Let $\phi$ be a convex, $C^1$-function and consider the functional: $$ (1)\qquad \mathcal{F}(\bf u)=\int_{\Omega} \phi (|\nabla \bf u|) \,dx $$ where $\Omega\subset \mathbb{R}^n$ is a bounded open set and $\bf u: \Omega \to \mathbb{R}^N$. The associated Euler Lagrange system is $$ -\mbox{div} (\phi' (|\nabla\bf u|)\frac{\nabla\bf u}{|\nabla\bf u|} )=0 $$ In a fundamental paper K.~Uhlenbeck proved everywhere $C^{1,\alpha}$-regularity for local minimizers of the $p$-growth functional with $p\ge 2$. Later on a large number of generalizations have been made. The case $1
{\bf Theorem.} Let $\bfu\in W^{1,\phi}_{\loc}(\Omega)$ be a local minimizer of (1), where $\phi$ satisfies suitable assumptions. Then $\bfV(\nabla \bfu)$ and $\nabla \bfu$ are locally $\alpha$-Hölder continuous for some $\alpha>0$.
We present a unified approach to the superquadratic and subquadratic $p$-growth, also considering more general functions than the powers. As an application, we prove Lipschitz regularity for local minimizers of asymptotically convex functionals in a $C^2$ sense.
We consider the time average of the (renormalized) current fluctuation field in one-dimensional weakly asymmetric simple exclusion.
The asymmetry is chosen to be weak enough such that the density fluctuation field still converges in law with respect to diffusive scaling. Remark that the density fluctuation field would evolve on a slower time scale if the asymmetry is too strong and that then the current fluctuations would have something to do with the Tracy-Widom distribution. However, the asymmetry is also chosen to be strong enough such that the density fluctuation field does not converge in law to an infinite-dimensional Ornstein-Uhlenbeck process, that is something non-trivial is happening.
We will, at first, motivate why studying the time average of the current fluctuation field helps to understand the structure of this non-trivial scaling limit of the density fluctuation field and, second, show how one can replace the current fluctuation field by a certain functional of the density fluctuation field under the time average. The latter result provides further evidence for the common belief that the scaling limit of the density fluctuation field approximates the solution of a Burgers-type equation
Abstract: The goal of this talk is to discuss threeproblems on fractional and related stochastic fields, in which wavelet methodshave turned out to be quite useful.
The first problemconsists in constructing optimal random series representations of Lévyfractional Brownian field; by optimal we mean that the tails of the seriesconverge to zero as fast as possible i.e. at the same rate as the l-numbers.Note in passing that there are close connections between the l-numbers of aGaussian field and its small balls probabilities behavior.
The secondproblem concerns a uniform result on the local Hölder regularity (the pointwiseHölder exponent) of multifractional Brownian motion; by uniform we mean thatthe result is satisfied on an event with probability 1 which does not depend onthe location.
The third problemconsists in showing that multivariate multifractional Brownian motion satisfiesthe local nondeterminism property. Roughly speaking, this property, which wasintroduced by Berman, means that the increments are asymtotically independentand it allows to extend to general Gaussian fields many results on the localtimes of Brownian motion.
TBA
The viability of a market impact model is usually considered to be equivalent to the absence of price manipulation strategies in the sense of Huberman & Stanzl (2004). By analyzing a model with linear instantaneous, transient, and permanent impact components, we discover a new class of irregularities, which we call transaction-triggered price manipulation strategies. Transaction-triggered price manipulation is closely related to the non-existence of measure-valued solutions to a Fredholm integral equation of the first kind. We prove that price impact must decay as a convex decreasing function of time to exclude these market irregularities along with standard price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest. Joint work with Aurélien Alfonsi, Jim Gatheral, and Alla Slynko.
Based on hep-th/0912.3249 by Arkani-Hamed et. al..
The significance of the effects of non-healing wounds has been the topic of many research papers and lectures during the last 25 years. Efforts have been made to understand the effects of long-standing venous hypertension, diabetes, the prevalence of wounds in such conditions with as well as the difficulties faced in managing such wounds with some success. Successful efforts to define standard care regimes have also been made. However, attempts to introduce innovative therapy have been much less successful. Is this merely because we have not understood the intricacies of the problem? And would system based modelling be an untried technique?
