ABSTRACTS

Anders Björner (KTH Stockholm):
Blockers and Vanishing Ideals of Subspace Arrangements

The blocker of a set family $A$ is the collection of inclusionwise minimal sets that intersect all sets in $A$. This construction is well-known in combinatorics and combinatorial optimization. The corresponding construction on set partitions (and more generally on geometric lattices) arises in the study of vanishing ideals of arrangements of linear subspaces in a vector space.

I will survey examples and properties of blocker duality from a combinatorial viewpoint, beginning with some new results on blocker duality in general posets. I will then describe the relevance of this concept for vanishing ideals that are generated by products of linear forms. The results also touch on some Turan-type problems from extremal combinatorics.

The talk is based on joint work with A. Hultman, I. Peeva and J. Sidman.
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Alexandr Gaifullin (Moscow):
Local Formulae for Pontrjagin Classes of Combinatorial Manifolds

Let $\mathcal T_n$ be the set of all isomorphism classes of oriented $(n-1)$-dimensional PL spheres. By $\mathcal T^n$ denote the Abelian group of all functions $f:\mathcal T_n\to \mathbb{Q}$ such that $f(-L)=-f(L)$ for any $L\in \mathcal{T}_n$, where $-L$ is the PL sphere $L$ with the inverse orientation. Let $K$ be a combinatorial manifold. Consider the cooriented simplicial chain $f_{\sharp}(K)=\sum f(\Lk\Delta)\Delta$, where the sum is over all $(\dim K-n)$-dimensional simplices $\Delta$. A function $f\in\mathcal{T}^n$ is called a local formula if the chain $f_{\sharp}(K)$ is a cycle for any $K$. We say that $f$ is a local formula for a rational characteristic class $p\in H^*(BPL;\mathbb Q)=H^*(BO;\mathbb Q)$ if the Poincar\'e dual of $f_{\sharp}(K)$ represents the cohomology class $p(K)$ for any $K$.

We prove that for any rational characteristic class $p$ there exists a local formula for $p$. We also prove that any local formula is a local formula for some characteristic class $p$. We find explicitly all local formulae for the first Pontrjagin class. We obtain some estimates on denominators of the values of such local formulae. The usage of bistellar moves is very important for our results concerning the explicit formulae for the first Pontrjagin class.

Let $\delta :\mathcal T^n\to \mathcal T^{n+1}$ be given by $(\delta f)(L) = \sum f(\Lk v)$, where the sum is over all vertices $v\in L$. Obviously, $\delta2=0$. Hence $\mathcal{T}^*$ is a cochain complex. A function $f$ is a local formula if and only if $f$ is a cocycle of this complex; $f$ is a coboundary if and only if $f_{\sharp}(K)$ is a boundary for any $K$. We prove that a local formula for a given rational characteristic class is unique up to the addition of a coboundary. Hence we have an isomorphism $H^*(\mathcal{T}^*)\cong H^*(BPL;\mathbb Q)=\mathbb{Q}[p_1,p_2,\ldots ]$.
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Vitaliy Kurlin (U Moscow):
A Programme to Compute Drinfeld Associators

Drinfeld associator is a key tool in computing the Kontsevich integral of knots. Le, Murakami, and Bar-Natan introduced a method to calculate the Kontsevich integral by a tangle decomposition of a given knot. This method uses a Drinfeld associator, namely an infinite series in two non-commutative variables. By definition a Drinfeld associator $\Phi$ is a solution of highly complicated algebraic equations (hexagons and pentagon). Formally, $\log(\Phi)$ lies in the graded completion of the Lie algebra $A_3$ generated by the symbols $t^{12},t^{23},t^{13}$ modulo the relation $[t^{12}+t^{23},t^{13}]=0$. The existence of such a solution was proved by Drinfeld analytically. The coefficients of a Drinfeld associator were expressed via values of multiple zeta-functions by Le and Murakami. Explicit rational expressions of these coefficients are still unknown.

