TOPICAL WORKSHOP ON GEOMETRY AND PHYSICS
ABSTRACTS:
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on the name of the corresponding speaker (if it is highlighted) as
listed in the programme.
| TOPICAL WORKSHOP ON GEOMETRY AND PHYSICS | |||||
|---|---|---|---|---|---|
| MONDAY | TUESDAY | WEDNESDAY | THURSDAY | FRIDAY | |
| 9.30 | Edwin Beggs | Beth Ruskai | Mikhail Shubin | Mike Eastwood | Paul Tod |
| 11.00 | Coffee | Coffee | Coffee | Coffee | Coffee |
| 11.30 | Beth Ruskai | Edwin Beggs | Jan Slovak | Paul Tod | Mike Eastwood |
| 1.00 | Lunch | Lunch | Wine Trip | Lunch | Yum Cha |
| 2.30 | Nalini Joshi | Adam Harris | Wine | Mikhail Shubin | |
| 3.30 | Coffee | Coffee | Wine | Coffee | |
| 4.00 | Jim McCarthy | Bert Green | Wine | Jan Slovak | |
| 4.30 | Manfred Scheunert | Ruibin Zhang | Wine | Matilde Marcolli | |
| 5.00 | Drinks | ||||
| 7.30 | Dinner |
Speaker Mike Eastwood
Speaker: Bert Green
Speaker: Adam Harris
Speaker: Nalini Joshi
Speaker: Jim McCarthy
Speaker: Matilde Marcolli
Speaker: Mary Beth Ruskai
It is well known that the uncertainty principle precludes the possibility
of orthonormal bases of the form $g_{kn}(x) = e^{i2 \pi kx} g(x-n)$ with
``nice'' $g$, i.e., with both $xg(x)$ and $g^{\prime}(x)$ [or
equivalently, $\omega \hat{g}(\omega)$] in $L^2(\bf{R})$.
Therefore, it was something of a surprise when Daubechies
showed one could construct orthonormal bases of wavelets of the form
$\psi_{kn}(x) = 2^{-k/2} \psi(2^{-k}x - n)$ with $\psi$ smooth
and compactly supported. However, the uncertainty principle still
precludes the possibility that both $\psi(x)$ and its Fourier transform
$\hat{\psi}(\omega)$ decay exponentially or that $\psi$ has sufficient
symmetry to permit orthonormal bases for either the half-line
$L^2(0,\infty)$ or the Hardy space $H^2(\bf{R})$ of functions
whose Fourier transform is supported on a half-line.
It may be even more surprising that wavelet ideas led
to the construction of new orthonormal bases of the form
$h_{kn} = \cos \left( \frac{k-1}{2} \pi\right) h(x-n)$
(with $k \geq 1$) where $h(x)$ can be chosen to have compact support
or so that both $h(x)$ and $\hat{h}(\omega)$ decay exponentially.
If one observes that $\cos \left( \frac{k-1}{2} \pi\right)$
alternates between $\cos (j\pi x)$ and $\sin (j\pi x)$
(as $k = 1, 2 \ldots$), this basis
seems very similar to $g_{jn}(x) = e^{i \pi jx} g(x-n)$. However, the
change from $2 \pi$ to $\pi$ in the exponent necessarily yields
a linearly {\em dependent} set (for $j \in {\bf Z}$) which cannot
possibly be orthonormal and restriction of $e^{i \pi jx} g(x-n)$
to $j = 0, 1, 2 \ldots$ can not span $L^2(\bf{R})$.
In these two lectures, we will try to use some ideas from physics to
shed some light on the subtle question of why the uncertainty principle
which forbids bases of the first type permits some of the others.
In doing so, we will see how wavelet ideas led naturally to other types
of orthonormal bases, such as the local Fourier bases and wavelet packets.
Speaker: Manfred Scheunert
Speaker: Jan Slovak
Speaker: Paul Tod
Speaker Ruibin Zhang
Wine Trip:
Trip to MacLaren Vale
Drinks:
"Informal mixer"
Title:
Involutive structures: local and global aspects
Abstract: Involutive structures provide a unified way of discussing
linear first order partial differential equations. Such structures
are also called formally integrable and the main local question is
to what extent these equations are really integrable. Hans Lewy's
famous example from 1957 shows that this is a non-trivial question.
