7,533 research outputs found
Novel topological beam-splitting in photonic crystals
We create a passive wave splitter, created purely by geometry, to engineer
three-way beam splitting in electromagnetism in transverse electric
polarisation. We do so by considering arrangements of Indium Phosphide
dielectric pillars in air, in particular we place several inclusions within a
cell that is then extended periodically upon a square lattice. Hexagonal
lattice structures more commonly used in topological valleytronics but, as we
discuss, three-way splitting is only possible using a square, or rectangular,
lattice. To achieve splitting and transport around a sharp bend we use
accidental, and not symmetry-induced, Dirac cones. Within each cell pillars are
either arranged around a triangle or square; we demonstrate the mechanism of
splitting and why it does not occur for one of the cases. The theory is
developed and full scattering simulations demonstrate the effectiveness of the
proposed designs
Affine Hecke algebras and generalized standard Young tableaux
This paper introduces calibrated representations for affine Hecke algebras
and classifies and constructs all finite dimensional irreducible calibrated
representations. The primary technique is to provide indexing sets for
controlling the weight space structure of finite dimensional modules for the
affine Hecke algebra. Using these indexing sets we show that (1) irreducible
calibrated representations are indexed by skew local regions, (2) the dimension
of an irreducible calibrated representation is the number of chambers in the
local region, (3) each irreducible calibrated representation is constructed
explicitly by formulas which describe the action of the generators of the
affine Hecke algebra on a specific basis in the representation space. The
indexing sets for weight spaces are generalizations of standard Young tableaux
and the construction of the irreducible calibrated affine Hecke algebra modules
is a generalization of A. Young's seminormal construction of the irreducible
representations of the symmetric group. In this sense Young's construction has
been generalized to arbitrary Lie type
Combinatorics in Schubert varieties and Specht modules
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, June 2011."June 2011." Cataloged from PDF version of thesis.Includes bibliographical references (p. 57-59).This thesis consists of two parts. Both parts are devoted to finding links between geometric/algebraic objects and combinatorial objects. In the first part of the thesis, we link Schubert varieties in the full flag variety with hyperplane arrangements. Schubert varieties are parameterized by elements of the Weyl group. For each element of the Weyl group, we construct certain hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial if and only if the Schubert variety is rationally smooth. For classical types the arrangements are (signed) graphical arrangements coning from (signed) graphs. Using this description, we also find an explicit combinatorial formula for the Poincaré polynomial in type A. The second part is about Specht modules of general diagram. For each diagram, we define a new class of polytopes and conjecture that the normalized volume of the polytope coincides with the dimension of the corresponding Specht module in many cases. We give evidences to this conjecture including the proofs for skew partition shapes and forests, as well as the normalized volume of the polytope for the toric staircase diagrams. We also define new class of toric tableaux of certain shapes, and conjecture the generating function of the tableaux is the Frobenius character of the corresponding Specht module. For a toric ribbon diagram, this is consistent with the previous conjecture. We also show that our conjecture is intimately related to Postnikov's conjecture on toric Specht modules and McNamara's conjecture of cylindric Schur positivity.by Hwanchul Yoo.Ph.D
Topology of Arrangements and Representation Stability
The workshop “Topology of arrangements and representation stability” brought together two directions of research: the topology and geometry of hyperplane, toric and elliptic arrangements, and the homological and representation stability of configuration spaces and related families of spaces and discrete groups. The participants were mathematicians working at the interface between several very active areas of research in topology, geometry, algebra, representation theory, and combinatorics. The workshop provided a thorough overview of current developments, highlighted significant progress in the field, and fostered an increasing amount of interaction between specialists in areas of research
Physics-based visual characterization of molecular interaction forces
Molecular simulations are used in many areas of biotechnology, such as drug design and enzyme engineering. Despite the development of automatic computational protocols, analysis of molecular interactions is still a major aspect where human comprehension and intuition are key to accelerate, analyze, and propose modifications to the molecule of interest. Most visualization algorithms help the users by providing an accurate depiction of the spatial arrangement: the atoms involved in inter-molecular contacts. There are few tools that provide visual information on the forces governing molecular docking. However, these tools, commonly restricted to close interaction between atoms, do not consider whole simulation paths, long-range distances and, importantly, do not provide visual cues for a quick and intuitive comprehension of the energy functions (modeling intermolecular interactions) involved. In this paper, we propose visualizations designed to enable the characterization of interaction forces by taking into account several relevant variables such as molecule-ligand distance and the energy function, which is essential to understand binding affinities. We put emphasis on mapping molecular docking paths obtained from Molecular Dynamics or Monte Carlo simulations, and provide time-dependent visualizations for different energy components and particle resolutions: atoms, groups or residues. The presented visualizations have the potential to support domain experts in a more efficient drug or enzyme design process.Peer ReviewedPostprint (author's final draft
Fermion condensation and super pivotal categories
We study fermionic topological phases using the technique of fermion
condensation. We give a prescription for performing fermion condensation in
bosonic topological phases which contain a fermion. Our approach to fermion
condensation can roughly be understood as coupling the parent bosonic
topological phase to a phase of physical fermions, and condensing pairs of
physical and emergent fermions. There are two distinct types of objects in
fermionic theories, which we call "m-type" and "q-type" particles. The
endomorphism algebras of q-type particles are complex Clifford algebras, and
they have no analogues in bosonic theories. We construct a fermionic
generalization of the tube category, which allows us to compute the
quasiparticle excitations in fermionic topological phases. We then prove a
series of results relating data in condensed theories to data in their parent
theories; for example, if is a modular tensor category containing
a fermion, then the tube category of the condensed theory satisfies
.
We also study how modular transformations, fusion rules, and coherence
relations are modified in the fermionic setting, prove a fermionic version of
the Verlinde dimension formula, construct a commuting projector lattice
Hamiltonian for fermionic theories, and write down a fermionic version of the
Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted
to three detailed examples of performing fermion condensation to produce
fermionic topological phases: we condense fermions in the Ising theory, the
theory, and the theory, and compute the
quasiparticle excitation spectrum in each of these examples.Comment: 161 pages; v2: corrected typos (including 18 instances of "the the")
and added some reference
- …