# Program

## Thursday, September 4, 2014

Time | Name | Title |
---|---|---|

Coffee | ||

11:15 - 12:00 | Fabien Friedli | Spectral zeta functions for graphs and the Riemann zeta function |

12:15 - 13:00 | Roberto Castellini | On deformations of singular curves |

Lunch break | ||

14:30 - 15:15 | Claudiu Valculescu | Discrete geometry from an algebraic point of view |

15:30 - 16:15 | Harry Schmidt | Relative Manin-Mumford in additive extensions |

Coffee break | ||

16:35 - 17:20 | Filip Misev | Cutting and glueing fibre surfaces |

19:45 | Conference dinner |

The colloquium dinner will take place on Thursday night at 19:45 in the restaurant Parterre, Klybeckstrasse 1B.

## Friday, September 5, 2014

Time | Name | Title |
---|---|---|

9:30 - 10:15 | Chidambaram Annamalai | Combinatorial Algorithm for Restricted Max-Min Fair Allocation |

Coffee break | ||

10:35 - 11:15 | Hanspeter Kraft | How to write a research proposal? |

11:30 - 12:15 | Manuela Utzinger | What cool things wavelets can be used for and why it is good to use the stuff adaptively |

Lunch break | ||

14:00 - 14:45 | Kay Werndli | Homotopy Excision and Cellularity |

15:00 - 15:45 | Thibault Pillon | Schur's Lemma, from unitary representations to affine isometric actions |

Coffee break | Vote for best talks and awards ceremony |

## Abstracts of the talks

Fabien Friedli

Spectral zeta functions for graphs and the Riemann zeta function

In this talk, I will explain how to define a spectral zeta function associated to a graph and discuss the special cases of Z and of the cycle with n vertices. I will show how these functions are related to each other and how we can deduce information about Riemann zeta function from this relation.

Based on joint work with A. Karlsson.

Roberto Castellini

On deformations of singular curves

Let C be a germ of a complex algebraic plan curve such that O is an isolated singular point. Two germs C and C' are equi-singular if the associated analytic functions are equivalent by a cascade blow analytic sequence, a notion developed for real germs, which, for complex ones, is equivalent to having isomorphic resolution spaces. For complex curves there are many different invariants, like the Eggers-Wall tree, the Enriques tree and the dual graph. Such invariants can be seen all together in a new invariant, the kite of a singularity.

In a paper published in 1973, A'Campo used deformations of complex curves to study monodromies, that is deformations defined by deforming the germ on a exceptional component and then contracting it. This method is principally graphical, therefore it is difficult to understand the topological types of such deformations. It can also be used to understand deformations of real germs.

In my thesis I developed a method for understanding such deformations. In the talk I will introduce some results I obtained about them and about their topological types.

The motivation for this research is to give a primary step in the understanding of deformations of singular locus of algebraic varieties.

Claudiu Valculescu

Discrete geometry from an algebraic point of view

The talk will outline some applications of elementary algebraic geometry in discrete geometry. This topic reached a peak with the solution that Guth and Katz gave in 2010 for the long-standing distinct distance problem in the plane, posed by Erdos in 1946. We introduce the general subject, present a theorem of Szemeredi and Trotter on incidences between points and lines and an argument of Elekes about sum sets and product sets. We conclude the talk by sketching the proof of the following result in this area: Let S be a finite set of n points on a constant degree algebraic curve C in the complex plane and a bilinear function B(p,q) of points p,q in the complex plane. Then, considering the asymptotic behaviour with respect to n, B takes at least n^{4/3} distinct values on S, unless C has a special form or B is degenerate. This is joint work with Frank de Zeeuw.

Harry Schmidt

Relative Manin-Mumford in additive extensions

We will discuss the relative Manin-Mumford conjecture for families of two dimensional commutative algebraic groups. These will depend on one complex parameter λ and we are especially interested in the case of an additive extension of the Legendre family E_{λ}. We then have an exact sequence

0 → G_{a} → G_{λ} → E_{λ} → 0

where G_{a} is the additive group (C,+). In this context the relative Manin-Mumford conjecture states that the intersection of a curve in G_{λ} with the set of torsion points is at most finite unless it is contained in a smaller family of algebraic subgroups in G_{λ}.

