Related papers: Generic and special constructions of pure O-sequen…

Using linear algebra methods we study certain algebraic properties of monomial rings and matroids. Let I be a monomial ideal in a polynomial ring over an arbitrary field. If the Rees cone of I is quasi-ideal, we express the normalization of…

Brown has shown that the Stanley-Reisner ring of the broken circuit complex of a graph has a linear system of parameters which is defined in terms of the circuits and cocircuits of the graph. Later on Brown and Sagan conjectured a special…

We develop a matrix model for the SO(32) Heterotic string with certain Wilson lines on the lightlike circle. This is done by using appropriate T-dualities. The method works for an infinite number of Wilson lines, but not for all. The matrix…

We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of…

We prove that for $d\geq 3$, the 1-skeleton of any $(d-1)$-dimensional doubly Cohen Macaulay (abbreviated 2-CM) complex is generically $d$-rigid. This implies the following two corollaries (by Kalai and Lee respectively): Barnette's lower…

We review the theory of strange attractors and their bifurcations. All known strange attractors may be subdivided into the following three groups: hyperbolic, pseudo-hyperbolic ones and quasi-attractors. For the first ones the description…

In this paper we show a partial answer the a question of C. Huneke and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity.…

We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such recurrence…

We study a sequence of connections which is associated with a Riemannian metric and an almost symplectic structure on a manifold. We prove that if this sequence is trivial (i.e. constant) or 2-periodic, then the manifold has a canonical…

Recently, a remarkable correspondence has been unveiled between a certain class of ordinary linear differential equations (ODE) and integrable models. In the first part of the report, we survey the results concerning the 2nd order…

Rota's basis conjecture, open since 1989, states that if B_1, B_2, ..., B_n are n bases of a vector space of rank n, then there is an nxn grid of vectors such that the vectors in the ith row are precisely the elements of B_i and such that…

We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we…

We show that the Golod property of a Stanley-Reisner ring can depend on the characteristic of the base field. More precisely, for every finite set $T$ of prime numbers we construct simplicial complexes $\Delta$ and $\Gamma$, such that…

The higher rank numerical ranges of generic matrices are described in terms of the components of their Kippenhahn curves. Cases of tridiagonal (in particular, reciprocal) 2-periodic matrices are treated in more detail.

We present some new results about the resonance varieties of matroids and hyperplane arrangements. Though these have been the objects of ongoing study, most work so far has focussed on cohomological degree 1. We show that certain phenomena…

We study rings of integral modular forms for congruence subgroups as modules over the ring of integral modular forms for the full modular group. In many cases these modules are free or decompose at least into well-understood pieces. We…

We generalize Baker-Bowler's theory of matroids over tracts to orthogonal matroids, define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, circuit sets, and orthogonal vector sets, and…

In this paper, we introduce the concept of $k$-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of $k$-clean ideals, we show…

Levelness and nearly Gorensteinness are well-studied properties of graded rings as a generalized notion of Gorensteinness. In this paper, we compare the strength of these properties. For any Cohen-Macaulay homogeneous affine semigroup ring…

Let $Q_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that $Q_n$ is non-singular with probability…