of the study of Geometric Optics in the1800s. Though the years it has repeatedly cropped up in many areas of mathematics, but only in thepast 30 years or so has it received serious attention. Recently there has been great progress inunderstanding contact structures. Depending on ones perspective contact structures sometimes seemlike topological objects, sometimes geometric objects and sometimes dynamical objects. In this talkI will begin by discussing how contact structures arise out of natural problems and how they have deep connections with topology and dynamics. Then after surveying a few topics about contact
structures in low dimensions I will define contact homology in certain situations. Contact homologyis a new invariant of contact structures (and/or certain submanifolds of them) that is similar, in
spirit, to Gromov-Witten invariants of symplectic manifolds or Floer homology of Lagrangiansubmanifolds in symplectic manifolds. I then will proceed to discuss applications of contact homology, in particular, I will describe how it yields potentially new invariants of submanifolds of
Euclidean space.

Many graphs arising in various information networks exhibit the power
law behavior the number of vertices of degree $k$ is proportional
to $k^{-eta}$ for some positive $eta$. Such graphs are called
power law graphs. In the study of the spectra of power law graphs,
there are basically two competing approaches. One is to prove
analogues of Wigner`s semi-circle law while the other predicts that
the eigenvalues follow a power law distributions. We will show that
both approaches are essentially correct if one considers the
appropriate matrices. We will prove that (under certain mild
conditions) the eigenvalues of the (normalized) Laplacian of a random
power law graph follow the semi-circle law while the spectrum of the
adjacency matrix of a power law graph obeys the power law. Our
results are based on the analysis of random graphs with given expected
degrees and their relations to several key invariants. These results
have implications on the usage of spectral techniques in many areas
related to pattern detection and information retrieval.

This is a joint work with Prof. Fan Chung and Prof. Van Vu.

Modern cryptography is very algebraic by nature. In this overview
talk we will explain the major secret and public key
cryptographic protocols. For secret key systems, Rijndael has
become the new standard and we will describe this protocol
through a sequence of algebraic operations in a finite ring $R$.

In the area of public key cryptography the major protocols are
the RSA protocol, the traditional Diffie-Hellman and the ElGamal
protocol. The last two protocols are based on the hardness of
the discrete logarithm problem in a finite group.

The discrete logarithm problem can be viewed as a semi-group
action on a set. This leads naturally to a generalized
Diffie-Hellman key exchange and a generalized ElGamal one-way
trapdoor function.

Using this point of view we will provide several interesting
semi-group actions on finite sets. Our main focus will be
examples of semi-ring actions on a semi-module. These examples
may lead to practical new one-way trapdoor functions.

The presented results constitute joint work with Gerard Maze and
Chris Monico.

We introduce covariate adjusted regression (CAR) for situations where both predictors and response in a regression model are not directly observable, but are contaminated with a multiplicative factor that is
determined by the value of an unknown function of an observable covariate. We demonstrate how the regression coefficients can be
estimated by establishing a connection to varying coefficient regression. The proposed covariate adjustment method is illustrated with an analysis of the regression of plasma fibrinogen concentration as response on serum transferrin level as predictor for 69 hemodialysis patients. In this example, both response and predictor are thought to be influenced in a multiplicative fashion by body mass index. A bootstrap hypothesis test enables us to test the significance of the regression parameters. We establish the asymptotic distribution of the parameter estimates for this new covariate adjusted regression model.

R E C R U I T M E N T

We consider estimation of a density from a sample that contains
measurement errors. This problem, known as a deconvolution problem, has
applications in many different fields such as astronomy, chemistry or
public health, since in real data applications, it happens quite often
that the observations are made with error. The contaminating density, or
error density, is often assumed to be known. In this context, a so-called
deconvolution kernel density estimator has been proposed in the literature
(see for example Carroll and Hall (1988) or Stefanski and Carroll (1990)).

The behavior of the deconvolution kernel density estimator depends
strongly on a smoothing parameter called the bandwidth. We discuss several
possible ways of choosing an appropriate bandwidth in practice.

We next consider deconvolution kernel estimation of a density with left
and/or right unknown finite endpoints. From the contaminated sample, we
estimate the boundary of the support by the value which maximizes a
certain diagnostic function. This function can for example be based on the
derivative of a deconvolution kernel estimator of the density. We
establish asymptotic properties of the proposed estimator and study the
practical aspects of the method via a simulation study.

This is joint work with Irène Gijbels.

Langlands conjectured that a continuous complex Galois
representation can
be associated to an automorphic ''representation'' such that their $L$-functions
agree. We reduce the problem for representations into $GL(4)$ with solvable
image into several cases. We prove that in certain cases, Langlands
modularity conjecture holds. In particular, we obtain a new case of
Artin s $L$-function conjecture.

T T O P $quad$ R E C R U I T M E N T

In recent years, vast quantities of
scientific data have been
generated in various disciplines,
thanks to the almost universal use of
computer technology. What are we to do with this data?
Can we find simple
ways of understanding complex data?
I discuss two areas of recent research
into this problem: ($1$) nonlinear dimensionality reduction;
($2$) topological estimation.

In Monte Carlo estimation of expected values of functionals of solutions of SDEs, we have to generate approximate sample paths, probably by time stepping. Typically, the bias is roughly proportional to a power of the time step, h. The power depends on the time stepper and on the functional. For a functional F(X(T)), so called weak error esimates give first order convergence for the forward Euler and Milstein methods. To bound the bias for more complicated functionals, we study the joint distribution of the path observed at the time step times: (X(h), x(2h), ... X(nh)), where nh=T. We compute the difference between the joint density for the time stepper and the exact joint density. For forward Euler this does not go to zero with h, but for Milstein it goes as sqrt(h). Our analysis leads to Runge Kutta method that is simpler than Milstein but has similar statistical properties. The talk will include background material including the definition of the Milstein method. This is joint work with Peter Glynn and Jose Antonio Perez.

I will give a general framework to study the topology of
infinite dimensional groups via their actions on contractible spaces known
as Buildings. The groups of interest to us will be loop groups, Kac-Moody
groups, symplectomorphism groups and similar transformation groups. I will
give examples of natural buildings associated to such groups and use them
to derive some consequences.

R E C R U I T M E N T

In this talk I will discuss some recent results on spectral invariants of
torsion type.

Spectral invariants of elliptic operators on compact manifolds are
important global geometric quantities. The Ray-Singer torsion is a
spectral invariant with significant topological implications. We discuss a
new spectral invariant which generalizes Ray-Singer`s construction. This
new invariant behaves nicely under holonomy restrictions. In particular,
it coincides with the BCOV torsion (first constructed in Mirror Symmetry
theory) when restricted to Calabi-Yau manifolds.

As an application in algebraic geometry, we prove a Shafarevich type
theorem for holomorphic moduli of polarized Calabi-Yau manifolds. We also
show some links between our new invariant and automorphic forms,
generalizing classical results in the two dimensional case.

Parts of the results are joint work with Lu and Yoshikawa.

Order related procedures (such as median, quantiles, and
nonparametric procedures) in one-dimensional data analysis and inference
have played such important roles that their analogues in high dimensions
have been sought for years (but without many satisfactory results). The
task is non-trivial because there is no natural and clear order principle
in high dimensions. On the other hand, data depth turns out to be a quite
promising tool for a center-outward ordering of multi-dimensional
observations. In this talk motivations of data depth are discussed.
Examples illustrating notions of data depth including half-space,
simplicial and projection depth are provided. Applications of data depth
in location, in regression, and in other settings are discussed. Depth
based procedures can outperform their competitors by maintaining a good
balance between efficiency and robustness. Computing issues and some
future research directions of data depth are briefly addressed.

I describe the topology of (the classifying space of) the
symplectomorphism groups of a family of symplectic 4-manifolds. In
particular, we calculate the integral cohomology of these classifying
spaces. We also study the space of compatible complex structures on these
symplectic manifolds and outline a proof showing that this space is
contractible.

The Sobolev space $W^{1,2}(M,N)$ between two Riemannian manifolds
$M$ and $N$ appears naturally in the calculus of variations. We will
discuss necessary and sufficient (topological) conditions for smooth maps
to be strongly or weakly dense in this space. These problems are of
analytical interest and closely related to the theory of harmonic maps.

It is shown that the commutant of the Pin-group action on
tensor powers of the spinor representation is already generated by
the elements which appear in the second tensor power, and a complete
set of relations is given. This can be used to completely classify
all tensor categories whose Grothendieck semiring is the one of a
classical group.

Motivated by a ``mass formula" due to Serre for local fields,
this talk will investigate the probability that a uniformly random
monic polynomial with coefficients in the p-adic integers factors
completely into linear factors.

I will discuss some attempts to recover topological information about geometric objects from ``point cloud data" sampled from the object. Examples will include data sets obtained from image data, and the problem of recognizing shapes, i.e. subcomplexes of Euclidean $3$-space.

