Abstracts

Abstract: In 1970s, T. Jorgensen characterized in his famous unfinished paper the combinatorial structures of the Ford domains of quasifuchsian punctured torus groups. In the paper he used the "geometric continuity" method which is roughly the method to characterize all possible changes in combinatorial structure while a group gradually deforms.
In this talk, I will explain the Jorgensen's work briefly and then two variations of it. One is to extend the method to the outside of the space of discrete groups to obtain the complete hyperbolic structures on hyperbolic two-bridge knots, which is done by my joint work with M. Sakuma, M. Wada and Y. Yamashita. The other is an attempt to study the Ford domains of hyperbolic structures on a manifold, which has a pair of punctured tori as boundary but is not the product of the punctured torus and the interval, by using the geometric continuity method.

Abstract: In this paper we showed that in order to a surface to be lightlike it is necessary that one of the parameter curves is null curve and the other is spacelike curve. Moreover, lightlike ruled surfaces in $\mathbb{R}_{1}^3=(\mathbb{R}^{3},dx^{2}+dy^{2}-dz^{2})$ are studied with respect to whether ruling curves are spacelike or null. It is seen that in the first case the Gaussian curvature of the ruled surfaces vanishes. In the second case the Gaussian curvature of the ruled surfaces is negative. In the second case lightlike ruled surfaces are totally umbilical. Furthermore, lightlike surfaces of revolution are shown to be only cones and the second case lightlike ruled surface.

Abstract: Coauthor: Mario Eudave-Muñoz
We are interested in knowing what type of manifolds are obtained by doing Dehn surgery on closed pure 3-braids in $S^3$. In particular, we want to determine when we get $S^3$ by surgery on such a link. In a previous paper we showed that for some families of closed pure 3-braids, most surgeries produce Haken manifolds (Topology Appl. 131, (2003), 255-272). Now we consider links which are small closed pure 3-braids; these are the closure of 3-braids of the form $(a^n)(b^m)(bab)^p$, where a, b are the generators of the 3-braid group and n, m, p are even integers. We study Dehn surgeries on these links, and determine exactly which links admit an integral surgery producing the 3-sphere. This problem is equivalent to determining which surgeries on a certain six components link L produce $S^3$. The link L is strongly invertible and its exterior double branch covers a certain configuration of arcs and spheres, which we call the Hexatangle. Now we have to determine which fillings of the spheres by rational tangles produce the trivial knot. This hexatangle is a generalization of the Pentangle, which is studied by Gordon and Luecke in a recent paper (Comm. Anal. Geom. 12 (2004), 417-485).

Abstract: There has been a continuous feedback between low dimensional geometric topology and geometric problems coming from algebraic and analytic geometry. It is a natural question to give a topological classification of embeddings of algebraic curves in the complex projective plane. The first obstruction in order to check if two curves have homeomorphic embeddings is of combinatorial type and it is explained using algebraic links and Waldhausen graph manifolds. It is known that this combinatorial obstruction is not complete and two curves which are combinatorially but not topologically equivalent form a so-called Zariski pair. In this talk we will treat the history, techniques and examples of Zariski pairs.

Abstract: Let us consider a category of embedded 1-dimensional objects like braids, links, tangles etc. There is a natural filtration on the free Z-module generated by the objects, coming from the singular objects with a given number of double points. The associated grading, which is called the diagrams algebra, can be explicitly computed and one defined universal finite type invariant a map from our category into some completion of the diagrams algebra, which induces an isomorphism at the graded level. For instance the celebrated Kontsevich integral is such a universal invariant with rational coefficients. Some years ago Papadima defined a universal finite type invariant with integer coefficient for the braid group $B_n$ in an algebraical way using the structure of pure braid groups.This work was thereafter extended to surface braids by Gonzalez-Meneses and Paris. However, the invariant proposed by Papadima is not known to be multiplicative (i.e. if it induces an isomorphism of Z-algebras at the graded level) and in the case of surface braid groups the author recently proved the non existence of such an invariant. The multiplicativity is an essential condition for extending the invariant from braids to links. In this talk we will give a survey on finite type invariants and generalised braids and we will introduce a new notion of finite type invariants on the so called braid-permutation groups which restreint on usual braids coincides with the usual definition of finite type invariants. Finally we will construct a universal multiplicative invariant for braid permutation groups.

