Algebraic Statistics

This thesis on algebraic statistics contains five papers. In paper I we define ideals of graph homomorphisms. These ideals generalize many of the toric ideals defined in terms of graphs that are important in algebraic statistics and commutative algebra.   In paper II we study polytopes from subgraph statistics. Polytopes from subgraph statistics are important for statistical models for large graphs and many problems in extremal graph theory can be stated in terms of them. We find easily described semi-algebraic sets that are contained in these polytopes, and using them we compute dimensions and get volume bounds for the polytopes.  In paper III we study the topological Tverberg theorem and its generalizations. We develop a toolbox for complexes from graphs using vertex decomposability to bound the connectivity.  In paper IV we prove a conjecture by Haws, Martin del Campo, Takemura and Yoshida. It states that the three-state toric homogenous Markov chain model has Markov degree two. In algebraic terminology this means that a certain class of toric ideals are generated by quadratic binomials.  In paper V we produce cellular resolutions for a large class of edge ideals and their powers. Using algebraic discrete Morse theory it is then possible to make many of these resolutions minimal, for example explicit minimal resolutions for powers of edge ideals of paths are constructed this way.Denna avhandling om algebraisk statistik innehåller fem artiklar. I artikel I definieras ideal av grafhomomorfier. Dessa ideal generaliserar ett flertal konstruktioner av ideal från grafer som är viktiga i algebraisk statistik samt kommutativ algebra. I artikel II behandlas polytoper från delgrafsstatistik. Dessa är viktiga för att förstå statistiska modeller som beskriver stora grafer och många problem om ytterlighetsgrafer kan formuleras med dem. Bland verktygen som används är att beskriva semi-algebraiska mängder i polytoperna och genom detta bestämma deras dimension samt begränsa volymen. I artikel III behandlas den topologiska tverbergssatsen med generaliseringar. Grafkomplexen förstås genom att begränsa sammanhängandegraden medelst hörnnedbrytbarhet. I artikel IV bevisas att ideal tillhörande markovkedjor med tre tillstånd är genererade i grad två, vilket förmodats av Haws, Martin del Campo, Takemura och Yoshida. I artikel V skapas cellulära upplösningar för en stor klass av kantideal samt deras potenser. Med algebraisk diskret morseteori görs dessa upplösningar minimala för kantideal från stigar.

An Algebraic Spin and Statistics Theorem

A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional space-time. Our main assumption is the requirement that the modular groups of the von Neumann algebras of local observables associated with wedge regions act geometrically as pure Lorentz transformations. Such a property, satisfied by the local algebras generated by Wightman fields because of the Bisognano-Wichmann theorem, is regarded as a natural primitive assumption.Comment: 15 pages, plain TeX, an error in the statement of a theorem has been corrected, to appear in Commun. Math. Phy

Noetherianity up to symmetry

These lecture notes for the 2013 CIME/CIRM summer school Combinatorial Algebraic Geometry deal with manifestly infinite-dimensional algebraic varieties with large symmetry groups. So large, in fact, that subvarieties stable under those symmetry groups are defined by finitely many orbits of equations---whence the title Noetherianity up to symmetry. It is not the purpose of these notes to give a systematic, exhaustive treatment of such varieties, but rather to discuss a few "personal favourites": exciting examples drawn from applications in algebraic statistics and multilinear algebra. My hope is that these notes will attract other mathematicians to this vibrant area at the crossroads of combinatorics, commutative algebra, algebraic geometry, statistics, and other applications.Comment: To appear in Springer's LNM C.I.M.E. series; several typos fixe