The $overlinepartial_b$-Neumann problem is the analog for CR manifolds of
the $overlinepartial$-Neumann problem. All positive results about this
problem so far have applied to domains with a defining function depending
only on the real and imaginary parts of a single CR function. The key
feature of such domains is that they are foliated (away from a
characteristic curve) by compact, strictly pseudoconvex CR submanifolds of
real codimension 2. I will describe a new approach to finding estimates
based on decomposing the operator into its tangential and transverse parts
with respect to this foliation. Estimates for the tangential parts follow
from known results about the tangential Cauchy-Riemann complex on compact CR
manifolds, while estimates for the transverse part reduce to elliptic
estimates in the plane. For certain domains in the Heisenberg group, my
student Robert Hladky has used this method to obtain sharp boundary
regularity, even near characteristic points.

The J-flow is a parabolic flow on compact Kahler manifolds with two Kahler metrics. It was discovered by S. Donaldson and X. X. Chen independently. Donaldson defined it in the setting of moment maps and symplectic geometry. Chen described the flow as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. The Mabuchi energy is an important functional on the space of Kahler potentials. Its critical points give constant scalar curvature metrics, and its lower boundedness is related to stability in the sense of geometric invariant theory. I will show that under a condition on the initial data, the J-flow converges to a critical metric. I will then explain how this implies the lower boundedness of the Mabuchi energy for an open set of Kahler classes on manifolds with negative first Chern class.

This talk will discuss some conjectures about the relationship between the algebraic K-theory of a field on the one hand and the so-called derived completed complex representation theory of the absolute Galois group of the field. I will also discuss the connection of this derived completion with the study of the space of deformations of a given representation.

Fix a real reductive group $G$. Suppose $mathcal{O}'$ is a
nilpotent orbit in $mathfrak{g}'$, the dual of the complexified Lie
algebra of $G$. To each $X' in mathcal{O}'$, one may associate an sl(2)
triple, say $X', Y'$, and $H'$. Since $(1/2)H'$ lives in a Cartan
subalgebra of $mathfrak{g}'$, it defines an infinitesimal character for
$G$. One piece of the Arthur conjectures predicts that the smallest
representations of $G$ with infinitesimal character $(1/2)H'$ should
appear as local components of automorphic forms; in particular, they
should be unitary. (Two good examples to keep in mind are the trivial
representation and limits of discrete series with zero infinitesimal
character; the former corresponds to the principal orbit $mathcal{O'}$
and the latter to the zero orbit.) In this talk, we explain how to prove
a large part of this conjecture for certain classical groups using the
theta correspondence.

One of the most well known questions of Erdos in discrete geometry is
the following: Given n points in $R^d$, what is the smallest number of
distinct distances among them ? Here d is fixed and n tends to inifnity.
We denote by $f_d(n)$ is smallest number of distince distances.

The problem of determining $f_d(n)$ has been attacked by many
researchers (including Erdos, Beck, Chung,
Trotter, Szemeredi, Beck, Ssekely, Solymosi-Toth, Sharir, etc) for
decades. In this talk, I will give a brief overview and also present a new result
(joint with Solymosi). This result gives an almost sharp estimate for
$f_d(n)$ for relatively large dimension $d$. The main tool is what we
call "decomposition technique", which appears to be useful in other problems as well.

In attempting to understand the "meat ax" of finite group theory,
Diaconis has formulated an urn model. In the simplest case, balls
numbered 0 and 1 are placed in an urn. At times
$n = 1,2,...,$ two balls are drawn with
replacement. Those balls are replaced in the
urn, and a new ball that contains the sum mod $2$ of the
numbers on the drawn balls is added to the urn. A conjecture
is that the fraction of balls numbered $1$ converges to $1/2$.
This conjecture and some generalizations are proved
as a two-fold application of the almost supermartingale
convergence theorem of Robbins and Siegmund $(1972)$.

This is joint research with Benny Yakir.

It is often useful in cryptography to be able to generate
an elliptic curve over a finite field with a given number of points.
This talk will explain the complex multiplication (CM) method for
constructing suitable elliptic curves and explain a variant which is
joint work with A. Agashe and R. Venkatesan. I will also explain the CM
method for generating genus 2 curves and some interesting problems which
arise.

Principal Component Analysis (PCA) is a tool used across the spectrum of scientific applications. In modern practice, it is often applied to $n imes p$ data matrices with $n$ and $p$ both large. Classical theory (Anderson
1963) fails to apply in this setting. Using random matrix theory, Johnstone
(2000) recently shed light on some
theoretical aspects of PCA in this setup. Specifically, when the entries of the
$n imes p$ matrix
$X$ are iid ${cal N}(0,1)$ and $n/p
ightarrow
ho in (0,infty)$, he showed
that
$lambda_{n,p}$ , the largest eigenvalue of the empirical covariance matrix
$X'X$, converges to the so-called Tracy-Widom distribution (after proper
recentering and rescaling).
We will show that the result holds when $n,p
ightarrow infty$ and
$n/p
ightarrow 0$ or $
infty$, in effect removing the need to worry about the limiting behavior of
$n/p$.
We will also present preliminary results for rates of convergence. Finally, we
will illustrate how these and
related theoretical insights might be used in practice.

*RECRUITMENT TALK*
We present a new approach to CFD and Computational Turbulence Modeling
using adaptive stabilized Galerkin finite element methods with duality
based a posteriori error control for chosen output quantities of interest,
with the output based on the exact solution to the Navier-Stokes
equations, thus circumventing introducing and modeling Reynolds stresses
in averaged Navier-Stokes equations. We refer to our methodology as
Adaptive DNS/LES, where automatically by adaptivity certain features of
the flow are resolved in a Direct Numerical Simulation DNS, while certain
other small scale turbulent features are left unresolved in a Large Eddy
Simulation LES. The stabilization of the Galerkin method giving a weighted
least square control of the residual acts as the subgrid model in the LES.
The a posteriori error estimate takes into account both the error from
discretization and the error from the subgrid model. A crucial observation
from computational examples is that the contribution from subgrid modeling
in the a posteriori error estimation can be small, making it possible to
simulate aspects of turbulent flow without accurate modeling of Reynolds
stresses. Using the a posteriori error estimates we further consider the
question of uniqueness of weak solutions to the Navier-Stokes equations,
where we give computational evidence of both uniqueness and non-uniqueness
in outputs of weak solutions.

JOINT SEMINAR WITH UCI. THIS SEMINAR IS AT UCI.

JOINT SEMINAR WITH UCI. THIS SEMINAR IS BEING HELD AT UCI.

*RECRUITMENT TALK*
Epitaxial growth is a modern technology to grow thin solid films by depositing atoms or molecules onto an existing layer of material. Microscopic processes in epitaxial growth include the deposition of atoms onto a surface, desorption of adsorbed atoms (adatoms) into the gas phase, surface diffusion of adatoms, attachment and detachment of adatoms to and from atomic steps, adatom island nucleation, and island coalescence. These processes are characterized by fluctuation, non-equilibrium, and multiple spatial and temporal scales. In this talk, I will first present a derivation of an island dynamics model for epitaxial growth that includes step kinetics. This is an improvement of the classical Burton-Cabrera-Frank model that assumes
the adatom equilibrium along steps. An adaptive finite element method for the new model will be described. I will then focus on a class of fourth-order diffusion equations that model the coarsening in the surface dynamics of growing films after the roughening transition. Such growth equations are gradient flows of certain free energies. I will
show bounds for coarsening rates and the decay of energy, derive the energy asymptotics in the large-system-limit, and predict the exact scaling laws for the coarsening under the hypothesis of realization.

A descent in a permutation a(1)a(2)...a(n) is an i such that a(i) >
a(i+1). The distribution of the number of descents is given by the well
known Eulerian numbers.

We define a new statistic, namely the maximum number of non-overlapping
descents in a permutation. Two descents i and j overlap if |i-j| = 1. We
find the distribution of this new statistic using partially ordered
generalized patterns (POGPs). The POGPs are generalizations of the
Babson-Steingrimsson patterns, which in turn generalize the classical
permutation patterns.
A segment pattern is a pattern whose occurrence in a permutation is
required to consist of consecutive letters of the permutation. As an
example, the number of descents corresponds to the segment pattern 21.

Let P be an arbitrary segment pattern. Using POGPs, we give the
exponential generating function (e.g.f.) for the entire distribution of
the maximum number of non-overlapping occurrences of P, provided we know
the e.g.f. for the number of permutations that avoid P.

We also discuss POGPs in words (with repeated letters), where we get results
similar to those for permutations.

