ABSTRACTS

Sergei Anisov (Utrecht U):
Convex Hulls of Z_p Orbits in R^4, Euclid Algorithm, and Lens Manifolds

Given the p-element cyclic group Z_p acting on $C^2=R^4$ by g(z,w)=(\xi z, \xi^q w) (where g is a generator of Z_p, numbers p and q are coprime, and \xi is a primitive p-th root of unity), we describe the convex hull of a generic orbit. "Genericity" means that z,w are nonzero and 1< q< p-1: otherwise the orbits are flat regular p-gons. We show that this polytope is simplicial (though it is not true that a hyperplane never contains more than 4 of its vertices) and the number of its facets equals p(E(p,q)-3), where E(p,q)=n_1+n_2+...+n_k, where n_1+1/(n_2+1/(...+1/n_k)...) is the continued fraction expansion of p/q. Moreover, the combinatorial type of this polytope can be completely described in terms of the Euclid algorithm and can be visualized in terms of the Farey tesselation of hyperbolic plane.

Eric Babson (U Washington):
The Topology of Graph Maps

I will discuss graph theory for the viewpoint of the poset introduced by Lovasz for every pair of graphs. One aspect of this work involves generalizing the original chromatic number motivation by relating the difference in chromatic number to the topology of the associated poset. This also fits with some work of Brightwell and Winkler in this direction. Another major aspect is more foundational and builds on some easy cases in which limits agree in Graphs and in the topological category. This allows us to introduce graph bundles and a notion of graph homotopy equivalence which hint at a possible model structure for the graph category.
This talk will involove joint work with Kozlov, Kahle and Dochterman.

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.
[for full abstract click here]

Gunnar Carlsson (Stanford U):
Topology of Point Cloud Data

In recent years, there has been an increased interest in techniques for the recovery of topological invariants of a geometric object from a finite but large set of points sampled from it (point cloud data). Such information gives large scale qualitative information about the data set, which can be quite useful in understanding the data set and organizing its analysis. In this talk, we will discuss two of the key ideas which are useful in obtaining homological information, namely persistence and landmarking. We will illustrate these ideas with an analysis of a particular data set coming from natural image data.

Corrado De Concini (U Roma I):
Nested Sets and Residues

Given an arrangement of hyperplanes, we explain how to construct a basis for the homology of the complement using the theory of nested sets. Each cycle in the basis can be represented by an embedded compact torus. This is applied to determine an explicit expression of the so called Jeffrey Kirwan cycles which in turn are used in various contexts such as toric geometry, determination of volumes of polytopes, computations of partition functions.

Ivan Dynnikov (Moscow State U):
Knots in a "Book"

Every knot in three-space admits a deformation after which it is contained entirely in a union of half-planes having common boundary. I will speak about algebraic and combinatorial constructions related to such embeddings, in particular, about finitely presented semigroups whose center classifies links up to ambient isotopy, and an algorithm for recognizing the unknot based on a monotonic simplification procedure.

Torsten Ekedahl (U Stockholm):
On the Length Function on Bruhat Intervals

I shall discuss some results on the length of elements in a Weyl group lying below a given element and show that some of the properties which are true for the whole Weyl group are true in a weakened form for such intervals.

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 $K$.
[for full abstract click here]

Tadeusz Januszkiewicz (Ohio State U):
Weighted $L^2$ Cohomology of Coxeter Groups

For (an infinite) Coxeter group we study weighted $L^2$ cohomology of a space on which a Coxeter group acts.
Motivation comes form $L^2$ cohomology of buildings. These spaces have dimensions related to growth functions of Coxeter groups, and possibly provide interesting invariants for (among other things) convex flag polytopes.
Joint work with M.W.Davis, J. Dymara and B. Okun.

Maxim Kazarian (Moscow):
Application of Global Singularity Theory to the Study of Hurwitz Spaces

Hurwitz spaces are moduli spaces of meromorphic functions on algebraic curves. The study of the cohomology rings of these spaces is closely related to the computation of Gromov-Witten invariants (of projective line as well as of more general manifolds) and to the study of cohomology rings of modula spaces of complex curves. We apply the approach based on the suggested by R. Thom theory of universal polynomials in Chern classes associated with singularities and its generalization to the case of multisingularities found by M. Kazarian. This approach leads to new relations for the cohomology classes of the strata represented by functions with prescribed singularities.
Join work with S. Lando.

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.
[for full abstract click here]

Peter Littelmann (Wuppertal U):
A Pieri-Chevalley Type Formula for the $K$-theory of Flag Varieties

The classical Pieri formula for Grassmann varieties expresses the product of the class of a line bundle with the class of a Schubert variety. Of course, one would like to have a similar kind of formula for the $K$-theory of the Grassmann variety or, more generally, for flag varieties. In the talk we will speak about several different aproaches in the recent years to obtain such a formula, and the connections between these approaches.

Frank Lutz (TU Berlin):
Hom Complexes and Graph Coloring Manifolds

Based on the Hom complex construction of Lovasz, Babson and Kozlov recently provided new tools for obtaining topological lower bounds on the chromatic number of a graph. In this talk, we will discuss how the Hom complex construction can be used to associate a series of graph coloring manifolds with any flag simplicial sphere. For some of the examples that arise this way it has been possible to show that they are spheres, sphere bundles over spheres, cubical surfaces, and others. (Joint work with P?ter Csorba.)