Abstract: In this talk we will discuss some questions related to the presentation of a 3-manifold as cyclic coverings of the 3-sphere branched along a knot. In particular we will show that an integral homology sphere is homeomorphic to the 3-sphere iff it is a cyclic branched covering of $S^3$ of degree p for four pairwise distinct odd prime integers p. This result is sharp. We will discuss some possible generalisation. This is a joint work with Luisa Paoluzzi (Univ. Bourgogne) and Bruno Zimmermann (Univ. Trieste).

Abstract: In 1990 Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph G a polynomial, denoted [G], in three variables, A, B and a, which coincides, under the change of variables z = A - B , with the two-variable Kauffman polynomial when restricted to links. In this talk we will use a simplified version of this polynomial (fixing $B=A^{-1}$ and $a=A$) and show that for a planar graph G we have $[G]=2^{c-1}(-A-A^{-1})^v$, where c is the number of connected components of G and v is the number of vertices of G. Thus we get a good test for the planarity of spatial four-valent graphs.

Abstract: Let $X_{0}^{+}(N)$ be the Atkin-Lehner quotient of the modular curve $X_{0}(N)$ associated to the Fricke involution wN. Assume N > 5 prime and endow the real locus $X_{0}^{+}(N)(\mathbb{R})$ with the real topology. In this talk we revisit a special case of a result due to Ogg on the connected components of $X_{0}^{+}(N)(\mathbb{R})$. Then we obtain a formula for the homology class of each connected component of $X_{0}^{+}(N)(\mathbb{R})$ in terms of Manin symbols.

Abstract: For a homogeneous surface in 3-dimensional unite sphere, there is a lower bound for the first non-zero eigenvalue of the Laplace operator if this surface satisfies some special conditiones. We state and prove these conditions.

Abstract: We say that a knot k in the 3-sphere $S^3$ is a (1,n)-knot, if n is the minimal number of bridges of the knot with respect to a standardly embedded torus T. It is well known that if k is a (1,1) -knot, then it has tunnel number one; but a tunnel number one knot could have high bridge number with respect to a torus. Examples have been given of tunnel number one knots which are (1,2) -knots (Morimoto and Sakuma, Eudave-Muñoz, Johnson, Ramirez-Losada and Valdez-Sanchez), and have been asked by several mathematicians if there exist tunnel number one knots which are (1,n) -knots, with n > 2 . Here we give the first examples of tunnel number one knots which are not (1,1) or (1,2) -knots. To do that, we study incompressible surfaces in the complement of (1,2) -knots.

We give several constructions which produce closed meridionally incompressible surfaces of genus 2 in the complement of some (1,2) -knots. Then we show that if in the complement of a (1,2) -knot k there is a closed meridionally incompressible surface S of genus 2 , then S and k have to be produced by one of the given constructions. Then we show explicit examples of tunnel number one knots which contain a closed meridionally incompressible surface of genus 2 , and by analyzing the pieces in which these surfaces divide $S^3$, we concluded that these cannot be produced by the given constructions, so these knots are not (1,2) -knots. By a previous result of the author, these knots cannot be (1,1) -knots. So such knots must have bridge number 3 or bigger with respect to a torus.

Abstract: We prove that if M is a CW-complex and * is a 0-cell of M, then the crossed module $\Pi_2(M,M^1,*)$ does not depend on the cellular decomposition of M up to free products with $\Pi_2(D^2,S^1,*)$, where $M^1$ is the 1-skeleton of M. From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms $\Pi_2(M,M^1,*) \to G$ (which is finite) can be re-scaled to a homotopy invariant $I_G(M)$ (i. e. not dependent on the cellular decomposition of M.) We describe an algorithm to calculate $\pi_2(M,M^{(1)},*)$ as a crossed module over $\pi_1(M^{(1)},*)$, in the case when M is the complement of a knotted surface in $S^4$ and $M^{(1)}$ is the 1-handlebody of a handle decomposition of M, which, in particular, gives a method to calculate the algebraic 2-type of M. In addition, we prove that the invariant $I_G$ yields a non-trivial, and very calculable, invariant of knotted surfaces, even in one of its simplest forms.

