Meeting:
Fridays at 11:30am, DM 163
Florida International University
For more information contact:
Phillipe Rukimbira,
Thomas
Leness, Tedi Draghici, or Gueo
Grantcharov
Schedule for 2004:
Friday, April 8th, 2004, 11am, DM 409
Professor Charles Boyer
University of New Mexico
Einstein Metrics on Spheres
Abstract: We describe a technique that combines
the use of Sasakian
geometry and algebraic geometry to prove the existence
of an abundance of
Einstein metrics on odd dimensional spheres, including
exotic spheres.
Monday, March 8th, 2004, 11am, DM 409
Professor Dusan Repovs
University of Ljubljana
Fifty years of the recognition problem for topological
manifolds.
Abstract: We shall present an historical survey
of the
geometric topology of generalized manifolds, i.e.
ENR
homology manifolds, from their early beginnings
in 1930's
to the present day, concentrating on those geometric
properties
of these spaces which are particular for dimensions
3 and 4, in
comparison with generalized ($n>4$)-manifolds.
In the second part of the talk, we shall present
the current
state of the main two problems concerning this class
of spaces
-the resolution problem (the work of
Bestvina-Daverman-Venema-Walsh,
Bryant-Lacher, Brin-McMillan, Lacher-Repovs, Thickstun,
and others) and the general position
problem (the work
of Bing, Brahm, Lambert-Sher, Daverman-Eaton, Lacher-Repovs,
Daverman-Thickstun, Daverman-Repovs, Brahm, and
others). We shall
list open problems and related conjectures.
Schedule for 2003:
Friday, January 31st, 2003, 4pm-5pm, DM 163
Professor Tedi Draghici
Florida International University
(Non)-existence of Einstein compatible metrics on
certain
symplectic manifolds
Schedule for 2002:
Friday, November 15th, 11:30-12:30 noon, DM 163
Professor Stefan Ivanov
University of Sofia
Strings and geometry of connection with torsion 3-form and special holonomy
Abstract: Riemannian manifolds admitting parallel
spinors with respect to a
metric connection with totally skew-symmetric torsion
recently
become a subject of interest in theoretical and
mathematical
physics. One of the main reasons is that the number
of preserving
supersymmetries in string theory depends essentially
on the number
of parallel spinors. The existence of parallel spinors
restricts
the holonomy group, since the spinor holonomy representation
has
to have a fixed point, and therefore reduces the
structure group
of the frame bundle. We consider the cases when
the holonomy group
is contained in $G=\{U(n), U(n)\times Id, G_2, Spin(7)\}$.
We find
necessary and sufficient conditions of the existence
of a
$G$-connection with torsion 3-form and prove that
it is unique and
study the underlying geometry. In the case of groups
$G_2$ and
$Spin(7)$ the existence of a $G_2$-connection and
$Spin(7)$-connection is sufficient for having parallel
spinors. In
the case $G={U(n),U(n)\times Id}$ additional conditions
on the
curvature should be imposed. If $G=SU(n)$ we obtain
a
generalization of Calabi-Yau manifolds. We show
the existence of
$SU(n)$-connection with torsion 3-form on some compact
complex
(non-Kaehler) manifold with zero first Chern class
and conjecture
that this phenomena holds in general.
Friday, November 8th, 11:30-12:30 noon, DM 163
Professor Thomas Leness
Florida International University
The homotopy Seiberg-Witten invariants of Bauer-Furuta, II
Abstract: A continuation of the previous talk.
Friday, October 18th, 11:30-12:30 noon, DM 163
Professor Thomas Leness
Florida International University
The homotopy Seiberg-Witten invariants of Bauer-Furuta
Abstract: .Bauer and Furuta have introduced
a refinement of the Seiberg-Witten invariant,
using homotopy data instead of homological
data. While the Seiberg-Witten invariants
vanish on connected sums, these invariants do not
and are able to distinguish between
connected sum of (up to three) exotic K3 surfaces.
This is the first talk in a series of lectures
on these invariants; we will begin by reviewing
the definition of the standard Seiberg-Witten
invariants and some necessary homotopy theory.
Friday, October 11th, 11:30-12:30 noon, DM 163
Dr. Heberto del Rio
Florida International University
The Yamabe problem for almost Hermitian manifolds, Part II
Abstract: Click here.
Friday, October 4th, 11:30-12:30 noon, DM 163
Dr. Heberto del Rio
Florida International University
The Yamabe problem for almost Hermitian manifolds
Abstract: Click here.
Friday, September 27th, 11:30-12:30 noon, DM 163
Professor Gueo Grantcharov
Florida International University
Hypercomplex geometry and HKT structures, Part III
Abstract: A continuation of last week's talk.
Friday, September 20th, 11:30-12:30 noon, DM 163
Professor Gueo Grantcharov
Florida International University
Hypercomplex geometry and HKT structures, Part Deux
Abstract: A continuation of last week's talk.
Friday, September 13th, 11:30-12:30 noon, Room TBA
Professor Gueo Grantcharov
Florida International University
Hypercomplex geometry and HKT structures
Abstract: HKT geometry is the geometry of
a hyper-Hermitian connection
whose torsion is a 3-form. In the talk will be given
the basic general
facts about HKT structures as well as methods for
construction of examples.
Schedule for 2001:
Friday, Dec. 7, 11:30 - 12:30 noon, Room DM 144
NOTE: special time and
place.
Johann Davidov
Institute of Mathematics and Informatics, Bulgarian
Academy of Sciences
Geometry of Hermitian surfaces
Abstract: A Hermitian surface is a two-dimensional
complex manifold endowed
with a Riemannian metric compatible with the complex
structure.