Roy Meshulam (Technion, Haifa):
Laplacians, Homology and Hypergraph Matching

We'll discuss some connections between the expansion constant of a graph and the topology of certain complexes associated with the graph. These results are related to Garland's theorem on the cohomology of p-adic groups. Applications include a lower bound on the homological connectivity of the independence complex, in terms of a new graph domination parameter defined via vector representations of the graph. This in turn implies Hall type theorems for matchings in hypergraphs.
Joint work with R. Aharoni and E. Berger.

Nikolai Mnev (Steklov, St.Petersburg):
On Combinatorial Fiber Bundles

We survey constructions of combinatorial models for the classifying spaces BPL(X), where X is a compact PL-manifold, BPL_n, BPL and related combinatorial fiber bundle theories.

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 full abstract click here]

Taras Panov (Moscow State U):
Some Results and Problems of Toric Topology

Since the pioneering work of Davis and Januszkiewicz, algebraic topologists have been drawn increasingly towards the study of spaces which arise from well-behaved actions of the torus T^n. Investigations are no longer confined to the properties of toric varieties and Davis and Januszkiewicz's toric manifolds, but have extended to related geometrical structures, such as moment-angle complexes, subspace arrangements, and torus manifolds of Hattori and Masuda, as well as the homotopy types of associated spaces and their rationalisations and localisations. We refer to this enlarged field of activity as toric topology.
The talk is based on several ongoing projects with Victor Buchstaber, Mikiya Masuda and Nigel Ray.

Mario Salvetti (U Pisa):
Cohomology with Local Coefficients of Artin and Coxeter Groups

We describe general methods to compute twisted-cohomology of the complement of (real) arrangements of hyperplanes in $C^n$. Such methods turn out to be particularly useful for arrangements associated to Coxeter reflection groups. In this case it is possible to write down (small) resolutions both for Artin and Coxeter groups and perform explicit computations of some interesting cohomology. We also discuss the relation with the cohomology of the Milnor fibre of the arrangement.

Boris Shapiro (U Stockholm):
Polynomial Solutions of Linear Ordinary Differential Equations and Algebraic Curves

We study the properties and the asymptotics of polynomial solutions to a certain kind of linear ordinary differential equations. We show that the asymptotic logarithmic derivative of these solutions satisfies the algebraic equation defined by the symbol of the differential operator. We present a number of conjectures and speculations about the (still essentially non existing) theory of differentials of order $k$ which should generalize the classical theory of quadratic differentials on algebraic curves.

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 $S7$.
[for full abstract click here]

Bernd Sturmfels (UC Berkeley):
The Maximum Likelihood Degree

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and, under suitable hypotheses, it equals the top Chern class of the sheaf of differential forms with logarithmic singularities. Exact formulas in terms of degrees and Newton polytopes are given for polynomials with generic coefficients. This is joint work with Fabrizio Catanese, Serkan Hosten and Amit Khetan.

Tibor Szabo (ETH Zurich):
Matching Transversals and $m$-Connectedness

Let $G$ be a $d$-regular graph with a vertex partition $V(G)=V_1\cup\ldots \cup V_r$. A subset $T$ of the vertices is called a {\em matching transversal} if it intersects each $V_i$ in exactly one vertex and the induced graph $G[T]$ is a (not necessarily perfect) matching.
We can prove that if each part $V_i$ is of size at least $\frac{3}{2}d$, then a matching transversal always exists. Our proof depends on the existence of certain triangulations of the simplex and Sperner's Lemma. On the way we also obtain a sufficient condition for the $m$-connectedness of the so-called {\em induced matching complex} of the graph. Numerous questions remain open, most importantly the constant $\frac{3}{2}$ above is not known to be optimal.
The talk represents joint work with G\'abor Tardos.

Anatoly Vershik (Steklov I.SPb.):
Finite Version of Universality

Remarkable theorem of Hrushevski about extensions of the partial isomorphisms of the graphs can be generalized to many situation like metric spaces, matrices with natural constraints and so on.
I will give a survey based on the recent papers (partially with P.Cameron).

Oleg Viro (U Uppsala):
Knot Homology Theories

I will give a short survey of knot invariants of a new type that appeared recently. Each of them is a collection of homology groups parametrized by two integers ("dimensions") and categorifies one of well-known polynomial knot invariants.

Sergey Yuzvinsky (U Oregon):
Propagation of Cohomology for Hyperplane Complements

We will discuss the problem of understanding the cohomology with local coefficients on complex hyperplane complements and its relations with the cohomology of Orlik-Solomon algebras. In particular we will state the recent result about a propagation of the cohomology and give an idea of a proof.

Rade Zivaljevic (Math. Inst. SANU, Belgrade):
Combinatorics and Topology of Partitions of Mass Distributions; A Singularity Approach

We approach the general problem of partitioning continuous mass distributions or clouds of points in Euclidean spaces by the so called "Configuration space/test map scheme" (Chapter 14 of "Handbook of Discrete and Computational Geometry", Chapman and Hall 2004, eds. E. Goodman, J. O'Rourke). This scheme allows us to relate the geometric problems to the question of the existence of G-equivariant maps $f:M \rightarrow V$ such that $Image(f) \cap A = \emptyset$ , where $V$ is a real G-representation and $A\subset V$ in general the union of an affine subspace arrangement. For a generic f, the subspace (G-submanifold) $f^{-1}(A) \subset M$ , called the singularity of f, is used for effective calculations of relevant equivariant obstructions.
Many of these results were obtained in collaboration with P. Blagojevic, P. Mani-Levitzka, and S. Vrecica (arXiv: math.CO/0310377 and math.AT/0403161).

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: July 1, 2004