Abstract: I present my work math.GT/0401345, where the relative Seiberg-Witten (SW) and Ozsvath-Szabo (OS) invariants, for surfaces in 4-manifolds, were studied. C. Taubes previously introduced such relative invariants for tori, and we consider the case of higher genus. Refining the classical, "absolute" SW and OS invariants, the relative invariants have a similar package of properties, which may look more natural in the relative case. The product formula looks as a usual product of polynomials (similar to the genus 1 case of Taubes), and the adjunction inequality that estimates genus of membranes on a given surface, has the most simple, "classical" form, without positivity assumption on the self-intersection of a membrane. As a consequence, we obtain minimality of symplectic and Lagrangian membranes (on Lagrangian and respectively symplectic surfaces), which seems to be a new observation.

Abstract: A surface with nodes X is hyperelliptic if there exists an involution h : X → X such that the genus of $X/\left\langle h\right\rangle $ is 0. We prove that this definition is equivalent, as in the category of surfaces without nodes, to the existence of a degree 2 morphism π : X → Y satisfying an additional condition where the genus of Y is 0. Other question is if the hyperelliptic involution is unique or not. We shall prove that the hyperelliptic involution is unique in the case of stable Riemann surfaces but is not unique in the case of Klein surfaces with nodes. Finally, we shall prove that a complex double of a hyperelliptic Klein surface with nodes could not be hyperelliptic.

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Ernesto Girondo

On a conjecture of Whittaker on the uniformization of hyperelliptic

Abstract: The unknotting number u(K) of a knot K is one of the oldest knot invariants, but it remains mysterious. We will survey some results on u(K), particularly those that come from the connection, established by Montesinos, between u(K) and Dehn surgery on the double branched cover of K. This includes some recent joint work with John Luecke on algebraic knots (in the sense of Conway) that have unknotting number 1.

Abstract: We will present a joint work with A. Ananin [1]. The main result of this work asserts that there exists a complex hyperbolic structure on a trivial disc bundle over a closed oriented surface of genus 2. This answers a long standing problem in complex hyperbolic geometry, see [2-3].
Reference:
1. A. Ananin, N. Gusevskii, Complex hyperbolic structures on disc bundles over surfaces. II. Example of a trivial bundle, preprint, 2005.
2. W. M. Goldman, Conformally flat manifolds with nilpotent holonomy and the uniformization problem for 3-manifolds, Trans. Amer. Math. Soc. 278, 1983, no. 2, 573-583.
3. Ya. Eliashberg, Contact 3-manifolds twenty years since J. Martinets work, Ann. Inst. Fourier (Grenoble) 42, 1992, no. 1-2, 165-192.

Abstract: The structure of the moduli space $\mathcal{M}_{g,n}$ (for low values of the genus) exhibits many different forms of geometric structure, some of them closely related to complex homogeneous spaces. I shall attempt to present an interesting sample.

Abstract: We describe those extended Kleinian groups whose index two orientation preserving half is a Schottky group.

Abstract: In this talk, we will present some topological properties of wild knots that are limit sets of conformal kleinian groups acting on $\mathbb{S}^{3}$. We show that, if the ``original knot'' fibers over the circle then the wild knot $\Lambda$ also fibers over the circle. As a consequence, the universal covering of $\mathcal{S}^{3}-\Lambda$ is $\mathcal{R}^{3}$. We prove that the complement of a dynamically-defined fibered wild knot can not be a complete hyperbolic 3-manifold. We also prove that this type of wild knots are homogeneous: given two points $p,\,\,q\in{K}$ there exists a homeomorphism $f$ of the sphere such that $f(K)=K$ and $f(p)=q$. We also show that if the wild knot is a fibered knot then we can choose an $f$ which preserves the fibers.

Abstract: Hurwitz groups are the finite groups attaining the upper bound of 84(g-1) for the number of automorphisms of a compact Riemann surface of genus g > 1; equivalently, they are the nontrivial finite images of the (2,3,7) triangle group, which is the even subgroup of the rank 3 Coxeter group [3,7], a group of isometries of the hyperbolic plane. It appears that the analogous groups in dimension 3 are the finite quotients of the rank 4 Coxeter group [3,5,3], which acts on hyperbolic 3-space. Many of these quotients are (or are closely related to) finite simple groups. I shall describe joint work with Alexander Mednykh and with Cormac Long on the finite quotients of this and some other rank 4 Coxeter groups, and on the 3-manifolds (such as the Poincarè dodecahedral space and the Weber-Seifert space) associated with them.

