Let $X$ be a smooth hypersurface in projective space over a field $K$ of characteristic zero and let $U$ denote the open complement. Then the elements of the algebraic de Rham cohomology group $H_{dR}^n(U/K)$ can be represented by $n$-forms of the form $Q \Omega / P^k$ for homogeneous polynomials $Q$ and integer pole orders $k$, where $\Omega$ is some fixed $n$-form. The problem of finding a unique representative is computationally intensive and typically based on the pre-computation of a Groebner basis. I will present a more direct approach based on elementary linear algebra. As presented, the method will apply to diagonal hypersurfaces, but it will clear that it also applies to families of projective hypersurfaces containing a diagonal fibre. Moreover, with minor modifications the method is applicable to larger classes of smooth projective hypersurfaces.
ABSTRACT "We give a short introduction to randomwalk in random environment
(RWRE) and some open problems connected to RWRE.
Then, in dimension larger than or equal to four we studyballisticity conditions and their interrelations. For this purpose, we dealwith a certain class of ballisticity conditions introduced by Sznitman anddenoted $(T)_\gamma.$ It is known that they imply a ballistic behaviour of theRWRE and are equivalent for parameters $\gamma \in (\gamma_d, 1),$ where$\gamma_d$ is a constant depending on the dimension and taking values in theinterval $(0.366, 0.388).$ The conditions $(T)_\gamma$ are tightly interwovenwith quenched exit estimates.
As a first main result we show that the conditions are infact equivalent for all parameters $\gamma \in (0,1).$ As a second main result,we prove a conjecture by Sznitman concerning quenched exit estimates.
Both results are based on techniques developed in a paperon slowdowns of RWRE by Noam Berger.
(joint work with Alejandro Ram\'{i}rez)"
We investigate a class of weakly interactive particle systems with absorption. We assume that the coefficients in our model depend on an "absorbing" factor and prove the existence and uniqueness of the proposed model. Then we investigate the convergence of the empirical measure of the particle system and derive the Stochastic PDE satisfied by the density of the limit empirical measure. This result can be applied to credit modelling. This is a joint work with Dr. Ben Hambly.
TBA
This paper models a firm’s rollover risk generated by con.ict of interest between debt and equity holders. When the firm faces losses in rolling over its maturing debt, its equity holders are willing to absorb the losses only if the option value of keeping the firm alive justifies the cost of paying off the maturing debt. Our model shows that both deteriorating market liquidity and shorter debt maturity can exacerbate this externality and cause costly firm bankruptcy at higher fundamental thresholds. Our model provides implications on liquidity- spillover effects, the flight-to-quality phenomenon, and optimal debt maturity structures.
Trevor Wishart writes "I realise 'irrational' means something very specific to a mathematician, and I'm not using the word in that sense."
Abstract:
Trevor Wishart will discuss the use of Digital Signal Processing as a tool in musical composition, ranging from the application of standard analysis procedures (e.g. windowed Fourier Transforms), and common time-domain methods (Brassage), to more unconventional approaches (e.g. waveset distortion, spectral tracing, iterative-extension). He will discuss the algorithms involved and illustrate his talk with musical examples taken from his own work.
This workshop is linked to a musical performance of "Two Women" and "Globalalia" by Trevor Wishart in the Jacqueline du Pre concert hall that evening (5th Feb) at 8pm as part of the Music Department's "New Music Forum". Tickets are £12 (or £8 concession) but if you are interested please let me know (Rebecca Gower, @email or 152312) as we may be able to negotiate a much lower price for members of the Mathematical Institute in a group associated with his workshop.
Trevor will also be giving two lectures in the Denis Arnold Hall, Faculty of Music on the 3rd and 4th Feb which are open to the public and admission is free.
I will discuss the special properties of dimension groups obtained by model-theoretic forcing
The FPU lattice is a coupled system of ordinary differential equations in which each atom in a chain is coupled to its nearest neighbour by way of a nonlinear spring.
After summarising the properties of travelling waves (kinks) we use asymptotic analysis to describe more complicate envelope solutions (breathers). The interaction of breathers and kinks will then be analysed. If time permits, the method will be extended to two-dimensional lattices.
In dynamical systems given by an ODE, one is interested in the basin
of attraction of invariant sets, such as equilibria or periodic
orbits. The basin of attraction consists of solutions which converge
towards the invariant set. To determine the basin of attraction, one
can use a solution of a certain linear PDE which can be approximated
by meshless collocation.
The basin of attraction of an equilibrium can be determined through
sublevel sets of a Lyapunov function, i.e. a scalar-valued function
which is decreasing along solutions of the dynamical system. One
method to construct such a Lyapunov function is to solve a certain
linear PDE approximately using Meshless Collocation. Error estimates
ensure that the approximation is a Lyapunov function.
The basin of attraction of a periodic orbit can be analysed by Borg’s
criterion measuring the time evolution of the distance between
adjacent trajectories with respect to a certain Riemannian metric.
The sufficiency and necessity of this criterion will be discussed,
and methods how to compute a suitable Riemannian metric using
Meshless Collocation will be presented in this talk.
One of important subjects in the study of transonic flow is to understand a global structure of flow through a convergent-divergent nozzle so called a de Laval nozzle. Depending on the pressure at the exit of the de Laval nozzle, various patterns of flow may occur. As an attempt to understand such a phenomenon, we introduce a new potential flow model called 'non-isentropic potential flow system' which allows a jump of the entropy across a shock, and use this model to rigorously prove the unique existence and the stability of transonic shocks for a fixed exit pressure. This is joint work with Mikhail Feldman.
Many interesting classes of projective varieties can be studied in terms of their graded rings. For weighted projective varieties, this has been done in the past in relatively low codimension.
Let $G$ be a simple and simply connected Lie group and $P$ be a parabolic subgroup of $G$, then homogeneous space $G/P$ is a projective subvariety of $\mathbb{P}(V)$ for some\\
$G$-representation $V$. I will describe weighted projective analogues of these spaces and give the corresponding Hilbert series formula for this construction. I will also show how one may use such spaces as ambient spaces to construct weighted projective varieties of higher codimension.
This work aims to present a new approach called Flow Curvature Method
that applies Differential Geometry to Dynamical Systems. Hence, for a
trajectory curve, an integral of any n-dimensional dynamical system
as a curve in Euclidean n-space, the curvature of the trajectory or
the flow may be analytically computed. Then, the location of the
points where the curvature of the flow vanishes defines a manifold
called flow curvature manifold. Such a manifold being defined from
the time derivatives of the velocity vector field, contains
information about the dynamics of the system, hence identifying the
main features of the system such as fixed points and their stability,
local bifurcations of co-dimension one, centre manifold equation,
normal forms, linear invariant manifolds (straight lines, planes,
hyperplanes).
In the case of singularly perturbed systems or slow-fast dynamical
systems, the flow curvature manifold directly provides the slow
invariant manifold analytical equation associated with such systems.
Also, starting from the flow curvature manifold, it will be
demonstrated how to find again the corresponding dynamical system,
thus solving the inverse problem.
Moreover, the concept of curvature of trajectory curves applied to
classical dynamical systems such as Lorenz and Rossler models
enabled to highlight one-dimensional invariant sets, i.e. curves
connecting fixed points which are zero-dimensional invariant sets.
Such "connecting curves" provide information about the structure of
the attractors and may be interpreted as the skeleton of these
attractors. Many examples are given in dimension three and more.
TBA
This talk will introduce various aspects of modern cryptography. After introducing RSA and some factoring algorithms, I will move on to how elliptic curves can be used to produce a more complex form of Diffie--Hellman key exchange.