I shall present the definitions and illustrate with examples.
One of the most well-understood examples of an involutive
structure is a complex structure. That the two notions coincide is
the content of Newlander-Nirenberg theorem. Locally, that is the end
of the story. Globally, however, life is much more interesting. The
theory of compact complex manifolds is rich and exciting. Similar
comments apply to the theory of foliations. I shall discuss the
global aspects of some other examples.
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Title:
Quantum Non-euclidean Geometry
Abstract: A geometry can be constructed in which the points are quantal
events, consisting of the emission or absorption of neutral particles, and
lines are paths followed by neutral particles between their source
and destination. A point is represented by a projection corresponding to the
state of the particle immediately after emission or immediately before .
absorption. A definition will be given of the join and meet of points, which
are in a repsentation of SO(2)xSO(5,1). A metric tensor and curvature
tensor will be defined on this basis, and some solutions of Einstein's
field equations will be discussed. There are some new predictions
concerning the time of transmission of neutrinos and a change of
chirality in a gravitational field.
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Title: Degeneracy Loci in Complex Deformations
Abstract: Let $\pi$ be a holomorphic mapping from a complex manifold
${\cal X}$ onto a complex manifold $Y$. Let $A$ be a subset of $Y$
defined locally by the vanishing of a finite number of holomorphic
functions. Assume that each fibre $X_{y}$ of $\pi$ is a compact
space, and that all fibres $X_{y}, \ y\in Y\setminus A$, are
isomorphic as complex manifolds. The aim of this talk is to present a
general condition under which {\em all } fibres of $\pi$ share this
property. A new class of examples in which the complex structure is
seen conversely to "jump" at those fibres above $A$ will also be
presented.(The "structure-jumping" phenomenon has long been recognized
as the stumbling block to a comprehensive theory of moduli of complex
structure, but this issue will not be discussed.) Moreover, the
conditions under which these new examples occur will be seen to
complement fairly closely the hypothesis of our general "removable
singularities theorem", in which the "degeneracy loci" of holomorphic
vector fields along each fibre play a crucial role.
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Title:
Existence and nonexistence results for integrable equations
Abstract: Integrable equations are nonlinear differential
equations that are solvable through an associated linear problem. A
conjecture that such integrability is related to a special singularity
structure called the \textit{Painlev\'e property} has been standing
for over twenty years. Here we give an overview and describe recent
existence and nonexistence results towards a proof of the global
Painlev\'e property of equations such as the the Korteweg-deVries
equation $u_t=uu_x+u_{xxx}$.
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Title:
Topological field theories related to supersymmetric Yang-Mills
theories in higher dimensions
Abstract: After a ten minute introduction to TFT, and
SYM in 10d Minkowski space, I discuss how Euclidean SYM theories can
be produced in certain spacetime dimensions bigger than 4
by either timelike reduction of the 10d theory or a "Euclidean
rotation" of the Minkowski space. I then explain how these theories
may be related to "TFT" both by twisting or not, on manifolds of
special holonomy.
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Title:
The geometry of free fermions on singular surfaces
(joint work with M. Spreafico)
Abstract: A large number of phenomena in Physics are described by means
of the geometry of surfaces immersed in higher dimensional spaces.
Consider for instance the dynamics of random surfaces, which occur in
Statistical Physics in the description of the three dimensional
Ising model or the non perturbative approach to two dimensional
quantum gravity and string theory in Theoretical Physics.
It is interesting to analyse the geometry underlying all these
physical models.
In particular, one can consider the case of free massless fermionic
fields moving on a surface immersed in the three dimensional euclidean
space with conical type singularities. A correction
term arises in the frame anomaly due of the presence of
singular points in the immersion.
The result is obtained following some recent contributions
to the Index Theory for Dirac operators on
spaces with conical singularities.