Abstract: Virtual knot theory is an extension of classical knot theory with both a combinatorial and a topological intepretation. Topologically a virtual knot is an embedding of a circle into a thickend surface taken up to ambient isotopy, homeomorphisms of the surface and stabilization by empty handle addition and subtraction (a handle is empty if the knot does not go through it). Combinatorially, virtual knot theory is the theory of oriented Gauss codes (or diagrams) taken up to the Reidemeister moves. The diagrams need not connote codes that have planar embeddings as classical knot diagrams. As a result, one can represent virtual knots by classical diagrams augmented by extra crossings that we call virtual crossings. These "virtual diagrams" can be used for a complete combinatorial description of the theory. This talk will discuss different invariants of virtual knots, including the fundamental group, the quandle, the Jones polynomial, quantum invariants, Vassiliev invariants and Khovanov homology. There are many interesting examples in this theory including infinitely many non-trivial virtual knots with unit Jones polynomial. It is an open problem whether any of these examples are classical.

Abstract: In this talk we settle down a Thurston's conjecture regarding convergence of freely decomposable Kleinian groups. He conjectured that Schottky groups converge algebraically if conformal structures at infinity converge to a projective measured lamination in Masur domain. We give a generalized solution to his conjecture. This is a joint work with C. Lecuire and K. Ohshika.

Abstract: A compact Riemann surface X is said to be doubly symmetric if it admits two antiholomorphic involutions with fixed points ( symmetries in short). By the classical result of Harnack the set of fixed points of a symmetry consists of at most g+1 ovals and if the symmetry has g+1-q ovals then we shall call it an (M-q)-symmetry. We remaind the bound of Bujalance-Costa-Singerman for the total number of ovals of two symmetries of a Riemann surface in terms of its genus g and the order of their product and we made its further study. We also find its version which takes into account the number of the fixed points of the product. We conclude that (M-q)- and (M-q')-symmetries of a Riemann surface of genus g commute for g ≥ q+q'+1 and furthermore we show that actually, with just one exception in any genus, q+q'+1 is the the minimal lower bound for g which guarantees their commutativity. Bujalance and Costa have found the upper bound for the degree of hyperellipticity of the product of commuting (M-q)- and (M-q')-symmetries. For g ≥ q+q'+1 we find necessary and sufficient conditions for an integer p to be such degree, taking into account separabilities of the symmetries.

Abstract: The talk is based on the paper published in Journal of Algebra 292 (2005),184-242. The Kontsevich integral is a powerful knot invariant, but it is hard to compute. A Drinfeld associator is an algebraic ingredient for computing a closed expression of the Konstevich integral starting from a plane diagram of a given knot. By definition the logarithm of a Drinfeld associator lives in the simple Lie algebra L generated by a, b, c modulo [a,b] = [b,c] = [c,a] and satisfies highly complicated algebraic equations (pentagon and hexagon) involving 5 and 6 exponentials. The author described explicitly all compressed Drinfeld associators in the quotient L/[[L,L], [L,L]] over commutators of commutators. The key tool is a computable compressed version of Campbell-Hausdorff formula, which is applicable for solving exponential equations in other cases.

Abstract: We study and attempt to classify Riemann Surfaces that admit an automorphism with just one fixed point. If time permits, we should extend this to automorphisms with few fixed points.

Abstract: For three-dimensional homology cobordisms, M. Goussarov and K. Habiro have conjectured that the information carried by finite-type invariants should be characterized in terms of ``cut-and-paste'' operations defined by the lower central series of the Torelli group of a surface. In this talk, we explain that this conjecture is an instance of a classical problem in group theory, namely the ``dimension subgroup problem.'' This observation allows us to prove, by purely algebraic methods, weak versions of the Goussarov-Habiro conjecture.

