| 31 March (at 4:15pm) |
Speaker: Jeff Viaclovsky (Wisconsin)
Title: Yamabe invariants and limits of self-dual hyperbolic monopole metrics (Note: Special Time)
Abstract:
Consider the self-dual conformal classes on n # CP^2
discovered by LeBrun. These depend upon a choice of
n points in hyperbolic 3-space, called monopole
points. I will discuss the limiting behavior of
various constant scalar curvature metrics in these
conformal classes as the points approach each other,
or as the points tend to the boundary of hyperbolic
space. There is a close connection to the orbifold
Yamabe problem (which I will show is not always
solvable, in contrast with the case for compact
manifolds).
|
| 7 April |
Speaker: Davi Maximo (Texas)
Title: Curvature conditions preserved (or not) under Ricci Flow
Abstract:
Ricci flow is a flow of Riemannian metrics designed to improve a
given initial metric. In particular, if the initial metric satisfies some
curvature condition, one wishes that its evolution will also satisfy the
same, if not a better, condition. That indeed is always the case with
many conditions (e.g. non-negative scalar curvature, non-negative
isotropic curvature), but there are also conditions that might be lost along
the flow (notably non-negative Ricci curvature). In this talk we will
survey some of the conditions known to be preserved by Ricci flow and also
construct a closed 4-dimensional manifold where the non-negative Ricci
condition is not preserved.
|
| 14 April |
Speaker: Brendan Guilfoyle (Institute of Technology, Tralee, Ireland)
Title: From Codazzi-Mainardi to Cauchy-Riemann
Abstract:
in this talk we discuss a co-dimension 2
initial boundary value problem in which one seeks to
attach holomorphic discs to Lagrangian surfaces in a
neutral Kaehler 4-manifold. In the case where the
4-manifold is the space of oriented affine lines of
Euclidean 3-space, we show how mean curvature flow in
the Kaehler manifold can be utilized to solve this
problem with sufficient flexibility to imply a bound on
a Keller-Maslov index along the boundary. This bound
implies an index bound on isolated umbilic points on
convex surfaces in Euclidean 3-space, thus establishing
the Caratheodory conjecture on the minimum number of
umbilic points on a closed convex surface.
|
| 21 April |
Speaker: No Seminar
Title:
Abstract:
|
| 28 April |
Speaker: Blake Temple (Davis)
Title: Self-similar Waves that Induce an Anomalous Acceleration into the Standard Model of Cosmology
Abstract:
In 1927, the American astronomer Edwin Hubble
showed that the Universe is expanding: distant
galaxies are receding at a rate proportional to
their distance. This confirmed the so-called
Standard Model of Cosmology , that the universe, on
the largest scale, is evolving according to a
Friedmann-Robertson-Walker spacetime. The starting
assumption in this model is the Copernican
Principle--that on the largest scale, we are not
in a special place in the universe--that the
universe is homogeneous and isotropic about every
point like the FRW spacetime. In 1998, more
accurate measurements of the recessional velocity
of distant galaxies based on Type 1a supernova
data, made the astounding discovery that the
Universe is actually accelerating relative to the
Standard Model. This is referred to as the
Anomalous Acceleration of the galaxies, and its
explanation is one of the great problems of
physics. The only way to preserve the FRW
framework and the Copernican Principle is to modify
the Einstein equations by adding an artificial
correction term called the Cosmological Constant.
Dark Energy, the physical interpretation of
the Cosmological Constant, is then an unknown
source of anti-gravitation that, for the model to
be correct, must account for some 70 percent of the
energy density of the universe. This is stated as
a fact on the NASA webpage. In this talk I discuss
a new family of self-similar expanding wave
solutions of the Einstein equations which author
introduced with Joel Smoller in the recent August
issue of PNAS. The family includes the Standard
Model of Cosmology during the radiation phase of
the expansion, but in addition there is an
adjustable free parameter which, by adjustment, can
speed up or slow down the expansion rate relative
to the Standard Model. These self-similar waves
would perturb a uniform background like waves
emanating from a rock thrown into a still pond. Our
purpose, then, is to explore the possibility that
they could account for the Anomalous Acceleration
of the galaxies within classical General
Relativity, without Dark Energy or the
Cosmological Constant.
|
| 5 May |
Speaker: Fernando Marques (IMPA)
Title: Deformations of the hemisphere that increase scalar curvature
Abstract:
Let $(M^n,g)$ be a compact Riemannian manifold of scalar curvature at
least $n(n-1)$
and totally geodesic boundary. A conjecture of Min-Oo asserts that if
the boundary is
isometric to a standard sphere of radius one, then $M$ is isometric to
the hemisphere.
In this talk we will describe the construction of counterexamples to
Min-Oo's conjecture
for all $n\geq 3$.
This is joint work with Simon Brendle and Andre Neves.
|
| Thursday 13 May (Note Special Date) |
Speaker: Xiuxiong Chen (Wisconsin)
Title: The space of Kahler metrics
Abstract:
In this talk, we will discuss geometric structure
in the infinite dimensional space of Kahler
potentials. In particular, we will discuss recent
progress in Kahler geometry (existence and
uniqueness of extremal Kahler metrics where Kahler
Einstein is a special case). Moreover, we will
discuss some problems in Kahler geometry which
might be useful to attack the existence problem of
extremal Kahler metrics via deformation methods.
|
| 19 May (3:15PM) in 380F |
Speaker: Dan Knopf (Texas)
Title: Ricci flow through singularities (Note: New time and location)
Abstract:
We construct smooth forward Ricci flow evolutions of
singular initial metrics resulting from rotationally
symmetric neckpinches, without performing an
intervening surgery. In the restrictive context of
rotational symmetry, this construction gives
evidence in favor of Perelman's hope for a
"canonically defined Ricci flow through
singularities". We also provide the asymptotic
profile of these solutions as they emerge from the
singularity. (This is joint work with Sigurd
Angenent and Cristina Caputo.)
|
| 19 May (4:15PM) |
Speaker: Henry Wente (Toledo)
Title: Exotic Capillary Tubes (Note: New time)
Abstract:
In contrast to the standard capillary tube, an
exotic capillary tube is a rotationally symmetric
tube of variable cross-section which if positioned
correctly in a vessel of fluid possesses a continuum
of equilibrium configurations. The controlling
variables are the capillary constant k =
(rho)g/(sigma) and the contact angle (gamma).
Lowering the tube slightly from its natural position
causes the tube to completely fill up while raising
the tube slightly forces the tube to drain out.
Other surprising consequences follow.
|
| 26 May |
Speaker: Joel Fish (Stanford)
Title: Generalizing and refining Gromov's compactness theorem for J-curves
Abstract:
In 1985, Gromov defined a J-curve to be a (pseudo) holomorphic map from
a closed Riemann surface to an (almost) complex manifold, and he proved
a notion of compactness for sequences of curves with uniformly bounded
energy. This talk will focus on a generalization of Gromov's result
which holds locally in the target manifold, which allows the sequence of
J-curves to have unbounded topology, and which regards J-curves as close
relatives of minimal surfaces.
|
| 2 June |
Speaker: Ben Weincove (UCSD)
Title: Contracting exceptional divisors by the Kahler-Ricci flow
Abstract:
We give a criterion under which a solution of the
Kahler-Ricci flow contracts exceptional divisors on
a compact manifold and can be uniquely continued on
a new manifold. This is a joint work with Jian Song.
|