To distinguish continuous from discontinuous evolutionary change, a relation of nearness between phenotypes is needed. Such a relation is based on the probability of one phenotype being accessible from another through changes in the genotype. This nearness relation is exemplified by calculating the shape neighborhood of a transfer RNA secondary structure and provides a characterization of discontinuous shape transformations in RNA. The simulation of replicating and mutating RNA populations under selection shows that sudden adaptive progress coincides mostly, but not always, with discontinuous shape transformations. The nature of these transformations illuminates the key role of neutral genetic drift in their realization.

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Continuity in evolution: on the nature of transitions.

We derive a spacetime formulation of quantum general relativity from (Hamiltonian) loop quantum gravity. In particular, we study the quantum propagator that evolves the three-geometry in proper time. We show that the perturbation expansion of this operator is finite and computable order by order. By giving a graphical representation in the manner of Feynman of this expansion, we find that the theory can be expressed as a sum over topologically inequivalent (branched, colored) two-dimensional (2D) surfaces in 4D. The contribution of one surface to the sum is given by the product of one factor per branching point of the surface. Therefore branching points play the role of elementary vertices of the theory. Their value is determined by the matrix elements of the Hamiltonian constraint, which are known. The formulation we obtain can be viewed as a continuum version of Reisenberger's simplicial quantum gravity. Also, it has the same structure as the Ooguri-Crane-Yetter 4D topological field theory, with a few key differences that illuminate the relation between quantum gravity and topological quantum field theory. Finally, we suggest that certain new terms should be added to the Hamiltonian constraint in order to implement a ``crossing'' symmetry related to 4D diffeomorphism invariance.

http://cds.cern.ch/record/316883/files/9612035.pdf

'Sum over surfaces' form of loop quantum gravity

We use the method of interlacing families of polynomials introduced to prove two theorems known to imply a positive solution to the Kadison--Singer problem. 
The first is Weaver's conjecture $KS_{2}$ \cite{weaver}, which is known to imply Kadison--Singer via a projection paving conjecture of Akemann and Anderson. The second is a formulation due to Casazza, et al., of Anderson's original paving conjecture(s), for which we are able to compute explicit paving bounds. 
The proof involves an analysis of the largest roots of a family of polynomials that we call the "mixed characteristic polynomials" of a collection of matrices.

Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem

https://www.math.toronto.edu/khesin/papers/kdvch.pdf

Euler equations on homogeneous spaces and Virasoro orbits

Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by Doplicher, Haag, and Roberts (DHR); and we give an alternative proof of Doplicher and Robert's reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to J. E. Roberts and the abstract duality theorem for symmetric tensor *-categories, a self-contained proof of which is given in the appendix.

/pdf/algebraic-quantum-field-theory-5dcyob49qu.pdf

Herbert Busemann

Papers

Advances in mathematics