Venous ulcers are the majority of lower extremity wounds, and a clinical challenge. A previously developed model of venous ulcers permits some understanding of why compression bandaging is successful but fails to accommodate complications such as exudate and infection. Could this experimental model be improved by system based modelling?
Chronic wounds need to be modelled however the needs for such models should be examined in order that the outcome permits advances in our thinking as well in clinical management.
The talks will discuss relations between two major conjectures in the theory of groups of finite Morley rank, a modern chapter of model theoretic algebra. One conjecture, the famous the Cherlin-Zilber Algebraicity Conjecture formulated in 1970-s states that infinite simple groups of finite Morley rank are isomorphic to simple algebraic groups over algebraically closed fields. The other conjecture, due to Hrushovski and more recent, states that a generic automorphism of a simple group of finite Morley rank has pseudofinite group of fixed points.
Hrushovski showed that the Cherlin-Zilber Conjecture implies his conjecture. Proving Hrushovski's Conjecture and reversing the implication would provide a new efficient approach to proof of Cherlin-Zilber Conjecture.
Meanwhile, the machinery that is already available for the work at pseudofinite/finite Morley rank interface already yields an interesting
result: an alternative proof of the Larsen-Pink Theorem (the latter says, roughly speaking, that "large" finite simple groups of matrices are Chevalley groups over finite fields).
We study the axisymmetric stretching of a thin sheet of viscous fluid
driven by a centrifugal body force. Time-dependent simulations show that
the sheet radius tends to infinity in finite time. As the critical time is
approached, the sheet becomes partitioned into a very thin central region
and a relatively thick rim. A net momentum and mass balance in the rim leads
to a prediction for the sheet radius near the singularity that agrees with the numerical
simulations. By asymptotically matching the dynamics of the sheet with the
rim, we find that the thickness in the central region is described by a
similarity solution of the second kind. For non-zero surface tension, we
find that the similarity exponent depends on the rotational Bond number B,
and increases to infinity at a critical value B=1/4. For B>1/4, surface
tension defeats the centrifugal force, causing the sheet to retract rather
than stretch, with the limiting behaviour described by a similarity
solution of the first kind.
We show that data assimilation using four-dimensional variation
(4DVar) can be interpreted as a form of Tikhonov regularisation, a
familiar method for solving ill-posed inverse problems. It is known from
image restoration problems that $L_1$-norm penalty regularisation recovers
sharp edges in the image better than the $L_2$-norm penalty
regularisation. We apply this idea to 4DVar for problems where shocks are
present and give some examples where the $L_1$-norm penalty approach
performs much better than the standard $L_2$-norm regularisation in 4DVar.
We will present a physical motivation of the SYZ conjecture and try to understand the conjecture via calibrated geometry. We will define calibrated submanifolds, and also give sketch proofs of some properties of the moduli space of special Lagrangian submanifolds. The talk will be elementary and accessible to a broad audience.
The Alexander polynomial of a link was the first link polynomial. We give some ways of defining this much-studied invariant, and derive some of its properties.
I will show that generating functions for certain non-compact Calabi-Yau 3-folds are modular forms. This is joint work with Hiroshi Iritani.
The famous theorem of Szemerédi says that for any natural number $k$ and any $a>0$ there exists $n$ such that if $N\ge n$ then any subset $A$ of the set $[N] =\{1, 2,\ldots , N\}$ of size $|A| \ge a N$ contains an arithmetic progression of length $k$. We consider the question of when such a theorem holds in a random set. More precisely, we say that a set $X$ is $(a, k)$-Szemerédi if every subset $Y$ of $X$ that contains at least $a|X|$ elements contains an arithmetic progression of length $k$. Let $[N]_p$ be the random set formed by taking each element of $[N]$ independently with probability $p$. We prove that there is a threshold at about $p = N^{-1/(k-1)}$ where the probability that $[N]_p$ is $(a, k)$-Szemerédi changes from being almost surely 0 to almost surely 1.
There are many other similar problems within combinatorics. For example, Turán’s theorem and Ramsey’s theorem may be relativised, but until now the precise probability thresholds were not known. Our method seems to apply to all such questions, in each case giving the correct threshold. This is joint work with Tim Gowers.
I will show that generating functions for certain non-compact
Calabi-Yau 3-folds are modular forms. This is joint work with Hiroshi
Iritani.
We consider Einstein-scalar field Lichnerowicz equations in the positive case in compact Riemannian manifolds. We discuss existence and stability issues for these equations
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.