The curvature operator of such a surface has a specific
decomposition corresponding to the irreducible orthogonal
decomposition of the space of four-tensors having
the same
symmetries as the Riemannian curvature tensor under
the action of
the group $U(2)$. The vanishing of some of the components
of the
curvature tensor singles out interesting classes
of Hermitian
surfaces which will be discussed in this talk. A
special attention
will be paid to the classification problem for compact
self-dual
Hermitian surfaces.
Wednesday, Dec. 5, 11:00-12noon, Room DM 144
NOTE: special time and place.
Oleg Mushkarov
Institute of Mathematics and Informatics, Bulgarian
Academy of Sciences
Harmonic almost-complex structures on twistor spaces
Abstract: If an even-dimensional Riemannian
manifold $(N,h)$ admits an
almost-Hermitian structure, it has many and it is
natural to seek
for "reasonable" criteria that distinguish some
of these
structures. A natural way to obtain such criteria
is to consider
the almost-Hermitian structures on $(N,h)$ as sections
of the
twistor bundle ${\cal T}$.
Motivated by the harmonic maps theory, C.Wood has
suggested to
consider as "optimal" the almost-Hermitian structures
$J:(N,h)\to
({\cal T},\tilde h)$ which are critical points of
the energy
functional under variations through sections of
${\cal T}$ where
$\tilde h$ is the natural Riemannian metric on ${\cal
T}$ induced
by $h$ and the standard metric of the fibre. These
critical points
are not harmonic maps in general but, by analogy,
they are
referred to as "harmonic almost-complex structures".
The main
result of this talk states that the Atiyah-Hitchin-Singer
and
Eells-Salamon almost-complex structures on the negative
twistor
space of an oriented Riemannian four-manifold $N$
(i.e. the
component of ${\cal T}$ whose sections are the almost-Hermitian
structures compatible with the opposite orientation
of $N$) are
harmonic in the sense of C.Wood if and only if the
base manifold
is, respectively, self-dual or self-dual and of
constant scalar
curvature. The stability of these almost-complex
structures will
be also discussed.
Friday, November 9th:
Professor Phillipe Rukimbira
Florida International University
Energy of unit vector fields with isolated
singularities (continued)
Abstract: This is a continuation of the previous talk.
Friday, October 5th:
Professor Phillipe Rukimbira
Florida International University
Energy of unit vector fields with isolated
singularities.
Abstract: On the standard 3-sphere, Hopf vector
fields minimize the
energy functional defined on the space of unit vector
fields. This is
not the case anymore in higher dimensions setting
where there is strong
indication that the minimum energy could be
realized by singular vector
fields.
In this short talk, minimality conditions will be
discussed for unit
vector fields with isolated singularities.
Friday, September 14th:
Professor Tedi Draghici
Florida International University
Local models and integrability of certain almost
Kahler 4-manifolds
Schedule For 2000:
Friday, November 10th:
Professor Tedi Draghici
Florida International University
Kahler manifolds with constant eigenvalues
of the Ricci tensor
Abstract: We study Kahler manifolds
whose Ricci tensor has two, distinct, constant
eigenvalues. Immediate examples are products
of Kahler-Einstein manifolds, but the
obvious question is: "Are there irreducible examples?"
We provide some answers,
showing also how the question relates to the celebrated
(still open) Goldberg conjecture:
the almost complex structure of a compact almost
Kahler Einstein manifold must be integrable
Friday, November 3rd:
Professor Graham Taylor
Florida International University
"SU(3) Donaldson Polynomial Invariants"
Abstract: The new ingredients needed to extend
Donaldson's polynomial
invariants for smooth four-manifolds to bundles
with structure group
SU(3) are outlined. In particular, the perturbations
required to
understand the moduli space of anti-self-dual connections
in a
neighborhood of those solutions which reduce to
connections on an SU(2)
bundle are described.
Friday, October 20th:
Professor Thomas Leness
Florida International University
"PU(2) monopoles and degeneracy loci"
Abstract: A continuation of last week's
talk.
Friday, October 13th:
Professor Thomas Leness
Florida International University
"PU(2) monopoles and degeneracy loci"
Abstract: We describe how a third family
of gauge theoretic invariants,
the spin polynomials of Pidstrigach and Tyurin,
can be related to the
Seiberg-Witten invariants by the PU(2) monopoles
program. In addition,
we describe a program for relating the spin polynomials
directly to
the Donaldson invariants.
Friday, October 6th:
Professor Thomas Leness
Florida International University
"PU(2) monopoles and degeneracy loci"
Abstract: Cancelled due to flooding.
Friday, September 29th:
Professor Philippe Rukimbira
Florida International University
"Critical unit vector fields: a brief survey"
Abstract: We present a brief
survey of critical vector fields for the volume and
energy functionals. Our survey includes some
new examples from flat contact
metric geometry.
Friday, September 22nd:
Professor Tedi Draghici
Florida International University
"Donaldson's moment map approach, III"
Abstract: Part III of the previous talk.
Friday, September 15th:
Professor Tedi Draghici
Florida International University
"Donaldson's moment map approach, II"
Abstract: Part II of the previous talk.
Friday, September 8th:
Professor Tedi Draghici
Florida International University
"Donaldson's moment map approach"
Abstract: In a recent paper
(Asian J. Math. Vol.3, No.1, 1-16, March 1999),
Donaldson pointed out that various infinite dimensional
problems can be recast into
a framework involving the moment map of the action
of a group on a symplectic
manifold. The framework is inspired from finite
dimensions, where it is known to be
valid; in infinite dimensions it is widely conjectural
at this point. Our goal is to look on
how it works in finite dimensions, and then present
how Donaldson fits this set up to
the problem of finding K\"ahler-Einstein and extremal
metrics on compact K\"ahler
manifolds.