Abstract: We describe a general approach for evaluation of volume for cone-manifolds and polyhedra in spaces of constant curvature. In the Euclidean space the Tartaglia formula for the volume of tetrahedron is well known. Recently, I. Kh. Sabitov suggested a recursive algorithm to find the volume formula for an arbitrary Euclidean polyhedron. In the hyperbolic and spherical spaces the main results in this direction were obtained in the papers by Mike Hilden, Maria Teresa Lozano and José Maria Montesinos. Two dimensional background for general theory of orbifolds and conemanifolds in hyperbolic space was developed by David Singerman, his students and collaborators. The starting point of our consideration is based on trigonometric identities between dihedral angles and edge lengths of polyhedron. Then the Schläfli formula is applied to find explicit integral formulae for the volume. We use this approach to calculate the volume of tetrahedron, octahedron, cube and other polyhedra. In a similar way the hyperbolic and spherical volumes of knot and link orbifolds and cone-manifolds can be obtained. A new kind of the Santaló type formula relating volume of spherical polyhedron, the volume of its dual and their shared middle curvature is also established.
Supported by INTAS (grant 03-51-3663), RFBR (grant 06-01-00153) and by Fondecyt (grants 7050189, 1060378)

Abstract: The boundary of a complex normal surface germ is a Waldhausen 3-manifold and its plumbing graph is well known. Let M(f) be the boundary of the Milnor fiber of a holomorphic germ f ,where f is defined at the origin of the 3-dimensional complex space. If f has a non-singular locus,we show ( in commun with A.Pichon) that M(f) is also a Waldhausen 3-manifold. We provide germs f such that M(f) is not the boundary of any normal surface germ.

Abstract: Satellites of a knot have proved very useful in extending the range of recent combinatorially-based knot invariants. Skein theory leads to a description of them using the linear skein of the annulus as a parameter space. The meridian map is an endomorphism of this linear space induced by placing an extra meridian loop around any diagram in the annulus. The eigenvalues of this map for the Homfly skein all occur with multiplicity 1. The corresponding eigenvectors, depending on a choice of two partitions, form a natural basis in many constructions of knot and manifold invariants, and they play a key role in the transition between the quantum SL(N,q) invariants of a knot and its Homfly invariants. I shall give an account of some of the simple skein theoretic features of the eigenvectors, and their resulting properties and uses.

Abstract: It is well-known that for any reciprocal Laurent polynomial f(t) with |f(1)|=1 there exists a knot whose Alexander polynomial is f(t). In this talk, we construct a knot of unknotting number one via a braidzel surface with a given Alexander polynomial. A braidzel surface is a kind of Seifert surface for a knot. It is introduced by L. Rudolph as a generalization of pretzel surfaces. We also show that this knot is fibered with a braidzel surface as a fiber if a given Alexander polynomial is monic. We also construct infinitely many fibered knots with the same Alexander polynomial from our knots according to Morton's construction. We show that all of them is different by HOMFLY polynomial.

Abstract: A seamed surface is a two-dimensional complex, appearing in modern mathematical physics. We extend the definition of Hurwitz numbers on seamed surfaces and we prove that these Hurwitz numbers form a system of correlators for a Klein topological field theory. We describe also a structure algebra corresponding to it. Non-trivial part of this algebra is a new associative algebra of bichromatic graphs. We prove that this algebra is isomorphic to the algebra of intertwining operators for the representation of symmetric group $S_n$ in the set of all partition of n elements to batches.

Abstract: The talk concerns the following particular case of the Hurwitz existence problem: for a given set $\Pi$ of partitions of a number $n$ to define whether there exists a $n$-fold covering of sphere by sphere for which $\Pi$ is the branch datum. For the case when $\Pi$ contains a partition consisting of asingle element the answer was given by R. Thom. In this talk we peresent the solution in the case when one of partitions contains two elements.

Abstract: This talk concerns joint work with Peter Buser. For a Riemann surface endowed with a hyperbolic metric, J. Birman and C. Series have shown that the set of all points lying on any simple closed geodesic is nowhere dense on the surface. (This set is sometimes referred to as the Birman-Series set). The talk will discuss the existence of a positive constant $C_g$, such that for any surface of genus g, the complementary region to the Birman-Series set allows an isometrically embedded disk with radius $C_g$. The behavior of $C_g$ in function of g, as well as some bounds will also be discussed.

Abstract: A TQFT is a monoidal functor between the category of cobordism (2+1) and the category of projective A module (with A an unitary and commutative ring or a field). It has bee studied the case of framing cobordism. In this case we consider manifold with additional structure ($spin$ or $spin_c$). We can consider the triangulation on a manifold, this approach leads to another category which is triangulate cobordism and thus we construct a monoidal functor on this category. This construction leads to triangulate TQFT, which are in one to one correspondence with the TQFT. This general construction leads to the construction of Wakui (in the case of finite group) and to the construction of Gelfand-Kazhadan, Turaev (in the case of category).

