Past

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.

TBA

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.

We introduce Outer space, a contractible finite dimensional topological space on which the outer automorphism group of a free group acts 'nicely.' We will explain what 'nicely' is, and provide motivation with comparisons to symmetric spaces, analogous spaces associated to linear groups.

Following work of Bridgeland in the smooth case and Chen in the terminal singularities case, I will explain a proposal that extends the existence of flops for threefolds (and the required derived equivalences) to also cover canonical singularities.  Moreover this technique conjecturally says much more than just the existence of the flop, as it shows how the dual graph changes under the flop and also which curves in the flopped variety contract to points without contracting divisors.  This allows us to continue the Minimal Model Programme on the flopped variety in an easy way, thus producing many varieties birational to the given input.    

Due to the scaling properties of the Navier-Stokes equations,

homogeneous initial data may lead to forward self-similar solutions.

When the initial data is small enough, it is well known that the

formalism of mild solutions (through the Picard-Duhamel formula) give

such self-similar solutions. We shall discuss the issue of large initial

data, where we can only prove the existence of weak solutions; those

solutions may lack self-similarity, due to the fact that we have no

results about uniqueness for such weak solutions. We study some tools

which may be useful to get a better understanding of those weak solutions.

Suppose that $C$ and $C'$ are cubic forms in at least 19 variables over a

$p$-adic field $k$. A special case of a conjecture of Artin is that the

forms $C$ and $C'$ have a common zero over $k$. While the conjecture of

Artin is false in general, we try to argue that, in this case, it is

(almost) correct! This is still work in progress (joint with

Heath-Brown), so do not expect a full answer.

As a historical note, some cases of Artin's conjecture for certain

hypersurfaces are known. Moreover, Jahan analyzed the case of the

simultaneous vanishing of a cubic and a quadratic form. The approach

we follow is closely based on Jahan's approach, thus there might be

some overlap between his talk and this one. My talk will anyway be

self-contained, so I will repeat everything that I need that might

have already been said in Jahan's talk.

Abstract: There has in the last year been much progresson the universality problem for the spectra of a Wigner random matrices, i.e.Hermitian or symmetric random matrices with independent elements. I will givesome background on this problem and also discuss what can be said when we onlyassume a few moments of the matrix elements to be finite.

A copolymer is a chain of repetitive units (monomers) that

are almost identical, but they differ in their degree of

affinity for certain solvents. This difference leads to striking

phenomena when the polymer fluctuates

in a non-homogeneous medium, for example made up by two solvents

separated by an interface.

One may observe, for exmple, the localization of the polymer at the

interface between the two solvents.

Much of the literature on the subject focuses on the most basic model

based on the simple symmetric random walk on the integers, but

E. Bolthausen and F. den Hollander (AP 1997) pointed out

the convergence of the (rescaled) free energy of such a discrete model

toward

the free energy of a continuum model, based on Brownian motion,

in the limit of weak polymer-solvent coupling. This result is

remarkable because it strongly suggests

a universal feature for copolymer models. In this work we prove that

this is indeed the case. More precisely,

we determine the weak coupling limit for a general class of discrete

copolymer models, obtaining as limits

a one-parameter (alpha in (0,1)) family of continuum models, based on

alpha-stable regenerative sets.