Spectral space | EPFL Graph Search

In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. Let X be a topological space and let K(X) be the set of all compact open subsets of X. Then X is said to be spectral if it satisfies all of the following conditions: X is compact and T0. K(X) is a basis of open subsets of X. K(X) is closed under finite intersections. X is sober, i.e., every nonempty irreducible closed subset of X has a (necessarily unique) generic point. Let X be a topological space. Each of the following properties are equivalent to the property of X being spectral: X is homeomorphic to a projective limit of finite T0-spaces. X is homeomorphic to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic (as a bounded lattice) to the lattice K(X) (this is called Stone representation of distributive lattices). X is homeomorphic to the spectrum of a commutative ring. X is the topological space determined by a Priestley space. X is a T0 space whose frame of open sets is coherent (and every coherent frame comes from a unique spectral space in this way). Let X be a spectral space and let K(X) be as before. Then: K(X) is a bounded sublattice of subsets of X. Every closed subspace of X is spectral. An arbitrary intersection of compact and open subsets of X (hence of elements from K(X)) is again spectral. X is T0 by definition, but in general not T1. In fact a spectral space is T1 if and only if it is Hausdorff (or T2) if and only if it is a boolean space if and only if K(X) is a boolean algebra. X can be seen as a pairwise Stone space. A spectral map f: X → Y between spectral spaces X and Y is a continuous map such that the of every open and compact subset of Y under f is again compact. The category of spectral spaces, which has spectral maps as morphisms, is dually equivalent to the category of bounded distributive lattices (together with morphisms of such lattices).

New Results in Integer and Lattice Programming

Christoph Hunkenschröder

An integer program (IP) is a problem of the form

\min \{f(x) : \, Ax = b, \ l \leq x \leq u, \ x \in \Z^n\}

, where

A \in \Z^{m \times n}

b \in \Z^m

l,u \in \Z^n

, and

f: \Z^n \rightarrow \Z

is a separable convex objective function. The problem o ...

EPFL2020

Arithmetic and geometric structures in cryptography

Benjamin Pierre Charles Wesolowski

We explore a few algebraic and geometric structures, through certain questions posed by modern cryptography. We focus on the cases of discrete logarithms in finite fields of small characteristic, the structure of isogeny graphs of ordinary abelian varietie ...

EPFL2018

A Discontinuous Galerkin Reduced Basis Element Method For Elliptic Problems

Alfio Quarteroni, Paolo Pacciarini

We propose and analyse a new discontinuous reduced basis element method for the approximation of parametrized elliptic PDEs in partitioned domains. The method is built upon an offline stage (parameter independent) and an online (parameter dependent) one. I ...

Edp Sciences S A2016