*RECRUITMENT TALK*
The current renaissance in the study of evolving faceted
crystal surfaces
was prompted by the discovery of nano-scale faceted pyramidal islands
(quantum dots) on Si films, as well as the appearance of novel in situ
imaging techniques.
We consider the coarsening dynamics of
faceted crystal surfaces that pertain to two distinct continuum models;
so-called thermal annealing (A) and net-growth (G) regimes.
We present a novel theoretical framework which
unites the two problems by recognizing
their common (leading-order) kinematic framework;
namely, piecewise-affine surfaces evolving on a suitably slow time scale.
Dynamic evolution laws, intrinsic to such surfaces, are then found
for each problem through novel matched asymptotic analysis.
Our theory resolves the long-standing annealing-to-growth enigma,
whereby the scaling law for the increase in time, t, of the
characteristic facet size, L, is observed to undergo
a transition when switching from (A) to (G).
In addition, the nature of the transition in surface morphology
between (A) and (G) follows readily from the stability
properties of the associated dynamical systems.
Large scale simulations, which are rich in both topological and
statistical terms, will also be presented.

In this talk, we will consider the problem of designing multi-focal
optical lenses. This problem has been actively studied over the past
20 years and the results obtained along the way of research have been
used directly in the lens industry.

We study a variational approach for the multi-focal lens design problem
and find out that the Euler-Lagrange equation associated with the
variational problem is a fourth-order nonlinear elliptic partial
differential equation. We consider two linearizations of the equation
along with three types of boundary conditions, and analyze the existence,
uniqueness as well as regularity of the solutions under these types of
boundary conditions. Finally, we propose a numerical method with a special
type of spline functions for solving the linearized partial differential
equations, and the method is shown to be very efficient.

We give a survey of a series of papers by Birget, Olshanskii, Rips and myself about asymptotic invariants of groups and connections between them and other areas of mathematics. In particular we give an algebraic characterization of groups with word problem in NP, construct manifolds with arbitrary isoperimetric functions $(say f(x)=x^{pi+e})$, and finitely presented counterexamples to von Neumann conjecture (a finitely presented non-amenable group without free subgroups).

I will present a joint work with Alicia Dickenstein and Laura
Matusevich. Algebraic functions (defined as solutions to algebraic
equations with general symbolic coefficients) are classically known
to satisfy certain systems of linear partial differential equations
with polynomial coefficients. In the talk I will consider a more
general class of systems of differential equations. We prove that
such systems are holonomic and that their complex holomorphic solutions
have moderate growth. We also provide an explicit formula for the
holonomic rank of these systems as well as bases in their spaces of
complex holomorphic and Puiseux polynomial solutions.

We study a few classes of Hilbert space operators which are complex
symmetric with respect to a preferred orthonormal basis. The existence of
this additional symmetry has notable implications and in particular it
explains from a unifying point of view some classical results.

Tracing the use of the term over $150$ years gives insight into the way mathematicians name things, and the things mathematicians name.

Put another way: is every polynomial in $2 n^2$ variables that vanishes
on a pair of commuting matrices, in the ideal generated by the obvious
$n^2$ quadratic relations?

Alas, we do not know (and I cannot answer the question either).
I will introduce several other related schemes that seem easier to study,
like the space of pairs of matrices whose commutator is diagonal,
which I will prove is a reduced complete intersection, one of whose
components is the commuting variety. Conjecturally, it has only one
other component, and I will explain where that one comes from.
Along the way we will also see a rather curious invariant of permutations,
and much simple linear algebra.

In this talk we describe our work on proportional hazards model with mixed effects (PHMM) for right-censored data. Our motivation came from a
multi-center clinical trial in lung cancer, where treatment effects were
found to vary substantially among the centers. We provide a general
framework for handling random effects in proportional hazards
regression, in a way similar to the linear, non-linear and generalized
linear mixed effects models that allow random effects of arbitrary
covariates. This general framework includes the frailty models as a
special case. Semi parametric maximum likelihood estimates of the
regression parameters, the variance components and the baseline hazard,
and empirical Bayes estimates of the random effects can be obtained via
an MCEM algorithm. Variances of the parameter estimates are approximated
using Louis formula. The model found interesting applications in
recurrent events, twin data and genetic epidemiology. Following the
introduction of the model, our recent work has included topics on model
diagnostics and model selection. We will elaborate on one of these
topics during the talk.

Lie groups provide a formalization of the concept of symmetry in the classical theory of special functions, combinatorics, and physics. From this viewpoint, DAHA describes abstract Fourier transforms, especially those making the Gaussian Fourier-invariant. The classical Fourier transform, the Hankel transform, and the one from the theory of Gaussian sums are well known examples. Thus DAHA formalizes an important part of the classical Fourier analysis.

The Verlinde algebras are the key finite-dimensional examples. There exist three different Verlinde algebras of type A 1:
1) the one connected with the Hankel transform, and, hopefully,
with the massless conformal field theory,
2) the major Verlinde algebra associated with the Kac-Moody
fusion and the massive CFT and,
3) the algebra presumably describing the fusion of the
(1,p)-Virasoro model.
They will be discussed in the lectures.

We will read a paper by Princeton graduate student Jacob Rasmussen, on Khovanov homology and the slice genus. It is a breakthrough paper, proving the so called Milnor conjectures by purely combinatorial means. These conjectures predict the 4-dimensional genus of torus knots, and their only known proof involves Gauge theory, or more precisely, Seiberg-Witten theory.

Rasmussen gives a proof using only methods from combinatorial skein theory and a certain spectral sequence. He obtains a concordance invariant for knots, which is absolutely mind blowing. It can show that certain knots with trivial Alexander polynomial are not smoothly slice, even though by Freedmans theorem they are topologically slice. Hence the invariant sees the difference between smooth and topological phenomena in dimension four.

As usual, the talks in this seminar will be given by the participants, with two survey lectures at the beginning given by Justin and Peter. Rasmussens paper, supplemented by some survey articles, will be used as reference for later talks.

Introduced 13 years ago, DAHA play now a solid role in modern representation theory with wide spectrum of applications from Harmonic Analysis and Combinatorics to Algebraic Geometry and Mathematical Physics.
I will begin with general motivation in the theory of spherical functions and the definitions in the rank one case. The talk will be mainly about the so-called nonsymmetric Verlinde algebras, which are in the focus of the new theory, including applications to the ''non-cyclotomic'' Gaussian sums and recent results on the diagonal coinvariants (the combinatorics of two sets of variables).

The space of ordered n-tuples of points on a projective line has a compactification, due to Grothendieck and Knudsen, with many remarkable properties: it has a natural moduli interpretation, namely it is the moduli space of stable n-pointed rational curves. It has a natural Mori theoretic meaning, namely it is the log canonical model of the interior.
For a curve in the interior, there is a description of the limiting stable n-pointed rational curve, due to Kapranov, in terms of the Tits tree of $PGL_2$. We study these properties for the higher-dimensional versions of the Grothendieck-Knudsen space, the Chow quotients of Grassmannians.

A discussion of results arising from Yang-Mills gauge theory on compact surfaces and the relationship between YM2 and Chern-Simons.

Lie groups provide a formalization of the concept of symmetry in the classical theory of special functions, combinatorics, and physics. From this viewpoint, DAHA describes abstract Fourier transforms, especially those making the Gaussian Fourier-invariant. The classical Fourier transform, the Hankel transform, and the one from the theory of Gaussian sums are well known examples. Thus DAHA formalizes an important part of the classical Fourier analysis.

The Verlinde algebras are the key finite-dimensional examples. There exist three different Verlinde algebras of type A 1:
1) the one connected with the Hankel transform, and, hopefully,
with the massless conformal field theory,
2) the major Verlinde algebra associated with the Kac-Moody
fusion and the massive CFT and,
3) the algebra presumably describing the fusion of the
(1,p)-Virasoro model.
They will be discussed in the lectures.

Positivity is a distinguishing property of the field of real numbers. Writing a polynomial as a sum of squares gives a certificate that it is positive. Hilbert showed that a positive homogeneous quartic polynomial in three variables (ternary quartic) is a sum of three squares of quadratic polynomials. He also showed that there are positive polynomials of every higher degree or greater number of variables with no such sum of squares representations. This led to his 17th problem - to determine whether a positive polynomial is a sum of squares of rational functions. This was answered in the affirmative by Artin in 1926.

Recently, positive polynomials have have undergone a revival. In the 1990s Lasserre realized that recent theoretical results from real algebraic geometry and semi-definite programming could be combined to give effective algorithms for solving a class of relaxations of hard optimization problems. The relaxation replaces positivity by sum of squares representation.

I will briefly survey the history of positive polynomials and these modern applications, and then discuss a recent strengthening of Hilberts Theorem on ternary quartics: a positive ternary quartic is a sum of squares in exactly 8 inequivalent ways.

The choice of a proper finite element space is essential to guarantee the
stability of a discretised system of partial differential equations. In
many cases, the differential geometric structure of the system can be
captured by a differential complex, and finding an appropriate discrete
differential complex may lead to a finite element space which yields a
stable discretisation. Two problems, one from electromagnetism and
the other from elasticity, will be discussed in this context.