Bar-Natan presented a programme to compute a Drinfeld associator up to degree 7. Unfortunately, this programme is rather complicated, degree 7 is its maximal achievement. We suggest another method leading to the first terms of a Drinfeld associator by using hand computations only. Our programme is to calculate explicitly the image of $\log(\Phi)$ in the quotient $\bar A_3=A_3/[[A_3,A_3],[A_3,A_3]]$. Computing the Kontsevich integral is not our final aim. It is simply a method to evaluate Vassiliev invariants by using known weight systems. Moreover, calculating coefficients of Drinfeld associator will lead to explicit values of multiple zeta functions.
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Gaiane Panina (St. Petersburg, RAS):
On A.D.Alexandrov's Hypothesis, Hyperbolic Polytopes and Hyperbolic Fans

Non-trivial hyperbolic virtual polytopes appeared as an auxiliary construction for various counterexamples to the following A.D.Alexandrov's hypothesis:
Let $K \subset \Bbb R^3$ be a smooth convex body. If for a constant $C$, in each point of $ \partial K$, we have $R_1 \leq C \leq R_2$, then $K$ is a ball. ($R_1$ and $R_2$ stand for the principal curvature radii of $\partial K$).

For a long time mathematicians were certain about correctness of the hypothesis but obtained only some partial results. Recently, Y.Martinez-Maure (2001) has given a counterexample. First, he demonstrated that each smooth hyperbolic herisson generates a desired counterexample. Next, he presented such an example, namely, a smooth hyperbolic surface with four horns (i.e., points where the surface is neither hyperbolic no smooth), given by an explicit formula.

Surprisingly, this counterexample proved to be not unique: a series of counterexamples was given by G.Panina (2003) (hyperbolic herissons with any even number of horns). Some later, it turned out that counterexamples are even more various. Advanced examples (in particular, with even number of horns) were obtained by G.Panina, 2004.

The fans of hyperbolic virtual polytopes have interesting combinatorial properties. The edges of such a fan admit a proper colouring, which encodes important properties of the virtual polytope.For example, a cell of the fan corresponds to a horn of the polytope, if and only if the colour changes two times as going around the cell.
Regular triangulations of hyperbolic fans are of particular interest. They lead to a refinement of A.D. Alexandrov's uniqueness theorem for convex polytopes.
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Arkadiy Skopenkov (Moscow):
Inertia Groups of Smooth Embeddings

This talk is based on my paper "Inertia groups of smooth embeddings"(joint with M. Kreck).
Let $N$ be a closed smooth $n$-manifold. Denote by $Emb^m(N)$ the set of smooth isotopy classes of smooth embeddings $N\to S^m$. For $k\ge3$ the set $C^{m-n}_n:=Emb^m(S^n)$ has an abelian group structure. E. g. Haefliger proved that $C^{m-n}_n=0$ for $2m\ge3n+4$ and $C_43\cong Z_{12}$. The group $C^{m-n}_n$ acts on $Emb^m(N)$ by embedded connected sum of a manifold and a sphere. For $C^{m-n}_n\ne0$ and $N$ distinct from disjoint union of spheres no results on this action and no complete description of $Emb^m(N)$ were known. Our main results are examples of the triviality and the effectivity of this action for embeddings of 4-manifolds into $S7$.
The proof is based on the first author's modification of surgery theory and on invention of a new embeding invariant. As a corollary we obtain the following complete description of $Emb7(N)$.
Let $N$ be a closed connected smooth 4-manifold such that $H_1(N;Z)=0$ and the signature $\sigma(N)$ of $N$ is not divisible by a square of an integer $s\ge2$ (e.g. $N=C P2$). Then there is a 1--1 correspondence $Emb7(N)=\{x\in H_2(N;Z)\ |\ x\mod2=w_2(N),\ x2=\sigma(N)\}.$
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For further information, please contact:

Eva Maria Feichtner Department of Mathematics ETH Zürich 8092 Zürich, Switzerland e-mail: feichtne@math.ethz.ch

Dmitry Kozlov Department of Mathematics KTH Stockholm Stockholm, Sweden e-mail: kozlov@math.kth.se

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last updated: Jun 24, 2004