A graph G has a C-k-decomposition if its edge set can be partitioned into cycles of length k. We show that if delta (G) >= 2 vertical bar G vertical bar/3 - 1, then G has a C-4-decomposition, and if delta (G) >= vertical bar G vertical bar/2, then G has a C-2k-decomposition, where k is an element of N and k >= 4 (we assume G is large and satisfies necessary divisibility conditions). These minimum degree bounds are best possible and provide exact versions of asymptotic results obtained by Barber, Kuhn, Lo and Osthus. In the process, we obtain asymptotic versions of these results when G is bipartite or satisfies certain expansion properties.

Recently there has been renewed interest in phylogenetic inference methods based on phylogenetic invariants, alongside the related Markov invariants. Broadly speaking, both these approaches give rise to polynomial functions of sequence site patterns that, in expectation value, either vanish for particular evolutionary trees (in the case of phylogenetic invariants) or have well understood transformation properties (in the case of Markov invariants). While both approaches have been valued for their intrinsic mathematical interest, it is not clear how they relate to each other, and to what extent they can be used as practical tools for inference of phylogenetic trees. In this paper, by focusing on the special case of binary sequence data and quartets of taxa, we are able to view these two different polynomial-based approaches within a common framework. To motivate the discussion, we present three desirable statistical properties that we argue any invariant-based phylogenetic method should satisfy: (1) sensible behaviour under reordering of input sequences; (2) stability as the taxa evolve independently according to a Markov process; and (3) explicit dependence on the assumption of a continuous-time process. Motivated by these statistical properties, we develop and explore several new phylogenetic inference methods. In particular, we develop a statistically bias-corrected version of the Markov invariants approach which satisfies all three properties. We also extend previous work by showing that the phylogenetic invariants can be implemented in such a way as to satisfy property (3). A simulation study shows that, in comparison to other methods, our new proposed approach based on bias-corrected Markov invariants is extremely powerful for phylogenetic inference. The binary case is of particular theoretical interest as-in this case only-the Markov invariants can be expressed as linear combinations of the phylogenetic invariants. A wider implication of this is that, for models with more than two states-for example DNA sequence alignments with four-state models-we find that methods which rely on phylogenetic invariants are incapable of satisfying all three of the stated statistical properties. This is because in these cases the relevant Markov invariants belong to a class of polynomials independent from the phylogenetic invariants.

Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the best known bounds on the minimum degree which ensures an edge-decomposition of an r-partite graph into r-cliques (subject to trivially necessary divisibility conditions). The case of triangles translates into the setting of partially completed Latin squares and more generally the case of r-cliques translates into the setting of partially completed mutually orthogonal Latin squares.

Let $F$ be a strictly $k$-balanced $k$-uniform hypergraph with $e(F)\geq |F|-k+1$ and maximum co-degree at least two. The random greedy $F$-free process constructs a maximal $F$-free hypergraph as follows. Consider a random ordering of the hyperedges of the complete $k$-uniform hypergraph $K_n^k$ on $n$ vertices. Start with the empty hypergraph on $n$ vertices. Successively consider the hyperedges $e$ of $K_n^k$ in the given ordering, and add $e$ to the existing hypergraph provided that $e$ does not create a copy of $F$. We show that asymptotically almost surely this process terminates at a hypergraph with $\tilde{O}(n^{k-(|F|-k)/(e(F)-1)})$ hyperedges. This is best possible up to logarithmic factors.

Let n be sufficiently large and suppose that G is a digraph on n vertices where every vertex has in- and outdegree at least n/2. We show that G contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is n/2+1. Our result is best possible and improves on an approximate result by Häggkvist and Thomason.

. Let n be sufficiently large and suppose that G is a digraph on n vertices where every vertex has in-and outdegree at least n/ 2. We show that G contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is n/ 2 + 1. Our result is best possible and improves on an approximate result by H¨aggkvist and Thomason.

We study the mathematical properties of probabilistic processes in which the independent actions of n players (‘causes’) can influence the outcome of each player (‘effects’). In such a setting, each pair of outcomes will generally be statistically correlated, even if the actions of all the players provide a complete causal description of the players' outcomes, and even if we condition on the outcome of any one player's action. This correlation always holds when n=2, but when n=3 there exists a highly symmetric process, recently studied, in which each cause can influence each effect, and yet each pair of effects is probabilistically independent (even upon conditioning on any one cause). We study such symmetric processes in more detail, obtaining a complete classification for all n≥3. Using a variety of mathematical techniques, we describe the geometry and topology of the underlying probability space that allows independence and influence to coexist.

Many of the stochastic models used in inference of phylogenetic trees from biological sequence data have polynomial parameterization maps. The image of such a map --- the collection of joint distributions for a model --- forms the model space. Since the parameterization is polynomial, the Zariski closure of the model space is an algebraic variety which is typically much larger than the model space, but has been usefully studied with algebraic methods. Of ultimate interest, however, is not the full variety, but only the model space. Here we develop complete semialgebraic descriptions of the model space arising from the k-state general Markov model on a tree, with slightly restricted parameters. Our approach depends upon both recently-formulated analogs of Cayley's hyperdeterminant, and the construction of certain quadratic forms from the joint distribution whose positive (semi-)definiteness encodes information about parameter values. We additionally investigate the use of Sturm sequences for obtaining similar results.