Let G be a connected, simply connected, simple group over
the complex numbers. Let K be the fixed point group of an involutive
automorphism of G and let P be a parabolic subgroup of G such that KP has
interior in G. Then we say that (G,K,P) is a spherical triple if there
is a Borel subgroup of K, B, such that B has an open orbit in G/P. This
condition is equivalent with having (degenerate) principal series for the
non-compact real form corresponding to K be multiplicity free. In this
talk we will give a classification of these triples for the exceptional
groups.

We will describe the arithmetic of zeta values in the
function field context. It is a curious mix of many analogies,
some strong theorems where corresponding number field statements
are conjectures, and some questions where the number field situation
is understood, but the function field situation is not even
conjecturally understood

The goal of this presentation is to explore the multiple definitions of design research in education, and particularly in mathematics education. In doing so, I revisit components of Brown, A. L. (1992)*, which highlight the issues for warrant. I argue that as yet her concerns have gone relatively unheeded by the education research community. I point to the need for mixed methodologies in support of design based insights and develop a validity framework that is inclusive of design adaptation issues.

* Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. Journal of the Learning Sciences, 2, 141-178.Brown, A. L. (1992)

The subgroup growth of a group is the study of the number of
its index n subgroups as a function of n. We will present some sharp
estimates on this growth for lattices L in higher rank semi-simple Lie
groups G. A surprising phenomenom is that this growth depends only on the
root system of G and not on L. In the most general case, the results
depend on the generalized Riemann Hyphothesis but a number of results can
be also proved unconditionally.

The scattering of solitons off each other and with radiation is studied for NLS in three or more dimensions. The general problem of finding the large time behavior of such equations, including the proof of Asymptotic Completeness, and stability is discussed.

Recent solution of this problem for NLS with small radiation initial data will be described

In this talk, I will describe a construction of the exceptional
Lie algebras of type F and E using the octonions and the principle of triality.

Featuring the following speakers:

Robert Hecht-Nielsen - Founder, HNC Software; Professor, Electrical and Computer Engineering UCSD

Beth Smith - Mathematics Assistant Professor, Grossmont
College

Saul Molina - Assistant Systems Administrator, Math Dept UCSD

Space is limited - reserve your place - email jbitmead@ucsd.edu

Ramanujan graphs are k-regular graphs with optimal bounds on their eigenvalues. They play a central role in various questions in combinatorics and computer secience. Their construction is based on the work of Deligne and Drinfeld on the Ramanujan conjecture for GL(2). The
recent work of Lafforgue which settles the Ramanujan conjecture for GL(n) over function fields opens the door to study of Ramanujan complexes: these are higher dimensional analogues which are obtained as quotients of the Bruhat-Tits building of PGL(n) over local fields.

We introduce and study new complete Kahler metrics on the moduli and the Teichmuller spaces of Riemann surfaces, the Ricci and the perturbed Ricci metric. They are asymptotically equivalent to the Poincare metric. The perturbed Ricci metric has bounded negative sectional and Ricci curvatures. As corollaries we prove the equivalence of these new metrics to several classical metrics such as the Kahler-Einstein metrics, proving a conjecture of Yau in the early 80s. Other consequences will also be discussed. This is joint work with K. Liu and S.-T. Yau.

There has been great interest in two-dimensional representations of Galois
groups, from conjectures of Artin concerning complex projective
representations of the symmetries of the Platonic solids, to conjectures
of Shimura and Taniyama concerning p-adic representations associated to
elliptic curves. Many of these conjectures were recently answered in the
affirmative by Wiles and Taylor using techniques from arithmetic algebraic
geometry. In this talk, we explain how these results can be extended even
further, and give some applications.

Saturday Schedule

10:30 - 11:00 AM Coffee and assorted danish

11:00 - 12:00 PM Gunther Uhlmann, University of Washington

Boundary rigidity and the Dirichlet-to-Neumann map

1:30 - 2:30 PM Chuu-Lian Terng, UCI & Northeastern University

Periodic and homoclinic orbits of the modified 2+1 chiral model

2:45 - 3:45 PM Oded Schramm, Microsoft Research

Conformally invariant random processes in two dimensions

3:45 - 4:15 PM Coffee break

4:15 - 5:15 PM Michael Lacey, Georgia Tech

Hilbert transforms and smooth families of lines

Sunday Schedule

9:30 - 10:00 AM Coffee and assorted danish

10:00 - 11:00 AM Jean-Pierre Rosay, University of Wisconsin-Madison

Pseudo-holomorphic discs and rough almost complex structures. (Isolated solutions of non-linear PDE)

11:15-12:15 PM Christoph Thiele, UCLA

Basic questions for the Nonlinear Fourier transform

Alex Postnikov recently gave a combinatorially explicit cell
decomposition of the totally nonnegative part of a Grassmannian, denoted $Gr_{kn}+$, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our
work is an explicit generating function which enumerates the cells in $Gr_{kn}+$ according to their dimension. As a corollary, we give a new proof that the Euler characteristic of $Gr_{kn}+$ is 1. Additionally, we use our result to produce a new q-analog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients.

We show that any complex submanifold in Hilbert space defined by global equations has a basis of pseudoconvex open neighborhoods, that holomorphic vector bundles are acyclic over smooth complete intersections in Hilbert space, and that such complete intersections are holomorphic retracts of some pseudoconvex open neighborhoods.

A tutorial overview of time-domain Bootstrap Methods for time series will be given with emphasis on block-bootstrap and subsampling. Comparisons of the methods
will be discussed, as well as the crucial issue of practical block size choice.

JOINT SEMINAR WITH UCI HELD at UC Irvine, MSTB 254

THIS SEMINAR IS NOT BEING HELD AT UCSD BUT AT UC IRVINE

JOINT SEMINAR WITH UC IRVINE HELD AT UCI, MSTB 254

THIS SEMINAR IS NOT BEING HELD AT UCSD BUT AT UC IRVINE

Let $0 < alpha < beta$ be two irrational numbers satisfying
$1/alpha + 1/beta = 1$. Then the sequences $a'_n = lfloor
{nalpha}rfloor$, $b'_n = lfloor{nbeta}rfloor$, $nge 1$, are
complementary over $IZ_{ge 1}$. Thus $a'_n = {rm mex_1}
{a'_i,b'_i : 1 le i< n}$, $n geq 1$ (${rm mex_1}(S)$, the
smallest positive integer not in the set $S$). Suppose that $c =
beta-alpha$ is an integer. Then $b'_n = a'_n+cn$ for all $n ge
1$.

We define the following generalization of the sequences $a'_n$,
$b'_n$: Let $c,;n_0inIZ_{ge 1}$, and let $XsubsetIZ_{ge 1}$
be an arbitrary finite set. Let $a_n = {rm mex_1}(Xcup{a_i,b_i
: 1 leq i< n})$, $b_n = a_n+cn$, $nge n_0$. Let $s_n =
a'_n-a_n$. We show that no matter how we pick $c,;n_0$ and $X$,
from some point on the {it shift sequence/} $s_n$ assumes either
one constant value or three successive values; and if the second
case holds, it assumes these values in a very distinct
fractal-like pattern, which we describe.

This work was motivated by a generalization of Wythoff's game to
$Nge 3$ piles.

This talk covers a variety of conjectures not resolved by Mark Haiman's
proof of the $n!$-conjecture. We also review what came to be called
"Science Fiction" which taken literally is blatantly false at the
representation theoretical level, yet it beautifully explains many of the
experimentally observed properties of the Macdonald $q,t$-Kostka
coefficients.

We give a proof of the Universality Conjecture
in Random Matrix Theory for orthogonal $(beta=1)$ and
symplectic $(beta=4)$ ensembles in the scaling limit
for a class of weights $w(x)=exp(-V(x))$
where V is a polynomial. For such weights
the associated equilibrium measure is supported on
a single interval.
Our starting point is Widom's representation
of the correlation kernels for the beta=1,4 cases
in terms of the unitary $(beta=2)$ correlation kernel
plus a correction term which involves orthogonal
polynomials $(OP's)$ with respect to the weight w introduced above.
We do not use skew orthogonal polynomials.
In the asymptotic analysis of the correction terms
we use amongst other things
differential equations for the derivatives
of OP's due to Tracy-Widom,
and uniform Plancherel-Rotach type asymptotics for OP's
due to Deift-Kriecherbauer-McLaughlin-Venakides-Zhou.
The problem reduces to a small norm problem
for a certain matrix of a fixed size
that is equal to the degree of the polynomial potential.

Quantum information theory has generated interesting conjectures concerning products of completely positive maps on matrix algebras. This talk will describe the background to these conjectures and also some recent proofs in special cases. Matrix inequalities have played an important role in these results, and the talk will describe applications of the Lieb-Thirring inequality and an extension of Hanner's inequality to this problem.