Abstract: Matveev's definition of the complexity c(M) of a (closed, irreducible) 3-manifold M gives a very natural measure of how complicated M is. In addition, c has very nice properties, including additivity under connected sum. This talk will describe variations of the definition of c which apply to the objects listed in the title. Along with the discussion of some properties of this extended notion of c, the talk will mention a subtlety which arises when trying to reproduce for 3-orbifolds the proof of Haken, Kneser and Milnor of existence and uniqueness of the splitting along spheres of a 3-manifold into irreducible ones. If time permits an account will also be given of work in progress with Hodgson and Pervova on computer tabulation of objects as listed in the title in increasing order of complexity.

Abstract: In 1989 Shigeyuki Morita gave a new construction of the Casson invariant. His construction amounted basically to show that the intersection form of a surface defines a trivial 2-cocycle on the Torelli group of a genus g surface. A well-chosen trivialization of this cocycle plus a correcting term was shown to coincide point-wise with the Casson invariant viewed as a function on the Torelli groups. Recall that the Torelli group is the kernel of the action of the mapping class group on the first homology of the underlying surface.
In this talk we will give a general procedure to construct invariants of homology spheres out of trivial cocycles on the Torelli groups. There is a unique obstruction for a trivial cocycles to have an invariant as a trivializing function. It is shown to be a cohomology class living in the first cohomology group of a precise subgroup of the mapping class group with coefficients in the universal torsion-free abelian quotient of the Torelli group.
If one applies this procedure to pull-backs of 2-cocycles defined on the universal torsion-free abelian quotient of the Torelli group then there is up to a multiplicative constant a unique candidate $J_{g}$ to produce an invariant. An elementary computation shows that the aforementioned obstruction for the cocycle $2J_{g}$ vanishes. We prove from its definition that the invariant thus obtained satisfies the surgery formulas that carachterize the Casson invariant. The presence of the factor 2 reflects the fact that the Casson invariant is an integral lift of the Rohlin invariant.

Abstract: The algebra of truncated polynomials $A_m = Z[x]/(x^m)$ plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free graph homology. It is not difficult to compute Hochschild homology of $A_m$ and the only torsion, $Z_m$, appears in grading $(i,\frac{m(i+1)}{2})$ for any odd i. We analyze here grading of graph cohomology which is producing torsion for a polygon. We find completely the cohomology $H^{1,v-1}_{A_2}(G)$ and $H^{1,2v-3}_{A_3}(G)$. In particular, we show that $H_0^{A_2}(G)$ is a ``symmetric" homology of a graph, that is the boundary of an edge is equal the sum of its endpoints.

Abstract: Which types of lens spaces are obtained by non-trivial Dehn surgeries on non-trivial knots? This is one of the most important problems on Dehn surgeries. We note that it is completely solved by Moser for Dehn surgeries on the torus knots. For Dehn surgeries on non-torus knots, Berge introduced the concept of doubly primitive knots and gave a certain integral surgery, which is a Dehn surgery along a ``surface slope'' of K, to obtain a lens space from any doubly primitive knot. For example, Fintushel and Stern have shown that 18-surgery on the (-2,3,7)-pretzel knot yields a lens space L(18,5). It is known that the (-2,3,7)-pretzel knot is hyperbolic and doubly primitive, and the surgery coefficient is determined by its surface slope. It is conjectured by Gordon that if a lens space is obtained by a non-trivial Dehn surgery on a non-torus knot, then the knot would be a doubly primitive knot and the surgery would be determined by its surface slope.
Also, there is a conjecture by Bleiler and Litherland: it would be impossible to obtain a lens space L(p,q) with |p| < 18 by a Dehn surgery on a non-torus knot. The speaker has proved that if Gordon's conjecture is true, then this is also true.
In this talk, we give a partial answer to the following question asked by Kazuhiro Ichihara: is it impossible to obtain a lens space L(p,q) with |q| < 5 by a Dehn surgery on a non-torus knot? In fact, we prove that if Gordon's conjecture is true, then this is positive. To this end, we completely list up all such lens spaces with |q| < 5 and prove that they are obtained only by torus knots.

