2,415 research outputs found
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
Mini-Workshop: Positional Games
Positional games is one of rapidly developing subjects of modern combinatorics, researching two player perfect information games of combinatorial nature, ranging from recreational games like Tic-Tac-Toe to purely abstract games played on graphs and hypergraphs. Though defined usually in game theoretic terms, the subject has a distinct combinatorial flavor and boasts strong mutual connections with discrete probability, Ramsey theory and randomized algorithms. This mini-workshop was dedicated to summarizing the recent progress in the subject, to indicating possible directions of future developments, and to fostering collaboration between researchers working in various, sometimes apparently distinct directions
Supersymmetry, lattice fermions, independence complexes and cohomology theory
We analyze the quantum ground state structure of a specific model of
itinerant, strongly interacting lattice fermions. The interactions are tuned to
make the model supersymmetric. Due to this, quantum ground states are in
one-to-one correspondence with cohomology classes of the so-called independence
complex of the lattice. Our main result is a complete description of the
cohomology, and thereby of the quantum ground states, for a two-dimensional
square lattice with periodic boundary conditions. Our work builds on results by
J. Jonsson, who determined the Euler characteristic (Witten index) via a
correspondence with rhombus tilings of the plane. We prove a theorem, first
conjectured by P. Fendley, which relates dimensions of the cohomology at grade
n to the number of rhombus tilings with n rhombi.Comment: 40 pages, 28 figure
Exact ground states of a staggered supersymmetric model for lattice fermions
We study a supersymmetric model for strongly interacting lattice fermions in
the presence of a staggering parameter. The staggering is introduced as a
tunable parameter in the manifestly supersymmetric Hamiltonian. We obtain
analytic expressions for the ground states in the limit of small and large
staggering for the model on the class of doubly decorated lattices. On this
type of lattice there are two ground states, each with a different density. In
one limit we find these ground states to be a simple Wigner crystal and a
valence bond solid (VBS) state. In the other limit we find two types of quantum
liquids. As a special case, we investigate the quantum liquid state on the one
dimensional chain in detail. It is characterized by a massless kink that
separates two types of order.Comment: 21 pages, 6 figures, v2: largely rewritten version with more emphasis
on physical interpretatio
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