Let A be a square matrix with rational entries. Then A can be written as the
sum S + N where S and N also have rational entries, S is semisimple, N is nilpotent, and
S and N commute. This representation is unique, and is called the Jordan decomposition
of A. It can be considered as a coordinate-free, and ambient-field-free, version of the
usual Jordan canonical form for matrices. This decomposition (in its multiplicative
version) is particularly useful in the study of algebraic groups.

If G is a finite group and a is an element of the rational group ring Q[G],
i.e. a is a linear combination of group elements with rational coefficients, then there
is an analogous decomposition: a = s + n where s and n lie in Q[G], s is semisimple, n
is nilpotent, and s and n commute (this representation is also unique).

Consider the integral versions of these decompositions: if the matrix A has
integer entries, need S and N have integer entries? if the element a in Q[G] has integer
coefficients, need s and n have integer coefficients? We give complete answers to these
questions. The multiplicative version of the integral group ring question is much more
subtle, however, and we only have partial results on this problem.

In joint work with J. Metcalfe and A. Stewart, we prove global and almost
global existence theorems for nonlinear wave equations with quadratic
nonlinearities in infinite homogeneous waveguides. We can handle both the
case of Dirichlet boundary conditions and Neumann boundary conditions. In
the case of Neumann boundary conditions we need to assume a natural
nonlinear Neumann condition on the quasilinear terms. The results that we
obtain are sharp in terms of the assumptions on the dimensions for the
global existence results and in terms of the lifespan for the almost global
results. For nonlinear wave equations, in the case where the infinite part
of wave guide has spatial dimension three, the hypotheses in the theorem
concern whether or not the Laplacian for the compact base of the wave guide
has a zero mode or not.

Ramsey's original theorem states that if one finitely colors the $k$
element subsets of ${\Bbb N}$ then there exists an infinite subsequence $M$
of ${\Bbb N}$ all of whose $k$ elements subsets have the same color.
This theorem and stronger versions entered into Banach space theory in the
1970's.
They were ideal for studying subsequences of a given sequence
$(x_i)\subseteq X$ (infinite dimensional separable Banach space).
We survey some of these applications and the following problem.
$X$ is said to satisfy the ultimate Ramsey theorem if for every finite
coloring $(C_i)_{i=1}^n$ of its unit sphere $S_X$ and $\varepsilon>0$ there exists
an infinite dimensional subspace $Y$ and $i_0$ so that
$S_Y\subseteq (C_{i_0})_\varepsilon =\{x:|x-z|<\varepsilon$ for some $z\in C_{i_0}\}$.
What spaces $X$ (if any) have this property?
We survey other results including Gowers' block Ramsey theorem for Banach
spaces.

Abstract: On of the most famous problems in number theory is to show that there are arbitrarily long arithmetic progressions, all of whose terms are prime numbers. At the start of the year, this was known to be true only for 3-term progressions. Recently, Ben Green and Terrence Tao have released a manuscript which purports to prove this conjecture. I will discuss some of the history of this problem, and outline their argument. I intend for this talk to be accessible to a broad audience, and hope that this will lead to a series of talks (not all by me) verifying the Green/Tao proof.

Abstract: An $(h,g)$ Sidon set is a set $S$ of integers with the property that the coefficients of
$(\sum_{s \in S} z^s)^h$
are bounded by $g$. These arose naturally is Simon Sidon's study of Fourier Series, and have become a standard topic in combinatorial number theory. I will present joint work with Greg Martin giving constructions of such sets, and discuss numerous open problems.

What is the image of the heat operator? To put it the other way around, what is the domain of the backward heat operator? That is, for which initial conditions is it possible to solve the backward heat equation (say, for some fixed time t)? Clearly, the initial conditions must be extremely regular, since the forward heat equation is highly smoothing. In the case of Euclidean space $R^{n}$, there is a simple answer: the initial conditions must admit an analytic continuation to $C^n$ with a certain (t-dependent) growth in the imaginary directions. For compact symmetric spaces (e.g. a sphere), there is a very similar answer. For noncompact symmetric spaces (e.g. hyperbolic space), the situation is much more complicated and we are just beginning to understand what is going on. I will describe some old and some new results in this area, which can be expressed in terms of a ‘‘eneralized Segal--Bargmann transform’’

Fifteen years ago, Bramble, Pasciak and Xu developed a framework
for analyzing the multigrid method (The analysis of multigrid
algorithms with nonnested spaces or noninherited quadratic forms,
Mathematics of Computation, 1991.) It was, known as BPX Framework,
widely used in the analysis of mutligrid and domain decomposition
methods. The framework was extended several times, covering less
regularity or non-symmetric, and other cases. However an apparent
limit of the framework is that it could not incorporate the number
of smoothings in the V-cycle analysis. Therefore, the framework
is limited in nonnested multigrid methods (as one-smoothing multigrid
methods won\'t converge for almost all nonnested cases), and
it produces variable V-cycle, relaxed coarse-level correction,
or non-uniform convergence rate V-cycle methods, or other non-optimal
results in analysis thus far. Therefore, most non-nested finite
element problems were show to converge only for W-cycles based
BPX framework, or an even earlier frame work of Bank and Dupont.

This paper completes a long time effort in extending the BPX Framework
so that the number of smoothings is included in the V-cycle analysis.
We will apply the extended BPX Framework to the analysis of many V-cycle
nonnested multigrid methods. Some of them were previously proven
for two-level and W-cycle iterations only.

Structure preserving numerical integrators aim to preserve as many of the physical invariants of a dynamical system as possible, since this typically results in a more qualitatively accurate simulation. Computational geometric mechanics is concerned with a class of structured integrators based on discrete analogues of Lagrangian and Hamiltonian mechanics.

Discrete theories of exterior calculus and connections on principal bundles provide some of the mathematical foundations of computational geometric mechanics, and address the question of how to obtain canonical discretizations that preserve, at a discrete level, the important properties of the continuous system.

Some recent progress on the construction of a combinatorial formulation of discrete exterior calculus based on primal simplicial complexes, and circumcentric dual cell complexes will be presented. These techniques have been used to systematically recover discrete vector differential operators such as the Laplace--Beltrami operator.

For discrete connections, the discrete analogue of the Atiyah sequence of a principal bundle is considered, and a splitting of the discrete Atiyah sequence is related to discrete horizontal lifts and discrete connection
forms. Continuous connections can be obtained by taking the limit of discrete connections in a natural way.

Examples spanning the work on exterior calculus and connections include the discrete Levi-Civita connection for a semidiscrete Riemannian manifold, and the curvature of an abstract simplicial complex endowed with a metric on the vertices.

For a given finite separable field extension $L/K$ we would like to find a generator $x \in L$ whose equation is as simple as possible. For example, one would like to have as many vanishing coefficients as possible. More generally, the transcendence degree of the subfield $k$ of $K$ generated by the coefficients of the equation should be as small as possible. It was shown by Buhler and Reichstein that the minimal transcendence degree one can reach has an interpretation in terms of rational covariants of the symmetric group $S_n$ where $n := [L:K]$. This number is called the essential dimension of $L/K$ or of $S_n$ it can be defined for every finite group. We will describe the relation between covariants and equations for field extensions and will explain the main results in this context.

A classical result from 1861 due to Hermite says that every separable equation of degree 5 can be transformed into an equation of the form $x^5 + b x^3 + c x + d = 0$. Later, in 1867, this was generalized to equations of degree 6 by Joubert. We show that both results can be understood as an explicit analysis of certain covariants of the symmetric groups $S_5$ and $S_6$. In case of degree 5, the classical invariant theory of binary forms of degree 5 comes into play whereas in degree 6 the existence of an outer automorphism of $S_6$ plays an essential r\^ole. Although these consequences for equations of degree 5 and 6 have been cited and used many times in the literature, it seems unclear if the methods and ideas of Hermite and Joubert have really been understood.

The loop space of a complex manifold M, consisting of all maps from the circle $S^1$ to M with some fixed $C^k$ or Sobolev regularity, is an infinite dimensional complex manifold. We identify an infinite dimensional subgroup of the Picard group of holomorphic line bundles on the loop space of the Riemann sphere, and show that the space of holomorphic sections of any such line bundle is finite dimensional. We also compute the (0,1) Dolbeault group of the loop space of the Riemann sphere (which is infinite dimensional).

We define another invariant for a finite group $G$, the so-called {\it covariant dimension of $G$} which is closely related to the essential dimension of $G$ which we introduced in the previous talk. It is the minimal possible dimension of the image of a faithful (regular) covariant of $G$. It turns out that many results for the essential dimension of a finite group $G$ can be carried over to the covariant dimension of $G$ where they admit surprisingly simple proofs. In this way we have been able to reconstruct the main results of Buhler and Reichstein for the essential dimension and their applications to field extensions. But still there remain a number of interesting open questions which are quite elementary and easy to understand. Some of them will be discussed at the end of the talk.

