On a Generalization of Zaslavsky's Theorem for Hyperplane Arrangements

We define arrangements of codimension-1 submanifolds in a smooth manifold which generalize arrangements of hyperplanes. When these submanifolds are removed the manifold breaks up into regions, each of which is homeomorphic to an open disc. The aim of this paper is to derive formulas that count the number of regions formed by such an arrangement. We achieve this aim by generalizing Zaslavsky's theorem to this setting. We show that this number is determined by the combinatorics of the intersections of these submanifolds.Comment: version 3: The title had a typo in v2 which is now fixed. Will appear in Annals of Combinatorics. Version. 2: 19 pages, major revision in terms of style and language, some results improved, contact information updated, final versio

On the log-concavity of the characteristic polynomial of a matroid

In this dissertation we address a long-standing conjecture, due to Heron, Rota and Welsh on the log-concavity of the characteristic polynomial of a matroid. After decades of attempts and a series of partial results, the conjecture was fully solved in 2018 by Adiprasito, Huh and Katz, using combinatorial analogues of several results in Algebraic Geometry concerning a particular cohomology ring called Chow ring. In February 2020, a new, simpler proof was announced by Braden, Huh, Matherne, Proudfoot and Wang. This dissertation is conceived to be a self-contained guide to support the reader in understanding these two papers, providing also the necessary background, a wide horizon ranging from Hodge Theory to Combinatorics to Toric Geometry. Moreover, we provide concrete and nontrivial examples of computations of Chow rings, of which we feel current literature is still lacking

Orlik-Solomon algebras and Hyperplane Arrangements

This work fits into the topic of hyperplane arrangements; this is a widely studied subject involving many different areas of mathematics (combinatorics, commutative algebra, topology, group theory, representation theory, etc..). To each hyperplane arrangement are associated certain combinatorial data; it turns out that the cohomology algebra of the complement of a complex hyperplane arrangement is uniquely determined by these combinatorial data. A famous isomorphic model for this algebra is the so-called Orlik-Solomon algebra. In the first part of the thesis (chapters 1-3) we study the topology of the complement of a complex hyperplane arrangement; in chapter 1 we review in detail the general theory of hyperplane arrangements, in chapter 2 we specialize to the braid arrangement and prove some explicit results and in chapter 3 we prove the isomorphism between the Orlik-Solomon algebra and the cohomology algebra of the complement. In the second part of the thesis (chapters 4 and 5) we study some cohomology representations of the symmetric group $S_n$. In chapter 4 we study the action of $S_n$ on the complement of the braid arrangement (the so-called pure braid space); we introduce an extended action of the symmetric group $S_{n+1}$ on $n+1$ elements that allows for a simple computation of the character of the action of $S_n$. We also study some properties of $H^*(M(B_n); C)$ as graded $S_n$-module. In chapter 5 we study the action of $S_n$ on the configuration space of $n$ points in $R^d$; it turns out that an argument similar to the case of the pure braid space applies. In particular we build an extended $S_{n+1}$-action and use it to study in detail the action of $S_n$. In the appendices we prove some fundamental results in algebraic topology and group actions that are important for the preceding discussion: the thom isomorphism and the theorem of transfer

Restricted Lie (super)algebras, central extensions of non-associative algebras and some tapas

The general framework of this dissertation is the theory of non-associative algebras. We tackle diverse problems regarding restricted Lie algebras and superalgebras, central extensions of different classes of algebras and crossed modules of Lie superalgebras. Namely, we study the relations between the structural properties of a restricted Lie algebra and those of its lattice of restricted subalgebras; we define a non-abelian tensor product for restricted Lie superalgebras and for graded ideal crossed submodules of a crossed module of Lie superalgebras, and explore their properties from structural, categorical and homological points of view; we employ central extensions to classify nilpotent bicommutative algebras; and we compute central extensions of the associative null-filiform algebras and of axial algebras. Also, we include a final chapter devoted to compare the two main methods (Rabinowitsch's trick and saturation) to introduce negative conditions in the standard procedures of the theory of automated proving and discovery

Foundations of Quantum Theory: From Classical Concepts to Operator Algebras

Quantum physics; Mathematical physics; Matrix theory; Algebr