Minimal weight in union-closed families

Let Omega be a finite set and let S be a set system on Omega. For x in Omega, we denote by d_{S}(x) the number of members of S containing x. A long-standing conjecture of Frankl states that if S is union-closed then d(x) \geq |S|/2 for some x in Omega. We consider a related question. Define the weight of S to be w(S)= \sum_{A in S} |A|. Suppose S is union-closed. How small can w(S) be? Reimer showed that w(S) \geq |S| \log_{2} |S| /2, and that this inequality is sharp. In this paper we show how his bound may be improved if we have some additional information about the domain Omega of S: if S separates the points of Omega, then w(S) \geq \binom{|\Omega|}{2}. This is stronger than Reimer's Theorem when Omega > \sqrt{|S|\log_2 |S|}. In addition we construct a family of examples showing the combined bound on w(S) is tight except in the region |\Omega|=\Theta (\sqrt{|S|\log_2 |S|}), where it may be off by a multiplicative factor of 2. Our proof also gives a lower bound on the average degree: if S is a point-separating union-closed family, then the average degree over its domain is at least 1/2 \sqrt{|S| \log_2 |S|}+ O(1), and this is best possible except for a multiplicative factor of 2.Comment: 16 page

The journey of the union-closed sets conjecture

We survey the state of the union-closed sets conjecture.Comment: Some errors fixed and update

A finiteness theorem on symplectic singularities

For positive integers N and d, there are only finite number of conical symplectic varieties of dimension 2d with maximal weights N, up to isomorphism. The maximal weight of a conical symplectic variety X is, by definition, the maximal weight of the minimal homogeneous generators of the coordinate ring R of X.Comment: Final version, to appear in Compositio Mat

On algebraic properties of matroid polytopes

A toric variety is constructed from a lattice polytope. It is common in algebraic combinatorics to carry this way a notion of an algebraic property from the variety to the polytope. From the combinatorial point of view, one of the most interesting constructions of toric varieties comes from the base polytope of a matroid. Matroid base polytopes and independence polytopes are Cohen--Macaulay. We study two natural stronger algebraic properties -- Gorenstein and smooth. We provide a full classifications of matroids whose independence polytope or base polytope is smooth or Gorenstein. The latter answers to a question raised by Herzog and Hibi

The GIT Compactification of Quintic Threefolds

In this article, we study the geometric invariant theory (GIT) compactification of quintic threefolds. We study singularities, which arise in non-stable quintic threefolds, thus giving a partial description of the stable locus. We also give an explicit description of the boundary components and stratification of the GIT compactification

The Union-Closed Sets Conjecture for Small Families

We prove that the union-closed sets conjecture is true for separating union-closed families $\mathcal{A}$ with $|\mathcal{A}| \leq 2\left(m+\frac{m}{\log_2(m)-\log_2\log_2(m)}\right)$ where $m$ denotes the number of elements in $\mathcal{A}$.Comment: 5 page

The structure of q-W algebras

We suggest two explicit descriptions of the Poisson q-W algebras which are Poisson algebras of regular functions on certain algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. To obtain the first description we introduce certain projection operators which are analogous to the quasi-classical versions of the so-called Zhelobenko and extremal projection operators. As a byproduct we obtain some new formulas for natural coordinates on Bruhat cells in algebraic groups.Comment: 20 page

Symmetrical laws of structure of helicoidally-like biopolymers in the framework of algebraic topology. II. {\alpha}-helix and DNA structures

In the framework of algebraic topology the closed sequence of 4-dimensional polyhedra (algebraic polytopes) was defined. This sequence is started by the polytope {240}, discovered by Coxeter, and is determined by the second coordination sphere of 8-dimensional lattice E8. The second polytope of sequence allows to determine a topologically stable rod substructure that appears during multiplication by a non-crystallographic axis 40/11 of the starting union of 4 tetrahedra with common vertex. When positioning the appropriate atoms tin positions of special symmetry of the staring 4 tetrahedra, such helicoid determines an {\alpha}-helix. The third polytope of sequence allows to determine the helicoidally-like union of rods with 12-fold axis, which can be compare with Z-DNA structures. This model is defined as a local lattice rod packing, contained within a surface of helicoidally similar type, which ensures its topological stability, as well as possibility for it to be transformed into other forms of DNA structures. Formation of such structures corresponds to lifting a configuration degeneracy, and the stability of a state - to existence of a point of bifurcation. Furthermore, in the case of DNA structures, a second "security check" possibly takes place in the form of local lattice (periodic) property using the lattices other than the main ones.Comment: 20 pages, 6 figure

Projective modules over polyhedral semirings

I classify projective modules over idempotent semirings that are free on a monoid. The analysis extends to the case of the semiring of convex, piecewise-affine functions on a polyhedron, for which projective modules correspond to convex families of weight polyhedra for the general linear group.Comment: 27 page

The Chow ring of the Cayley plane

We give a full description of the Chow ring of the complex Cayley plane, the simplest of the exceptional flag varieties. We describe explicitely the most interesting of its Schubert varieties and compute their intersection products. Translating our results in the Borel presentation, i.e. in terms of Weyl group invariants, we are able to compute the degree of the variety of reductions $Y_8$ introduced in our related preprint math.AG/0306328.Comment: 16 page