Decompositions of graphs have a number of applications, for example, programming for multiple processor computers. In particular, this makes decompositions of high dimensional cubes into hamiltonian cycles of interest. These decompositions are most easily approached by looking at decompositions of more general cartesian products. For the undirected case, these decompositions are fairly well understood, but I want to present a new way of looking at these constructions. This new outlook allows one to solve the directed case as well.

Lynch's vanishing theorem for generalized Whittaker modules
applies to generic homomorphisms of the nilradical of a class of parabolic subalgebras which we have named nice. In this talk we describe the classification of nice parabolic subalgebras of simple Lie algebras. Our results give rise to a new characterisation of even nilpotents. We will also give a normal form of Richardson elements for nice parabolic subalgebras in the classical case.

In this talk we will discuss CY manifolds that are double covers of $CP^3$ ramified over 8 planes. We will show that the moduli space of those CY manifolds is a locally symmetric space. The proof of this fact is based on the proof that the Yukawa coupling does not admit quantum corrections. The meaning of this statement will be explain.

Have you ever seen someone juggle and wonder how he or she does
it? Or, are you able to juggle but have wondered how the process might be described mathematically? In this talk, I will introduce the concept of a juggling sequence and explain how juggling sequences can be used to describe simple juggling patterns. I will discuss some of the mathematical questions related to juggling sequences. Finally, I will
illustrate some simple juggling patterns by juggling them (when I'm not
picking up the balls off the floor, that is) and exhibit a pattern that
was unknown prior to the discovery of juggling sequences.

Refreshments will be provided.

Qian, Luscombe, and Gerstein (2001) introduced a model of gene
duplication that we may formulate as follows. Consider a multitype
Yule process starting with one individual in which there are no deaths
and each individual gives birth to a new individual at rate one.
When a new individual is born, it has the same type as its parent
with probability $1 - r$ and is a new type, different from all previously
observed types, with probability r. We refer to individuals with
the same type as families and provide an approximation to the joint
distribution of family sizes when the population size reaches $N$.
We also show that if $1 << S << N^{1-r}$, then the number of families
of size at least S is approximately $CNS^{-1/(1-r)}$, while if
$N^{1-r} << S$ the distribution decays more rapidly than any power.

Over the last 20 years, efficient algorithms have been developed to count the number of points of a given elliptic curve over a finite field. We discuss the inverse problem of constructing elliptic curves with a given number of points over a finite field. The difficulty of the problem depends on its exact wording. We present a solution to the problem that easily handles curves of the size occurring in cryptographic practice, and explain why it should be expected to do so.

Let $F$ be an open family of circles in the complex plane and let $P$ be the union of all the circles from $F$. Suppose that $f$ is a continuous function on $P$ which extends holomorphically from each circle $C$ belonging to $F$ (that is, its restriction to $C$ extends holomorphically through the disc bounded by $C$). Is $f$ holomorphic on $P$? We show that this question is naturally related to a question in $C^2$ and then, for certain families $F$, we show how some standard facts from several complex variables can be used to deal with this question.

The 'small world phenomenon' as observed by the psychologist Stanley Milgram in the 1960s is that most pairs of people are connected by a short chain of acquaintances. More generally we say a graph exhibits the small world phenomenon if the average distance between vertices is small due to a few random-like edges combined with a highly regular local structure. Randomly generated graphs with a power law degree distribution are often used to model large real-world networks. While these graphs have small average distance, they do not have the local structure associated with the small world phenomenon. We define a hybrid model which combines a global graph (a random power law graph) with a local graph (a graph with high local connectivity defined by network flow). We also present an efficient algorithm to extract a highly connected local graph from a given realistic network. We show that the hybrid model is robust in the following sense. Given a graph generated by the hybrid model, we can recover a good approximation to the original local graph in polynomial time.

In this talk we will compare the primes (2,3,5,7,11,...) with primes in a
graph which are classes of closed paths in the graph. In the process, we will look at the Riemann zeta function from number theory and the Ihara zeta function from graph theory and meditate on the Riemann hypothesis.

(Joint work with Denis Charles)

The nth modular polynomial parameterizes pairs of elliptic curves with
an isogeny of degree n between them. Modular polynomials provide the
defining equations for modular curves, and are useful in many different
aspects of computational number theory and cryptography. For example,
computations with modular polynomials have been used to speed elliptic
curve point-counting algorithms.

The standard method for computing modular polynomials consists of
computing the Fourier expansion of the modular j-function and solving a
linear system of equations to obtain the integral coefficients. The
computer algebra package MAGMA incorporates a database of modular
polynomials for n up to 59.

The idea of the talk is to compute the modular polynomial directly
modulo a prime p, without first computing the coefficients as integers.
Once the modular polynomial has been computed for enough small primes,
our approach can also be combined with the Chinese Remainder Theorem
(CRT) approach to obtain the modular polynomial with integral
coefficients or with coefficients modulo a much larger prime using
Explicit CRT. Our algorithm does not involve computing Fourier
coefficients of modular functions.

In many two-phase flows and moving interface problems,
the flux and the solution of the governing PDE often
have finite jumps across the interface. Such jumps often result from source distributions along the interface. One famous application is Peskin's immersed boundary model/method which has been widely used. Using a level set function to represent the interface, we have proposed a new method that can transfer an interface with discontinuity in the flux and/or in the solution to an interface problem with a smooth solution if the coefficient PDE is continuous. The new formulation is based on extensions of the jumps along the normal line of the interface. If the coefficient of the PDE is also discontinuous, then the transformation leads to a new interface problem with homogeneous, also called natural, jump conditions. This will greatly simplify the immersed interface method because no surface derivatives of the jump conditions is needed anymore. Theoretical and numerical analysis including implementation details and numerical example will also
be presented.

The talk will begin with a short survey of some of the main enumerative
results in the subject of restricted (or pattern-avoiding) permutations.

Next I will discuss some recent work concerning statistics on restricted
permutations, motivated in part by the surprising result of Zeilberger et
al. which states that the number of fixed points has the same distribution
in 321-avoiding permutations as in 132-avoiding permutations. I will give two combinatorial proofs of this result, and generalize it to other
statistics. Bijections to Dyck paths play an important role in the proofs.

Finally I will consider permutations avoiding several patterns
simultaneously, as well as generalized patterns (i.e., with the
requirement that some elements occur in adjacent positions). Using similar bijections we obtain generating functions enumerating these restricted permutations with respect to several statistics.

I will discuss links between a certain class of Pythagorean triples and
certain aspects of modern number theory, such as the theory of elliptic
curves, their L-functions, and the various still unsolved conjectures
predicting their behavior.

During the last decade several stochastic stratified processes were introduced in the probability literature. They include the fiber Brownian motion introduced by Bass and Burdzy as a process that switches between two-dimensional and one-dimensional evolution, a Markov process on a `whiskered sphere' described by Sowers, processes on trees like the Walsh process, the Evans process, spider martingales, and Brownian snakes. We study the rate of weak convergence to one such process. Using a Wasserstein-type metric as the distance that metrizes weak convergence, we show that the rate of this convergence can be controlled by the rates of convergence of other related processes.

For a related classical example of a Hamiltonian system on a cylinder, the corresponding estimates on the rate of convergence are obtained using two different methods. One of these methods reveals an explicit expression for the Wasserstein distance in the classical case.

If $f$ is a nonzero complex-valued function defined on a finite abelian group $A$ and $\hat f$ is its Fourier transform, then $|f|| \hat f| \ge |A|$, where $f$ and $\hat f$ are the supports of f and $\hat f$. In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group $A$ is replaced by a transitive right $G$-set, where $G$ is an arbitrary finite group. We obtain stronger inequalities when the $G$-set is primitive and we determine the primitive groups for which equality holds. We also explore connections between inequalities of this type and a result of Chebotarev on complex roots of unity, and we thereby obtain a new proof of Chebotarev’s theorem.

This talk will discuss the mathematical challenges of the control-oriented
analysis of Navier-Stokes systems in the following three areas:
\vskip.1in
1. FUNDAMENTAL PERFORMANCE & STABILIZATION LIMITATIONS,
\vskip.1in
2. GRADIENT-BASED OPTIMIZATION, and
\vskip.1in
3. MODEL-BASED LINEAR FEEDBACK CONTROL.

I will discuss links between a certain class of Pythagorean triples and
certain aspects of modern number theory, such as the theory of elliptic
curves, their L-functions, and the various still unsolved conjectures
predicting their behavior.

Refreshments will be provided.

We consider the following urn model proposed by Diaconis. There is an urn with $n$ balls, each ball has value 0 or 1. Pick two random balls and add a new ball with their sum (modulo 2) to the urn. (Thus the number of balls increases by one each time). What is the limiting distribution?

In general, let $G$ be a finite additive group. We begin with an $a$ set
$g_1, \dots, g_n$ of (not necessarily different) elements of $G$. (In fact, typically $n$ is much larger than $|G|$). At time $i=0,1,2,\dots,$ we choose two random elements $a$ and $b$ from the set $g_1, \dots, g_{n+i}$ and add the sum $g_{n+i+1} = a+b$ to the set.