It is possible to prove this by following the strategy employed by Masser and Zannier in their proof of the relative Manin-Mumford conjecture for the product of two Legendre families.

Filip Misev

Cutting and glueing fibre surfaces

A knot K can be studied via the embedded surfaces whose boundary is K. We will explain what it means for such a surface to be a "fibre surface" and how to make new fibre surfaces out of old.

Chidambaram Annamalai

Combinatorial Algorithm for Restricted Max-Min Fair Allocation

We study the basic allocation problem of assigning resources to players so as to maximize fairness. This is one of the few natural problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, a certain configuration-LP can be used to estimate the value of the optimal allocation to within a factor of 4 + ε. In contrast, however, the best known approximation algorithm for the problem has an unspecified large constant guarantee. In this paper we significantly narrow this gap by giving a 13-approximation algorithm for the problem. Our approach develops a local search technique introduced by Haxell [Hax95] for hypergraph matchings, and later used in this context by Asadpour, Feige, and Saberi [AFS12]. For our local search procedure to terminate in polynomial time, we introduce several new ideas such as lazy updates and greedy players. Besides the improved approximation guarantee, the highlight of our approach is that it is purely combinatorial and uses the configuration-LP only in the analysis.

Hanspeter Kraft

How to write a research proposal?

Manuela Utzinger

What cool things wavelets can be used for and why it is good to use the stuff adaptively

In many applications in natural sciences and engineering one comes across partial differential equations (PDEs). Since we do not know an exact solution to most of them, we need to have efficient methods at hand to be able to calculate a numerical solution. Living in a three-dimensional world, we need to discretize the three-dimensional domain of interest. This can be done for example by using the finite element method (FEM), leading - among other problems - to sparse but extremely large linear systems (curse of dimensionality). Some PDEs can then be reformulated as boundary integral equations, leading to a problem which is then to be solved on the domains boundary (BEM - boundary element method). This drasticially reduces the dimensionality of the problem but of course does not come without a certain cost: The appearing matrices become full, thus the cost for solving the linear system increases considerably. That is where the wavelets come in. By choosing those special basis functions, the matrix can be compressed, meaning that many matrix entries can be neglectad without compromising the accuracy. For geomerties with edges and corners the solution admits singularities. In order to resolve these singularities, we need to choose a very high level of refinement in certain areas of the domain. Since we do not want to invest that much memory and/or computational power for uniform mesh refinement, we are in need of an adaptive approach. Once these theoretical aspects are introduced, some numerical examples will be shown in order to validate the method.

Kay Werndli

Homotopy Excision and Cellularity

The main reason homology groups are more calculable than homotopy groups -- other than being abelian groups in all dimensions -- lies in the existence of a long exact sequence in homology, assignable to any quotient of spaces (homotopy theorists like calling these “cofibre sequences”). Homotopy groups have long exact sequences as well but these involve fibres (or again more grandiosely: “fibre sequences”) instead of quotients. The homotopy excision theorem of Blakers-Massey makes possible the building of a cut-off long exact sequence in homotopy from a cofibre sequence, allowing the reasoning about homotopy groups, at least within a certain range. We will see how this important classical theorem fits into the modern framework of Bousfield classes, which studies the localisation of categories (the category of spaces in our case) at some object and the spaces that become contractible under it; in analogy with elements that become invertible when localising a ring.

Thibault Pillon

Schur's Lemma, from unitary representations to affine isometric actions

Representation theory is a fundamental tool that spans mostly all areas of mathematics : Finite groups theory, Harmonic analysis, Geometry, Number theory and more. In this talk, I'll introduce the notion of a representation and the essential concept of irreducibility. Schur's Lemma states that a unitary representation is irreducible if and only if its commutant is the simplest, thus bridging a geometric concept to an algebraic one. I will give insights into the proof of the Lemma and will show how to produce a similar statement in the case of affine isometric actions. This theorem is a joint work with B.Bekka and A. Valette.

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