The arithmetical rank of the edge ideals of graphs with pairwise disjoint cycles

We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes

Homological Invariants of Monomial and Binomial Ideals

In this dissertation, we study numerical invariants of minimal graded free resolutions of homogeneous ideals in a polynomial ring R. Chapters 2, 3 and 4 deal with homological invariants of edge ideals of bipartite graphs. First, in Chapter 2, we relate regularity and depth of bipartite edge ideals to combinatorial invariants of the graphs. Chapter 3 discusses arithmetic rank, and shows that some classes of Cohen-Macaulay bipartite edge ideals define set-theoretic complete intersections. It is known, due to G. Lyubeznik, that arithmetic rank of a square-free monomial ideal I is at least the projective dimension of R/I. As an application of the results in Chapter 2, we show in Chapter 4 that the multiplicity conjectures of J. Herzog, C. Huneke and H. Srinivasan hold for bipartite edge ideals, and that if the conjectured bounds hold with equality, then the ideals are Cohen-Macaulay and has a pure resolution. Chapter 5 describes joint work with G. Caviglia, showing that any upper bound for projective dimension of an ideal supported on N monomials counted with multiplicity is at least 2N/2. We give the example of a binomial ideal, whose projective dimension grows exponentially with respect to the number of monomials appearing in a set of generators. Finally, in Chapter 6, we study Alexander duality, giving an alternate proof of a theorem of K. Yanagawa which states that for a square-free monomial ideal I, R/I has Serre's property (Si) if and only if its Alexander dual has a linear resolution up to homological degree i. Further, if R/I has property (S2) , then it is locally connected in codimension 1

Stronger arithmetic equivalence

Motivated by a recent result of Prasad, we consider three stronger notions of arithmetic equivalence: local integral equivalence, integral equivalence, and solvable equivalence. In addition to having the same Dedekind zeta function (the usual notion of arithmetic equivalence), number fields that are equivalent in any of these stronger senses must have the same class number, and solvable equivalence forces an isomorphism of adele rings. Until recently the only nontrivial example of integral and solvable equivalence arose from a group-theoretic construction of Scott that was exploited by Prasad. Here we provide infinitely many distinct examples of solvable equivalence, including a family that contains Scott's construction as well as an explicit example of degree 96. We also construct examples that address questions of Scott, and of Guralnick and Weiss, and shed some light on a question of Prasad.Comment: Published version; 23 page

Rees algebras, Monomial Subrings and Linear Optimization Problems

In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.Comment: PhD thesis, Cinvestav-IPN, June 201

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Number Theory, Analysis and Geometry: In Memory of Serge Lang

Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life