The symplectic geometry of higher Auslander algebras: Symmetric products of disks

We show that the perfect derived categories of Iyama’s d-dimensional Auslander algebras of type ${\mathbb {A}}$ are equivalent to the partially wrapped Fukaya categories of the d-fold symmetric product of the $2$ -dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk and those of its $(n-d)$ -fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type ${\mathbb {A}}$ . As a by-product of our results, we deduce that the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk organise into a paracyclic object equivalent to the d-dimensional Waldhausen $\text {S}_{\bullet }$ -construction, a simplicial space whose geometric realisation provides the d-fold delooping of the connective algebraic K-theory space of the ring of coefficients.

Toric Geometry ofSL2(ℂ) Free Group Character Varieties from Outer Space

Culler and Vogtmann defined a simplicial spaceO(g), calledouter space, to study the outer automorphism group of the free groupFg. Using representation theoretic methods, we give an embedding ofO(g) into the analytification of X(Fg,SL2(ℂ)), theSL2(ℂ) character variety ofFg, reproving a result of Morgan and Shalen. Then we show that every pointvcontained in a maximal cell ofO(g) defines a flat degeneration of X(Fg,SL2(ℂ)) to a toric varietyX(PΓ). We relate X(Fg,SL2(ℂ)) andX(v) topologically by showing that there is a surjective, continuous, proper map Ξv:X(Fg,SL2(ℂ)) →X(v). We then show that this map is a symplectomorphism on a dense open subset of X(Fg, SL2(ℂ)) with respect to natural symplectic structures on X(Fg, SL2(ℂ)) andX(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell ofO(g), and we show that eachvdefines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition ofX(PΓ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell inO(g) all arise as divisorial valuations built from an associated projective compactification of X(Fg, SL2(ℂ)).

Segal’s multisimplicial spaces

Some sufficient conditions on a simplicial space X : ?op ? Top guaranteeing that X1 ? ?|X| were given by Segal. We give a generalization of this result for multisimplicial spaces. This generalization is appropriate for the reduced bar construction, providing an n-fold delooping of the classifying space of a category.

PITCHING TENTS IN SPACE-TIME: MESH GENERATION FOR DISCONTINUOUS GALERKIN METHOD

Space-time discontinuous Galerkin (DG) methods provide a solution for a wide variety of numerical problems such as inviscid Burgers equation and elastodynamic analysis. Recent research shows that it is possible to solve a DG system using an element-by-element procedure if the space-time mesh satisfies a cone constraint. This constraint requires that the dihedral angle of each interior mesh face with respect to the space domain is less than or equal to a specified angle function α(). Whenever there is a face that violates the cone constraint, the elements at the face must be coupled in the solution. In this paper we consider the problem of generating a simplicial space-time mesh where the size of each group of elements that need to be coupled is bounded by a constant number k. We present an algorithm for generating such meshes which is valid for any nD×TIME domain (n is a natural number). The k in the algorithm is based on a node degree in an n-dimensional space domain mesh.