Abstracts
Fuji (Kyoto)
Hyperbolic Geometry and Open Topological Strings
In this talk, I will discuss the relation between topological invarinants of knot compliment space and the partition function of open topological string based on the volume conjecture.
Hikami (Tokyo)
From the Quantum Dilogarithm Function to the A-polynomial
In recent studies (e.g. volume conjecture) towards geometrical interpretation of quantum invariants, the A-polynomial plays an important role. I will explain how to compute the A-polynomial based on the quantum dilogarithm function.
Kohnert (Bayreuth, Germany)
Constructive methods for the computation of Schubert polynomials
In this talk I will first give a combinatorial description of Schubert polynomials connected to the symmetric group. This description is a nice generalization of the known description of the spezial case of Schur polynomials. I will also show how this decription can be generalized to the case of two sets of variables. In the sceond part of my lecture I will show how this combinatorial description can be used to compute with Schubert polynomials but also how it can be used to derive certain properties of Schubert polynomials. In the last part I will show which algorithms are available in the public domain software package SYMMETRICA.
Lu (Adelaide)
The gamma-hat genus, multiple zeta values and the equivariant cohomology of the free loop space
In recent years, the gamma function has appeared as the generating function for a multiplicative genus, in the sense of Hirzebruch. This genus, unlike the classical A-hat-genus (in the Atiyah-Singer index theorem) and L-genus (in Hirzebruch's signature theorem), takes values involving multiple zeta values. In this talk, we describe the derivation of a new, but closely related, genus from considerations in the equivariant cohomology of the free loop space. We show how this derivation can be viewed from the perspective of recent developments in the study of the algebraic structure of multiple zeta values.
Maillard (Paris 6, France)
Towards a geometrical interpretation of Ising model n-fold integrals and the theory of elliptic curves
We recall Ising model n-fold integrals corresponding, respectively, to the $n$-particle contribution of the susceptibility, the (lattice) form factors, and the correlation functions, of the isotropic square Ising model. We also recall the Fuchsian linear differential equations we found on these various n-fold integrals, as well as the russian-doll and direct sum structures we discovered on these differential operators. We then show that these differential operators are very selected Fuchsian linear differential operators, and that these remarkable properties have a deep geometrical origin. The central role played by the theory of elliptic curves, modular curves, and complex multiplication of elliptic curves will also be underlined.
Molev (Sydney)
Littlewood-Richardson polynomials
We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood-Richardson polynomials. We give a combinatorial rule for their calculation which is manifestly positive in the sense of W. Graham. We apply this rule for the calculation of the product of equivariant Schubert classes on Grassmannians and describe the multiplication rule in the algebra of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants.
Norbury (Melbourne)
Counting lattice points in the moduli space of curves
I will describe how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus $g$ curves with $n$ labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space.
Rubinstein (Melbourne)
Hyperplane arrangements and learning theory
Concept classes in learning theory are sets of binary strings, considered as subsets of a large binary cube. The Vapnik Chervonenkis (VC) dimension of a class is the dimension of the largest coordinate cube, so that coordinate projection of the class maps onto this smaller cube. A concept class is learnable if and only if it has bounded VC dimension. Learnable classes are those for which the information from a sample determines the class up to an error which converges to zero at a fast rate as the sample size increases. An important problem is to determine if learnability is equivalent to having a bounded size compression scheme - so that instead of needing the large number of labels of the binary strings, only a bounded amount of information is required for each string.
In studying this problem, only largest classes with a given VC dimension need to be considered. Such a class is called maximum if it has the most vertices with VC dimension d. Otherwise it is called maximal. We show that maximum classes correspond to certain arrangements of hyperplanes in hyperbolic space of the same dimension as the binary cube. By sweeping across hyperbolic space, using a generic hyperplane, we prove that all maximum classes have bounded size compression schemes, This result had been established previously, but our argument gives several important additional properties of compression schemes, including new information on the complexity of dealing with the remaining maximal classes. It also suggests new ways of thinking about hyperplane arrangements.
Wildberger (UNSW)
A new look at hyperbolic geometry
Hyperbolic geometry can be developed purely algebraically. This simplifies the basics of the subject, as well as many formulas, allows one to contemplate the outside of the light cone on the same footing as the inside, thereby bringing duality into the picutre and clarifying many theorems that otherwise don't really work. The Cayley Klein model becomes the crucial one, and the connections with projective geometry come to the fore. And everything works over a general field, not of characteristic two.
Yonezawa (Nagoya)
Matrix factorization and a categorification of MOY link invariant
M.Khovanov and L.Rozansky constructed a categorification of HOMFLY polynomial which is the quantum link invariant associated with U_q(sl_n) and its n-dimensional vector representation. The natural question is can we categorify the other quantum link invariants. I will talk about a idea of how to construct a categorification of the quantum link invariant associated with U_q(sl_n) and its fundamental representations called MOY link invariant.