Siegmund and Yakir studied the distribution of the set $g_1,g_2\dots$ and showed that it converges to the uniform distribution. This result leaves open the question: How quickly does this convergence occur? In this talk we address this issue.

In this talk, I will report on a recent joint work with Joseph A. Shalika and Yuri Tschinkel on the distribution of rational points of bounded height on certain group compactifications.

In mixed linear models with nonnormal data, the Gaussian Fisher-information matrix is called quasi information matrix (QUIM). The QUIM plays an important role in evaluating the asymptotic covariance matrix of the estimators of the model parameters, including the variance components. Traditionally, there are two ways to estimate the information matrix: the estimated information matrix and the observed one. Because the analytic form of the QUIM involves parameters other than the variance components, for example, the third and fourth
moments of the random effects, the estimated QUIM is not available. On the other hand, because of the dependence and nonnormality of the data, the observed QUIM is inconsistent. We propose an estimator of the QUIM consisting partially of an observed form and partially of an estimated one. We show that this estimator is consistent and computationally very easy to operate. The method is used to derive large sample tests of statistical hypotheses that involve the variance
components in a non-Gaussian mixed linear model. Finite sample
performance of the test is studied by simulations and compared with
the delete-group jackknife method that applies to a special case of non-Gaussian mixed linear models.

I extend the definition of concurrence into a family of entanglement
monotones, which I call concurrence monotones. I will discuss their
properties and advantages as computational manageable measures of entanglement. I will then show that the concurrence monotones provide bounds on quantum information tasks. As an example, I will discuss their applications to remote entanglement distributions (RED) such as entanglement swapping and Remote Preparation of Bipartite Entangled States (RPBES). I will present a powerful theorem which states what kind of (possibly mixed) bipartite states or distributions of bipartite states can not be remotely prepared. The theorem establishes an upper bound on the amount of G-concurrence (one member in the concurrence family) that can be created between two single-qudit nodes of quantum networks by means of tripartite RED. For pure bipartite states the bound on the G-concurrence can always be saturated by RPBES.

Let $K$ be a field and $GL_n(K)$ the group of invertible n by n matrices over $K$. About a century ago, I. Schur described the irreducible polynomial representations of $GL_n(K)$ for the field $K$ of complex numbers. The same problem over a field $K$ of prime characteristic is still widely open. In this context, G. James proved in 1981 a so-called column removal formula. It relates polynomial representations in different homogeneous degrees. Combinatorially, it is based on removing the first column of an associated Young diagram. We generalize this operation of column removal to considering ``complements" of Young diagrams. Algebraically this establishes equivalences between categories of polynomial representations in different degrees. As by-product it establishes and explains certain repeating patterns in decomposition numbers of general linear and symmetric groups. This work is related to work by Beilinson-Lusztig and MacPherson. It is joint work with Fang and Koenig.

The exponential map plays an important role in Lie theory; it allows one to linearize a Lie group in the neighborhood of the identity element, thus reducing many questions about Lie groups to (more tractable) questions about Lie algebras. Unfortunately (at least for an algebraic geometer),
the exponential map is not algebraic; it is given by an infinite series and thus cannot be defined in the setting of algebraic groups.
\vskip.2in
\noindent The next best thing is to linearize the conjugation action of $G$ on itself in a Zariski neighborhood of the identity element. For special orthogonal groups $SO_n$ this is done by the classical Cayley map, which has been used in place of the exponential map in some applications. In the 1980s D. Luna asked which other simple algebraic groups admit a ``Cayley map". In this talk, I will discuss the background of this problem and a recent solution, obtained jointly with N. Lemire and V. L. Popov.

Galois theory tells us that that some polynomials \vskip .1in

$f(x) = x^n + a_1 x^{n-1} + ... + a_{n-1} x + a_n$ \vskip .1in

\noindent of degree $n > 4$ cannot be solved in radicals. Equivalently, some $S_n$-covers cannot be split by a solvable base extension. J. Tits asked whether an analogous assertion remains true if $S_n$ is replaced by a connected group $G$. In this talk I will discuss the background of this problem and recent results (obtained jointly with V. Chernousov and $P$. Gille) which indicate that solvability in radical may, indeed be possible in this setting. In particular, I will explain a connection we found between Tits' question and a variant Hilbert's 13th problem.

Order parameters such as the surface gradient in thin film growth can be unbounded as the size of an underlying system increases. Such unbounded order parameters can be modeled by a variational problem in which the effective free energy consists of a negative logarithmic function of the order parameter and a usual regularizing term. In this talk, we will first describe such a model for unbounded order parameters and compare it with a usual Ginzberg-Landau type model for domain walls. We will also give heuristic arguments to show why low energy configurations should have large value of the order parameter. Rigorous results will then be presented, and proved using the direct method in the calculus of variations. We will conclude the talk by a discussion on the related dynamics.

If $G_1 x G_2$ is a subgroup of a finite group $H$, one can restrict a representation of $H$ to $G_1 x G_2$ and examine how it decomposes. One obtains in this way a function from irreducible representations of $G_1$ to (possibly reducible, possibly zero) representations of $G_2$. In certain cases, this function turns out to be very nice. We examine a particular case of this, when the groups involved are finite groups of Lie type. A large portion of the talk will be on recalling the classification of irreducible representations of such groups due to Deligne-Lusztig and Lusztig.

Worldvolumes of $D$-branes on Calabi-Yau threefolds give rise to
supersymmetric gauge theories. If the $D$-brane is marginally stable
these theories are quiver gauge theories. Using the derived category
of coherent sheaves to analyze such $D$-branes, the superpotential of
these gauge theories may be determined systematically.

In a recent preprint ``Combinatorial Hopf algebras and generalized
Dehn-Sommerville relations"

\noindent(math.CO/0310016), Aguiar, Bergeron, and Sottile set up a framework for studying combinatorial invariants encoded by quasisymmetric functions. They show that every character (i.e., multiplicative linear functional) on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character, both of which can in theory be computed from the even and odd parts of the ``universal character" on the Hopf algebra of quasisymmetric functions. \vskip .1in

\noindent In my talk I will introduce some of these ideas and then go on to give explicit formulas for the even and odd parts of the universal
character. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers:
$$C(m,n)=(2m)!(2n)!/(m!(m+n)!n!).$$ I will explain how properties of characters and of quasisymmetric functions can be used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients. \vskip .1in

\noindent This work is joint with Marcelo Aguiar.

I will talk about factorization of integers into primes, public key
cryptography, and more. \vskip .1in

\noindent Refreshments will be served!

We define a new spectrum for compact length spaces and Riemannian manifolds called the ``covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers $\delta>0$ which identify the distinct $\delta$ covers of the space. We investigate the relationship between this covering spectrum, the length spectrum, the marked length spectrum and the Laplace spectrum. We analyze the behavior of the covering spectrum under Gromov-Hausdorff convergence. This is a joint work with C. Sormani.

We consider the problem of best approximating a given function in
$L^2$ of a subset of the unit circle by the trace of an $H^2$ function whose norm on the complementary set is bounded by a prescribed constant. It is known that the solution can be obtained by solving a spectral equation for a certain Toeplitz operator. We show how diagonalization of such an operator ‡ la Rosenblum-Rovnyak allows to estimate the rate of convergence. We also consider the same problem in $L^\infty$ rather than $L^2$ norm, and present its solution that involves some unbounded Toeplitz operator.

I'll review how the stability of simulations in numerical relativity is related to having a well-posed continuum problem, and why well-posedness is not a property of the Einstein equations as such, but of the way in which they are formulated as an initial-boundary value (time evolution) problem. After reviewing the related concepts of well-posedness, strong hyperbolicity and symmetric hyperbolicity, I discuss applications to the Einstein equations, and in particular recent work. I conclude with an outlook on what strategies seem most promising to achieve well-posedness.

Building on the work of Gromov and Schoen, Kollar and Simpson we
will show that the universal covering of a smooth projective varieties
with a linear fundamental groups is holomorphically convex. At the end we will connect this result with some conjectures of Zelmanov's on
Burnside type of groups.

Projectively simple rings are those positively graded rings which are as close to being simple as possible. We show that under suitable hypotheses, such rings can be described using constructions from algebraic geometry. This naturally leads to an interesting geometric question: which varieties have automorphisms which fix no proper subvarieties? This is joint work with James Zhang and Zinovy Reichstein.

We show that the Grothendieck semiring of $O(N)$ is a quotient of the one of $O(\infty)$. Using this, we can express restriction multiplicities from $GL(N)$ to $O(N)$ via an alternating sum formula over the elements of a reflection group in terms of classical multiplicites due to Littlewood.