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Title:
Overcoming the Uncertainty Principle: Wavelets and Their Offshoots
Abstract:
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Title:
Deformations of enveloping algebras of Lie superalgebras
Abstract: Formal deformations (in the sense of Gerstenhaber) of an
associative algebra A can be investigated by means of the Hochschild
cohomology of A. If A = U(L) is the enveloping algebra of a Lie algebra L,
the Hochschild cohomology may be replaced by the cohomology of L with values
in U(L).
With the appearence of quantum algebras (in the sense of Drinfeld and Jimbo)
the problem of deforming U(L) became a matter of central interest. If L is a
simple Lie algebra, cohomology theory implies that U(L) (regarded as an
associative algebra, without its additional Hopf structure) has no non-trivial
deformations.
In the present talk I intend to briefly review the aforementioned classical
material and to show how much of it has been generalized to certain simple
Lie superalgebras.
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Title:
1. Examples of Parabolic Geometries (Morning)
Abstract:Manifolds equipped with parabolic geometries can be viewed as
deformations of the homogeneous spaces G/P, where G and P are suitable real
forms of semisimple complex groups G_c and their parabolic subgroups P_c. The
best known examples are the conformal Riemannian geometries on m-dimensional
manifolds corresponding to G=SO(m+1,1,R), P < G the Poincare conformal
subgroup. In the talk, a general scheme will be suggested introducing
analogies to scales (i.e. the choices of metrics in the conformal class),
Weyl and Cotton-York tensors, Weyl geometries, conformal circles, etc., for
all these geometries. Then the latter concepts will be illustrated on
several interesting examples related to G=SL(m,R).
Title:
2. Semi-holonomic jets, frames, and Cartan Connections (Afternoon)
Abstract:The so called `Cartan geometries' generalize the concept of a
homogeneous space in a similar way how Riemannian structures generalize the
standard Euclidean space. I will discuss the role played by the so called
Cartan connections and by the higher-order semi-holonomic frame bundles in this
context.
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Title:
1. Local heterotic geometry and self-dual Einstein-Weyl spaces
Abstract: We consider the local heterotic geometry of Delduc and Valent,
which arises in (4,0)-supersymmetry, and the self-dual Einstein-Weyl spaces of
Pedersen and Swann. Both of these are hypercomplex, but one has a
metric-preserving connection with torsion while the other is torsion-free
but not metric preserving. By a consideration of spinors, we are able to
find the relationship between them. When there is a symmetry, the
equations lead to an interesting family of 3-dimensional Einstein-Weyl spaces.
Title:
2. Four-dimensional D'Atri spaces subject to extra conditions
Abstract:We consider D'Atri spaces, which is to say Riemannian spaces
satisfying the D'Atri condition. After briefly reviewing the subject we use
spinors to show that 4-dimensional D'Atri spaces which are also
conformally-flat or Kahler or Einstein are necessarily locally symmetric.
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Title:
Quantum supergroups and their applications to knots and
three-manifolds
Abstract:The Reshetikhin - Turaev theory is extended to a
supersymmetric setting, yielding general methods for constructing topological
invariants of knots and three - manifolds using the theory of quantum
supergroups. We will discuss these methods in some detail, and also consider
explicit examples of new invariants of knots and three - manifolds produced
by these methods. The existing theory of quantum supergroups will also
be reviewed very briefly.
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Schedule: We will leave as soon as possible -- between
12:30 and 1:00pm on Wednesday July 9th. We will be going in cars
unless there is unreasonable demand -- which will be gauged at the
Monday "welcome". Anyone who is prepared to drive is asked to
contact one of the organizers. We will take in a number of the great
wineries, including Chapel Hill, Hugo, of the MacLaren Vales region,
approximately 1 hour south of Adelaide. Probably we will have
lunch in Woodstock, at least of Volodja has booked it! Our
plan is to be back in Adelaide by between 6:00 and 7:00pm so that
we have plenty of time to get to the dinner. It is expected that
interested parties will share the cost of petrol for the drivers.
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Schedule: All interested will head to the Staff Club in the
University of Adelaide at approximately 5:00pm on Monday evening.
Everyone buys their own, no free booze (sorry but I said informal!).
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