Special Session: Schedule and abstracts

Susan Williams: Mahler measures of Alexander polynomials

The Mahler measure of a d-variable polynomial is the geometric mean of its modulus over the multiplicative d-torus obtained by restricting each variable to the unit circle. When d=1, this is simply the absolute value of the leading coefficient times the product of the moduli of the roots that lie outside the unit circle. D.H. Lehmer asked in 1933 if an integer polynomial can have Mahler measure arbitrarily close to, but greater than, 1. The nearest known value occurs as the Mahler measure of the Alexander polynomial of a knot. The Mahler measure of the Alexander polynomial has topological significance, and links for which this value is small have intriguing geometric properties.

Daniel Silver: Links and Lehmer's Question

Lehmer's Question asks whether the product of the moduli of the roots of a monic integral polynomial can be arbitrarily close to but different from 1. The question, which remains unanswered despite much effort, is equivalent to one about fibered links. We discuss this approach and likely connections with hyperbolic 3-manifolds.

Igor Nikolaev: New arithmetic invariants of hyperbolic 3-manifolds

Hyperbolic 3-manifold $M$ can often be presented as surface bundle over the circle with a pseudo-Anosov monodromy $\varphi$. The action of $\varphi$ on fibre $S$ fixes a lamination ${\cal L}$ on $S$, as it was demonstrated by W.~P.~Thurston. We introduce the notion of ``slope'' $\theta\in {\Bbb R}$ of ${\cal L}$ on $S$ and show that $\theta$ is a quadratic irrationality, given by periodic continued fraction. The number field $K=Q(\theta)$ absorbs critical data on geometry, topology and combinatorics of manifold $M$. Our theory extends Reidemeister-Conway classification of the rational knots and tangles on the one hand, and complements Bianchi's theory of hyperbolic orbifolds over imaginary quadratic number fields, to the other.

Mark de Cataldo: The Hodge theory of algebraic maps

Report on joint work with L. Migliorini at Bologna where we introduce new structures on the singular cohomology of projective varieties.

Tom Braden: Toric Koszul Duality

We describe a "Koszul duality" functor on the derived category level between constructible sheaves with unipotent monodromy on an affine toric variety associated to a cone $\sigma$, and $T$-equivariant sheaves on the toric variety associated to the dual cone $\sigma^\vee$. It shares the same formal properties as the Koszul duality defined by Beilinson, Ginzburg, and Soergel for sheaves on flag varieties; in particular, it takes simple perverse sheaves to projectives and vice-versa. We describe some combinatorial consequences of this duality, and speculate on its relation with mirror symmetry.

David Massey: Milnor Monodromy in the Perverse Category

The Milnor fiber is of fundamental importance in the study of singular complex analytic hypersurfaces. However, even in the case where the hypersurface has a one-dimensional singular set, there is no known general procedure for calculating the cohomology groups of the Milnor fiber. In fact, as one varies the hypersurface, it is unknown which graded groups can possibly appear as the cohomology of the Milnor fibers . Certainly, there are a number of well-known restrictions on such Milnor graded groups. In this talk, we will describe how the additional structure provided by the category of perverse sheaves with a monodromy action places further restrictions on the cohomology of Milnor fibers.

David Ben-Zvi: Calogero-Moser and KP systems

We discuss relations between moduli spaces of D-modules on curves and integrable systems. In particular we will explain the relation between the Calogero-Moser many-body systems and the KP hierarchy.

Ruth Kellerhals: On the growth of hyperbolic Coxeter groups

After a short survey of the two dimensional case, we look at various aspects of the growth exponents for Coxeter groups acting by reflections on hyperbolic space of dimension bigger than or equal to three. We discuss arithmetical properties of the associated Poincar\'e series in function of the combinatorial-topological properties of the Poincar\'e fundamental polytope.

Chris Leininger: On groups generated by two positive multi-twists

Following Thurston, we study subgroups of the mapping class group generated by two positive multi-twists. We classify the configurations of curves for which the corresponding groups exhibit certain exceptional behaviors. We also identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number, and show that this is minimal for the groups under consideration. Connections with Coxeter groups, billiards, and knot theory are also observed.

Jianqiang Zhao: q-Analogs of Multiple Zeta Functions and Multiple Polylogarithms

We shall define the q-analogs of multiple zeta functions and multiple polylogarithms in this talk and study their properties, based on the work of Kaneko et al. and Schlesinger, respectively. In particular we will consider two kinds of shuffle relations resulted from the series expansions and the iterated integral representations, respectively.

David Metzler: Presentation of orbifolds and equivariant algebraic topology

We present some questions in equivariant algebraic topology arising out of the question of presenting a noneffective orbifold as a quotient of a manifold by an action of a compact Lie group. We will also note the relationship between these questions and the Brauer conjecture in algebraic geometry.

Daniel Matei: Formality of Complex Hypersurface Complements

Let X be the complement of a complex projective hypersurface S. From works of Deligne, Griffiths, Morgan, Sullivan and Kohno the following are known: The higher Massey products in the rational cohomology of X vanish. Moreover, the Malcev Lie algebra of the fundamental group of X it is isomorphic to the completed holonomy Lie algebra of the rational homology of X. In particular, if S has a smooth hyperplane section at infinity then X it is rationally formal. We show that in general X is not formal over prime fields Z/p by exhibiting non-vanishing Massey products in the Z/p-cohomology of X. We also investigate to what extent various Z/p-Lie algebras associated to the fundamental group of X are determined by the Z/p-holonomy Lie algebra of X.

Gian Mario Besana: The dimension of the Hilbert scheme of special threefolds

The classification of complex projective manifolds of a fixed,low degree, has been conducted in the recent past as a three-step process. Maximal lists of possible manifolds are first compiled, according to the admissible values of their numerical invariants. The second step deals with establishing the actual existence of manifolds in the lists, looking for effective constructions of explicit examples. Finally, the Hilbert scheme of existing manifolds with given Hilbert polynomial is investigated. In this work, the Hilbert scheme of 3-folds in $\mathbb{P}^N,$ $N\ge 6,$ that are scrolls over the projective plane or over a smooth quadric surface, or that are quadric or cubic fibrations over the projective line is studied. All known such threefolds of degree $7 \le d \le 11$ are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed. Some of the known examples are {\it good determinantal schemes} according to a definition due to Kleppe, Migliore, Miro Roig, Nagel and Peterson. A Zariski open subset of the component of the Hilbert scheme we investigate turns out to coincide with the locus of good determinantal varieties studied by these authors.