Joint Seminar with UCI

Joint Seminar with UCI

Suppose one has a polynomial equation in two variables, $f(x,y)=0$. This defines a curve, but if we work over the complex numbers, it in fact
defines a surface in $C^2$, which is difficult to visualize. I will discuss
ways of visualizing these surfaces by introducing the concepts of amoebas, and their relationship to a new notion known as tropical geometry. \vskip .2in

\noindent Refreshments will be served!

The existence of recurrent stochastic processes with stationary independent increments in an arbitrary locally compact group has remained a challenging research topic over a period of four decades. On the other hand, there exists an elaborate theory of transient processes in groups leading to interesting problems in potential theory. In the present expository talk, which is designed to be more or less self contained, the speaker will revisit the well-known Kesten-Spitzer transience criterion for processes in {\it abelian} locally compact groups, and will describe a purely measure theoretic approach to it.

The class numbers of the real cyclotomic fields ${\bf Q}(\cos(2\pi/{p^n}))$ are very difficult to compute. Indeed, they are not known for a single prime $p>67$. We analyze these class numbers using the Cohen-Lenstra heuristics on class groups and are led to make the following conjecture: For all but finitely many primes $p$, the class number of ${\bf Q}(\cos(2\pi/ {p^n}))$ is equal to the class number of ${\bf Q}(\cos(2\pi/ {p}))$ for all positive integers $n$. It is possible that there are no exceptional primes $p$ at all. Work in progress to test this conjecture empirically will also be discussed. This is joint work with Joe
Buhler and Carl Pomerance.

We will tell the story of the solution of several famous problems of Painlev´e, Ahlfors and Vitushkin. \vskip .1in

\noindent Essentially, the theory of nonhomogeneous Calder´on-Zygmund (CZ) operators is the topic of the lecture. The main cornerstone of the theory of CZ operators turns out not to be a cornerstone at all. Namely, one can completely get rid of
homogeneity of the underlying measure. The striking application of this theory is the solution of the series of problems of Painlev´e, Ahlfors and Vitushkin on the borderline of Harmonic Analysis and Geometric Measure Theory. \vskip .1in

\noindent We will show how the ideas from nonhomogeneous CZ theory interplay with Tolsa’s ideas of capacity theory with Calder´on-Zygmund kernels to give Tolsa’s solution of the famous Vitushkin conjecture of semiadditivity of analytic capacity. We also show what changes should be maid if we want to grow the dimension and to prove the semiadditivity of Lipschitz harmonic capacity in $Rn, n > 2$, where the wonderful tool of Menger-Melnikov’s curvature extensively used in $2D$ is “cruelly missing”.

A combinatorial game is a two-player game with perfect information and no chance elements. The players take turns making moves under clearly defined rules, and the last player able to make a move wins the game. NIM is one of the most well-known combinatorial games, and many people know that optimal play in Nim involves an addition operation which is just an XOR operation in disguise. Nim has a richer structure than this, however, as there is also a multiplication operation, and some unexpected and delightful things pop up such as Fermat powers (numbers of the form $2^{2^n}$). We'll explore nim fields -- finite fields of objects that aren't numbers, but NIMBERS, and really bizarre things happen -- even if I could convince you that 4 times 4 is 6, could you ever believe that the cube root of the nimber 2 is infinity? \vskip .1in

\noindent Refreshments will be served!

Higher composition laws in number theory were discovered by M.
Bhargava several years ago. They may be viewed as a generalization of
Gauss's law of composition of binary quadratic forms. M. Bhargava also
discovered a mysterious connection between higher composition laws and exceptional Lie groups. \vskip .1in

\noindent In our talk we will describe an unexpected relation between higher composition laws and the Freudenthal construction in the theory of Jordan algebras. We will show how this construction can be used to
shed additional light on existing composition laws, as well as provide
new examples of spaces with similar properties.

Young tableaux, beloved of combinatorialists, tolerated by representation theorists and geometers, seem at first glance to be an
unruly combinatorial set. I'll define a simplicial complex in which
they index the facets, and slightly more general objects (Buch's
``set-valued tableaux'') label the other interior faces. The theorem that says we're on a right track: This simplicial complex is homeomorphic to a ball. I'll explain why this is surprising, useful, and shows why Buch didn't discover the exterior faces too. Finally, I'll explain how algebraic geometry forced these definitions on us (or, ``How I made my peace with Young tableaux''). This work is joint with Ezra Miller and Alex Yong.

We first give an introduction to biostatistics, starting with what statistics is all about. We will elaborate on three areas of biostatistics: clinical trials, survival analysis and computational biology (bioinformatics). Time permitting, we will also briefly describe the biostatistics graduate programs around the country, and the excellent career opportunities open to those with a biostatistics degree.

In classical potential theory on $R^n (n >= 3)$, the condenser theorem states the following. Let $U$ and $V$ be open subsets of $R^n$ with disjoint closures, where $U$ is relatively compact. There exists a Newtonian potential $p$ of a signed measure $mu = mu^+ - mu^-$ such that: \vskip .1in

\noindent 1. $0 <= p <= 1$ \vskip .1 in

\noindent 2. $p = 1 on U, p = 0 on V$ \vskip .1in

\noindent 3. $mu^+$ is supported in the closure of $U$, $mu^-$ is supported in the closure of $V$ \vskip .1in

\noindent The fundamental proof of this theorem was given by A. Beurling and J. Deny in the framework of Dirichlet spaces. Later K. L. Chung and R. K. Getoor studied the condenser problem in the probabilistic context of Markov processes. They showed that the condenser potential at a point x is simply the probability that Brownian motion starting at $x$ hits $U$ before it hits $V$. In this talk we discuss the condenser problem in the potential-theoretic framework of balayage spaces. We introduce the notion of a fine condenser potential, for which existence and uniqueness are proved for arbitrary superharmonic functions on sets $U, V$ which are not necessarily open. The probabilistic interpretation carries over to this setting by replacing the hitting times by the penetration times. These results are joint work with Jürgen Bliedtner.

Motivated by studying the amoeba of a system of polynomial equations, we associate to each matroid $M$ a polyhedral complex $B(M)$, called the ``Bergman complex". I will describe the topology and combinatorics of this complex. Somewhat surprisingly, the space of phylogenetic trees is (essentially) a Bergman complex, and we obtain some new results about it as a consequence. If $M$ is oriented, the Bergman complex $B(M)$ has a ``positive part" $B+(M)$, which I will also describe. \vskip .1in

\noindent If time allows, I will show that for a Coxeter arrangement $A$, $B(A)$ is closely related to de Concini and Procesi's compactification of the complement of $A$, and $B+(A)$ is dual to a known Coxeter generalization of the associahedron. \vskip .1in

\noindent My talk will assume no previous knowledge of matroids and arrangements. \vskip .1in

\noindent Parts of this work are joint with Carly Klivans, Lauren Williams, and Vic Reiner.

The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation. \vskip .1in

\noindent Curiously, while the geometric structure of mechanical systems plays a critical role in the construction of geometric control algorithms, these algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. \vskip .1in

\noindent Geometric integration is the field of numerical analysis that focuses on developing geometric structure-preserving integrators, and computational geometric mechanics focuses on developing geometric integrators for dynamical systems arising from mechanics. \vskip .1in

\noindent This talk will introduce some of the discrete differential geometric tools necessary to implement control algorithms in a manner that respects and preserves the geometry of the problem. These tools include discrete analogues of Lagrangian mechanics, the exterior calculus of differential forms, and connections on principal bundles. \vskip .1in

\noindent This is joint work with Mathieu Desbrun (CS, Caltech), Anil Hirani (JPL), Jerrold Marsden (CDS, Caltech), Alan Weinstein (Math, Berkeley).

We say that a sequence of dense graphs $G_n$ is convergent if for every fixed graph $F$ the density of copies of $F in G_n$ tends to a limit $f(F)$. Many theorems and conjectures in extremal graph theory can be formulated as inequalities for the possible values of the function $f$. We prove that every such inequality follows from the positive definiteness of the so-called connection matrices. Moreover we construct a natural limit
object for the sequence $G_n$ namely a symmetric measurable function on the unit square. Along the line we introduce a rather general model of random graphs which seems to be interesting on its own right. \vskip .1in

\noindent Joint work with L. Lovasz (Microsoft Research).

A standard approach to reduced-order modeling of higher-order
linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for reduced-order modeling of first-order systems. While this approach results in reduced-order models that are optimal in a Pade sense, in general, these models do not preserve the form of the original higher-order system. \vskip .1in

\noindent In this talk, we present a new approach to reduced-order modeling of higher-order systems based on projections onto suitably
partitioned Krylov basis matrices that are obtained by applying Krylov-subspace techniques to an equivalent first-order system. We show that the resulting reduced-order models preserve the form of the original higher-order system. Moreover, possible additional properties such as passivity or reciprocity are also preserved. While the resulting reduced-order models are no longer optimal in the Pade sense, we show that they still satisfy a Pade-type approximation property. We also discuss some implementation details and present some numerical examples.