11:30
11:30
Instanton - a window into physics of M5-branes
Abstract
Instantons and W-bosons in 5d N=2 Yang-Mills theory arise from a circle
compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding
self-dual strings, respectively. We study an index which counts BPS
instantons with electric charges in Coulomb and symmetric phases. We first
prove the existence of unique threshold bound state of U(1) instantons for
any instanton number. By studying SU(N) self-dual strings in the Coulomb
phase, we find novel momentum-carrying degrees on the worldsheet. The total
number of these degrees equals the anomaly coefficient of SU(N) (2,0) theory.
We finally propose that our index can be used to study the symmetric phase of
this theory, and provide an interpretation as the superconformal index of the
sigma model on instanton moduli space.
12:00
Solitons from geometry.
Abstract
Solitons are localised non-singular lumps of energy which describe particles non perturbatively. Finding the solitons usually involves solving nonlinear differential equations, but I shall show that in some cases the solitons emerge directly from the underlying space-time geometry: certain abelian vortices arise from surfaces of constant mean curvature in Minkowski space, and skyrmions can be constructed from the holonomy of gravitational instantons.
M-theory dualities and generalised geometry
Abstract
In this talk we will review M-theory dualities and recent attempts to make these dualities manifest in eleven-dimensional supergravity. We will review the work of Berman and Perry and then outline a prescription, called non-linear realisation, for making larger duality symmetries manifest. Finally, we will explain how the local symmetries are described by generalised geometry, which leads to a duality-covariant constraint that allows one to reduce from generalised space to physical space.
Self-scaled barriers for semidefinite programming
Abstract
I am going to show that all self-scaled barriers for the
cone of symmetric positive semidefinite matrices are of the form
$X\mapsto -c_1\ln\det X +c_0$ for some constants $c_1$ > $0,c_0 \in$ \RN.
Equivalently one could state say that all such functions may be
obtained via a homothetic transformation of the universal barrier
functional for this cone. The result shows that there is a certain
degree of redundancy in the axiomatic theory of self-scaled barriers,
and hence that certain aspects of this theory can be simplified. All
relevant concepts will be defined. In particular I am going to give
a short introduction to the notion of self-concordance and the
intuitive ideas that motivate its definition.
Some properties of thin plate spline interpolation
Abstract
Let the thin plate spline radial basis function method be applied to
interpolate values of a smooth function $f(x)$, $x \!\in\! {\cal R}^d$.
It is known that, if the data are the values $f(jh)$, $j \in {\cal Z}^d$,
where $h$ is the spacing between data points and ${\cal Z}^d$ is the
set of points in $d$ dimensions with integer coordinates, then the
accuracy of the interpolant is of magnitude $h^{d+2}$. This beautiful
result, due to Buhmann, will be explained briefly. We will also survey
some recent findings of Bejancu on Lagrange functions in two dimensions
when interpolating at the integer points of the half-plane ${\cal Z}^2
\cap \{ x : x_2 \!\geq\! 0 \}$. Most of our attention, however, will
be given to the current research of the author on interpolation in one
dimension at the points $h {\cal Z} \cap [0,1]$, the purpose of the work
being to establish theoretically the apparent deterioration in accuracy
at the ends of the range from ${\cal O} ( h^3 )$ to ${\cal O} ( h^{3/2}
)$ that has been observed in practice. The analysis includes a study of
the Lagrange functions of the semi-infinite grid ${\cal Z} \cap \{ x :
x \!\geq\! 0 \}$ in one dimension.
On the convergence of interior point methods for linear programming
Abstract
Long-step primal-dual path-following algorithms constitute the
framework of practical interior point methods for
solving linear programming problems. We consider
such an algorithm and a second order variant of it.
We address the problem of the convergence of
the sequences of iterates generated by the two algorithms
to the analytic centre of the optimal primal-dual set.
Combinatorial structures in nonlinear programming
Abstract
Traditional optimisation theory and -methods on the basis of the
Lagrangian function do not apply to objective or constraint functions
which are defined by means of a combinatorial selection structure. Such
selection structures can be explicit, for example in the case of "min",
"max" or "if" statements in function evaluations, or implicit as in the
case of inverse optimisation problems where the combinatorial structure is
induced by the possible selections of active constraints. The resulting
optimisation problems are typically neither convex nor smooth and do not
fit into the standard framework of nonlinear optimisation. Users typically
treat these problems either through a mixed-integer reformulation, which
drastically reduces the size of tractable problems, or by employing
nonsmooth optimisation methods, such as bundle methods, which are
typically based on convex models and therefore only allow for weak
convergence results. In this talk we argue that the classical Lagrangian
theory and SQP methodology can be extended to a fairly general class of
nonlinear programs with combinatorial constraints. The paper is available
Modelling bilevel games in electricity
Abstract
Electricity markets facilitate pricing and delivery of wholesale power.
Generators submit bids to an Independent System Operator (ISO) to indicate
how much power they can produce depending on price. The ISO takes these bids
with demand forecasts and minimizes the total cost of power production
subject to feasibility of distribution in the electrical network.
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Each generator can optimise its bid using a bilevel program or
mathematical program with equilibrium (or complementarity) constraints, by
taking the ISOs problem, which contains all generators bid information, at
the lower level. This leads immediately to a game between generators, where
a Nash equilibrium - at which each generator's bid maximises its profit
provided that none of the other generators changes its bid - is sought.
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In particular, we examine the idealised model of Berry et al (Utility
Policy 8, 1999), which gives a bilevel game that can be modelled as an
"equilibrium problem with complementarity constraints" or EPCC.
Unfortunately, like bilevel games, EPCCs on networks may not have Nash
equilibria in the (common) case when one or more of links of the network is
saturated (at maximum capacity). Nevertheless we explore some theory and
algorithms for this problem, and discuss the economic implications of
numerical examples where equilibria are found for small electricity
networks.
Spreading fronts and fluctuations in sedimentation
Abstract
While the average settling velocity of particles in a suspension has been successfully predicted, we are still unsuccessful with the r.m.s velocity, with theories suggesting a divergence with the size of
the container and experiments finding no such dependence. A possible resolution involves stratification originating from the spreading of the front between the clear liquid above and the suspension below. One theory describes the spreading front by a nonlinear diffusion equation
$\frac{\partial \phi}{\partial t} = D \frac{\partial }{\partial z}(\phi^{4/5}(\frac{\partial \phi}{\partial z})^{2/5})$.
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Experiments and computer simulations find differently.