Morse theory of the moment map for representations of quivers

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-H\"older filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.Comment: 48 pages, small revisions from previous version based on referee's comments. To appear in Geometriae Dedicat

Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for x\in\RR^d, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold (\RR^d,{\bf g}), with a metric ${\bf g}$ which is conformal to the standard \RR^d metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker--Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of $m$.Comment: 37 page

Coronary artery surgery in a man with achondroplasia: a case report

<p>Abstract</p> <p>Introduction</p> <p>Achondroplasia is a musculoskeletal disorder associated with short stature. Despite an estimated prevalence of 1:25,000 in the general population, there is little literature concerning the diagnostic and treatment challenges faced by doctors dealing with a heart operation on a patient with this condition.</p> <p>Case presentation</p> <p>We present the case of a 41-year-old Caucasian man of Greek ethnicity with achondroplasia, who underwent bypass heart surgery.</p> <p>Conclusions</p> <p>The surgery was successful and did not present particular difficulties, showing that heart surgery can be safely performed on people with achondroplasia.</p

Existence of Ricci flows of incomplete surfaces

We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.Comment: 20 pages; updated to reflect galley proof correction

How many lobes do you see?

Accessory fissures represent a variation of the normal lung anatomy. Incomplete development or even the absence of the major or minor fissures can lead to confusion in distinguishing adjacent lobes. This report aims to present a rare intraoperative finding of an anatomic malformation of the right lung in a 19-year old male patient with recurrent pneumothorax who underwent a surgical repair. An accessory fissure which was separating the superior segment of the lower lobe from the basal segments gave to the whole lung the unique image of a four-lobed one. A profound knowledge of the accessory fissures, even if they are incidentally discovered, is of pivotal importance for the thoracic surgeon and leads to optimal operative assessment and strategic planning

From wellness to medical diagnostic apps: the Parkinson's Disease case

This paper presents the design and development of the CloudUPDRS app and supporting system developed as a Class I medical device to assess the severity of motor symptoms for Parkinson’s Disease. We report on lessons learnt towards meeting fidelity and regulatory requirements; effective procedures employed to structure user context and ensure data quality; a robust service provision architecture; a dependable analytics toolkit; and provisions to meet mobility and social needs of people with Parkinson’s

Measures on Banach Manifolds and Supersymmetric Quantum Field Theory

We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The first example is a family $\mu_P^{s,t}$ of measures on a space of functions on the two-torus, parametrized by a polynomial $P$ (the Wess-Zumino-Landau-Ginzburg model). The second is a family \mu_\cG^{s,t} of measures on a space \cG of maps from $\P^1$ to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family $\mu_{M,G}^{s,t}$ of measures on the product of a space of connection s on the trivial principal bundle with structure group $G$ on a three-dimensional manifold $M$ with a space of \fg-valued three-forms on $M.$ We show that these measures are positive, and that the measures \mu_\cG^{s,t} are Borel probability measures. As an application we show that formulas arising from expectations in the measures \mu_\cG^{s,1} reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures $\mu_{M,SU(2)}^{s,t},$ where $M$ is a homology three-sphere, will yield the Casson invariant of $M.$Comment: Minor correction

Double auricles of the right atrium: a unique anatomic deformity

<p>Abstract</p> <p>Background</p> <p>Anatomic deviations, especially those detected during the course of an operation, are medically intriguing, as they raise concerns about their clinical significance and putative complications.</p> <p>Case presentation</p> <p>We present, to our knowledge, for the first time a case of an anatomic deviation in the form of a second right atrial auricle in a 70 year-old, coronary bypass-operated male Caucasian patient of Greek origin. No complications were noted intra-or postoperatively.</p> <p>Conclusions</p> <p>A second right atrial auricle was found intraoperatively, without causing any clinical complications, or obstructing the normal course of a surgical procedure.</p