Abstract: A knotted surface in 4-space is called a surface-knot, which is described by a diagram through a projection of 4-space onto 3-space. Such a diagram is regarded as a disjoint union of compact connected surfaces called sheets. The sheet number of a surface-knot is the minimal number of sheets for all possible diagrams of the surface-knot. We give a lower bound of the sheet number by using the quandle cocycle invariant of a surface-knot, and prove that the sheet number of the 2- and 3-twist-spun trefoils are equal to 4 and 5, respectively.

Abstract: Euclidean tilings of surfaces have aroused great interest in recent years. The study of hyperbolic tilings on the other hand, has mostly been concerned by triangular tilings. The aim of this talk is to show why tilings by hyperbolic quadrangles are of great interest. This will be done by considering in detail some examples.

Abstract: We study the action of the mapping class group M(F) on the complex of curves of a non-orientable surface F. By using a result of Brown, we obtain a presentation for M(F) defined in terms of the mapping class groups of the complementary surfaces of collections of curves, provided that F is not sporadic, i.e. the complex of curves of F is simply connected. We also compute a finite presentation for the mapping class group of each sporadic surface.

Abstract: A compact Riemann surface X of genus g ≥ 2 is said to be p-hyperelliptic if X admits a conformal involution &rho, called a p-hyperelliptic involution, for which X/ρ is an orbifold of genus p. If in addition X admits a q-hypereliptic involution then we say that X is pq-hyperelliptic. Here we give a necessary and sufficient conditions for p,q and g to exist a pq-hyperelliptic Riemann surface of genus g. We give also some conditions under which p- and q-hyprelliptic involutions of a Riemann surface commute or are unique. These results can be enlarge to Klein surfaces and here the necessary conditions turns out to be also sufficient for the existence of surfaces with commuting p- and q-hyperelliptic involutions. We describe topological type of such surfaces.

Abstract: Via "rational-fold branched coverings", spatial graphs in the 3-sphere can be constructed from strongly invertible knots. In this talk, the following question will be discussed: from which knots do hyperbolic spatial graphs appear? This is a joint work with Kazuhiro Ichihara at Osaka Sangyo University, Japan.

Abstract: A smooth foliation of the 5-sphere by complex surfaces will be described as well as it space of deformations a la Kuranishi, Kodaira-Spencer. This is joint work with Laurent Meersseman (Universite de Bourgogne).

Abstract: The talk will be about a yet unproved "conjecture of Tait" which says roughly that any negatively achiral alternating knot has a minimal diagram on which the symmetry is accomplished by an involution. This is joint work with Cam Van Quach Hongler.

Abstract: Hurwitz curves are the compact Riemann surfaces of genus g > 1 with the - according to Hurwitz - maximal possible order 84(g-1) . For long time, the only known example was Klein's quartic in genus 3 . Macbeath found in the 60th a series of examples whose automorphism groups are special linear groups over finite fields. The talk will give a survey on three new results found during the last ten years:
(1) Galois orbits and fields of definition of the Macbeath-Hurwitz curves (M. Streit),
(2) Construction of their surface groups as congruence subgroups in a quaternion algebra (A. Dzambic),
(3) Upper and lower bounds for the number of all (isomorphism classes of) Hurwitz curves of genera < g (J.-Chr. Schlage-Puchta and J.W.).

Abstract: A closed Riemann surface x which can be realised as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism.

Accola (1984) showed that a p-gonal Riemann surface of genus greater than $(p-1)^2$ admits one unique p-gonal morphism. In this talk we will show that this bound is sharp and we will show the existence of a punctured Riemann surface of genus $(p-1)^2$ with two different trigonal morphisms. This is a joint work with Antonio Costa and Milagros Izquierdo.

Abstract: Consider the surface braid group $B_{n}(M)$ of n strings as a normal subgroup of the isotopy group $G(M,\widetilde{n})$ of homeomorphisms of the surface M permuting n fixed distinguished points. A classical theorem of Nielsen says that $Out(\pi _{1}(M))=Out(B_{1}(M))$ is isomorphic to G(M) if M is a closed surface which is not a sphere. Exploiting presentations of $B_{n}(M)$ and the pure braid group $SB_{n}(M)$, we investigate a generalization of Nielsen's theorem, that is, whether an automorphism of $B_{n}(M)$ or $SB_{n}(M)$ is geometric in the sense that it is induced by a homeomorphism of M in the form of conjugate action. Here are some of our results.