We study ideals whose primary decomposition specifies the relevant structural zeros of certain conditional independence models. The ideals we study generalize the class of ideals considered by Fink (2011) [5] in a way distinct from the generalizations of Herzog, Hibi, Hreinsdottir, Kahle, and Rauh (2010) [10] and Ay and Rauh (2011) [1]. We introduce switchable sets to give a combinatorial description of the minimal prime ideals, and for some classes we describe the minimal components. We discuss possible interpretations of the ideals we study, including as 2×2 minors of generic hypermatrices. We also introduce a definition of diagonal monomial orders on generic hypermatrices to compute some Gröbner bases.

We investigate Buchbaum and Eisenbud's construction of the second symmetric power S-R(2) (X) of a chain complex X of modules over a commutative ring R. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following version of a result of Avramov, Buchweitz, and Sega: let R -> S be a module-finite ring homomorphism such that R is noetherian and local, and such that 2 is a unit in R. Let X be a complex of finite rank free S-modules such that X-n = 0 for each n < 0. If boolean OR(n) Ass(R)(H-n (X circle times(S) X)) subset of Ass(R) and if X-p similar or equal to S-p for each p is an element of Ass(R), then X similar or equal to S.

A detailed description of Daniel Erman’s lectures: Boij and Soderberg recently proposed the radical notion that the numerics of graded free resolutions are easier to understand if one works “up to scalar multiplication” [BS08b]. The subsequent proof of their conjectures [EFW08], [ES09], [BS08a], [ES10] represents a breakthrough in our understanding of the structure of graded free resolutions, and has already led to a number of applications. The resulting Boij–Soderberg theory is an exciting subject for a graduate summer school, as it is a new tool which is already important in the study of graded free resolutions. In addition, the theory is so different from previous approaches to studying free resolutions, that there are a large number of opportunities for graduate students to find accessible research problems. The lectures would roughly be organized as follows:

We give a set of multidegrees that support all the numerical information for a monomial ideal that can be reverse searched and hence is parallelizable and has space complexity that is polynomial in the size of the input. Our approach uses a new definition of closed sets for simplicial complexes that may be useful in other contexts.

We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext(R)(n) (B, C) for n >> 0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Sega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext(R)(n) (B, C) = 0 for all n >> 0. We introduce and investigate an equivalence relation approximate to on the set of isomorphism classes of semidualizing complexes. This relation is defined in terms of a natural action of the derived Picard group and is well-suited for the study of semidualizing complexes over nonlocal rings. We identify numerous alternate characterizations of this relation, each of which includes the condition Ext(R)(n) (B, C) = 0 for all n >> 0. Finally, we answer our original question in some special cases.

This article is based on five lectures the author gave during the summer school, Interactions between Homotopy Theory and Algebra, from July 26-August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Grothendieck concerning the connectedness of the spectrum of certain rings. Several applications are given, including a theorem of Fulton and Hansen concerning the connectedness of intersections of algebraic varieties. In an appendix written by Amelia Taylor, an another application is given to prove a theorem of Kalkbrenner and Sturmfels about the reduced initial ideals of prime ideals.

Let R be a polynomial ring over a field with r variables. Let P be a homogeneous ideal of R such that all the variables of R are not zero divisors modP. Assume that the initial ideal of P is strongly stable. It is proven that if the irrelevant ideal is an associated prime ideal of the initial ideal of P, then the ideal generated by the first r-1 variables is also an associated prime ideal of this initial ideal.

The theory of the integral closure of ideals has resisted direct approaches to some of its basic questions (membership and completeness tests, and constructed). We mainly treat the membership problem in the monomial case by exploiting the connection with multiplicities and its linkage to the computation of volumes of polyhedra. We discuss several existent software packages and introduce our own contribution, a Monte Carlo based approach to the computation of volumes. Finally, we make comparisons of multiplicities of general ideals and of their initial ideals.

Our main purpose is to give multiple examples for using the available implementations for computing the normalization of an affine ring, computing the minimial generators of the normalization as an algebra over the original ring and integral closures of ideals. Some such examples have been published for Singular, but not for Macaulay 2 and we present both in this paper. We also briefly describe the implementations.

We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated prime ideal of dimension one larger.

Generic linkage is used to compute a prime ideal such that the radical of the initial ideal of the prime ideal is equal to the radical of a given codimension two monomial ideal that has a Cohen-Macaulay quotient ring.

We investigate a construction of the second symmetric power S 2 R (X) of a chain complex X of modules over a commutative ring R. Our construction has the advantage of being relatively straightforward to define, as it is the cokernel of a certain morphism X ⊗ R X → X ⊗ R X, defined for any R-complex. We prove that, when 2 is a unit in R, our construction respects homotopy equivalences. We explicitly describe the modules occurring in S 2 R (X), and this description allows us to characterize the complexes X for which S 2 R (X) is trivial or has finite projective dimension. Finally, we provide several explicit computations and examples; for instance, we show that our construction differs from others in the literature.