9 research outputs found
The universal coefficient theorem and quantum field theory
During the end of the 1950's Alexander Grothendieck observed the importance of the coefficient groups in cohomology. Three decades later, he presented his ``Esquisse d'un Programme" to the main french funding body. This program also included the use of different coefficient groups in the definition of various (co)homologies. His proposal was rejected. Another three decades later, in the 21st century, his research proposal is considered one of the most inspiring and important collection of ideas in pure mathematics. His ideas brought together algebraic topology, geometry, Galois theory, etc. becoming the origin for several new branches of mathematics. Today, less than one year after his death, Grothendieck is considered one of the most influential mathematicians worldwide. His ideas were important for the proofs of some of the most remarkable mathematical problems like the Weil Conjectures, Mordell Conjectures and the solution of Fermat's last theorem. Grothendieck's dessins d'enfant have been used in mathematical physics in various domains. Seiberg-Witten curves, N=1 and N=2 gauge theories and matrix models are a few examples where his insights are relevant. In this thesis I try to connect the idea of cohomology with coefficients in various sheaves to some areas of modern research in physics. The applications are manifold: the universal coefficient theorem presents connections to the topological genus expansion invented by 't Hooft and applied to quantum chromodynamics (QCD) and string theory, but also to strongly coupled electronic systems or condensed matter physics. It also appears to give a more intuitive explanation for topological recursion formulas and the holomorphic anomaly equations. The counting of BPS states may also profit from this new perspective. Indeed, the merging of cohomology classes when a change in coefficient groups is implemented may be related to the wall-crossing formulas and the phenomenon of decay or coupling of BPS states while crossing stability walls. The groups appearing in universal coefficient theorems may be regarded as obstructions characterizing the phenomena occurring when BPS stability walls are being crossed. Another important aspect is the existence of dualities. These are the non-perturbative analogue of symmetry transformations. Until now, they were discovered more by accident or by educated guesswork. I show in this thesis that there exists an underlying structure to the dualities, a structure that connects them the number fields used as coefficients in (co)homologies. This observation makes a nontrivial connection between number theory and physics
Identification of cardiomyopathy associated circulating miRNA biomarkers in patients with muscular dystrophy using a complementary cardiovascular magnetic resonance and plasma profiling approach
Deep vein thrombosis following the treatment of lower limb pathologic bone fractures – a comparative study
Repair and tissue engineering techniques for articular cartilage
Chondral and osteochondral lesions due to injury or other pathology commonly result in the development of osteoarthritis, eventually leading to progressive total joint destruction. Although current progress suggests that biologic agents can delay the advancement of deterioration, such drugs are incapable of promoting tissue restoration. The limited ability of articular cartilage to regenerate renders joint arthroplasty an unavoidable surgical intervention. This Review describes current, widely used clinical repair techniques for resurfacing articular cartilage defects; short-term and long-term clinical outcomes of these techniques are discussed. Also reviewed is a developmental pipeline of acellular and cellular regenerative products and techniques that could revolutionize joint care over the next decade by promoting the development of functional articular cartilage. Acellular products typically consist of collagen or hyaluronic-acid-based materials, whereas cellular techniques use either primary cells or stem cells, with or without scaffolds. Central to these efforts is the prominent role that tissue engineering has in translating biological technology into clinical products; therefore, concomitant regulatory processes are also discussed