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.