On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors \BC_1>0 and \BC_2=0 comprising the set U=\{\BC_1,\BC_2\} and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor \BC_2=0 corresponds to a material which is void. (For technical reasons \BC_2 is actually taken to be nonzero and we take the limit \BC_2\to 0). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.Comment: 39 pages, 11 figure

Causal Fermion Systems -- An Overview

The theory of causal fermion systems is an approach to describe fundamental physics. We here introduce the mathematical framework and give an overview of the objectives and current results.Comment: 54 pages, LaTeX, 1 figure, minor improvements (published version

Symmetry classes of disordered fermions

Building upon Dyson's fundamental 1962 article known in random-matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds. The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V, a space of endomorphisms in End(V+V*), a bilinear form on V+V* which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V*. Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way. This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.Comment: 52 pages, dedicated to Freeman J. Dyson on the occasion of his 80th birthda

Finite Fields: Theory and Applications

Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation