Winter Term 2014/2015
Description
In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
Speakers
All seminars will take place the seminar room in the Villa, Arnimallee 2, 14195 Berlin.
| Speaker | Date | Time | Title and Abstract |
|---|---|---|---|
| Aldo Guzmán Sáenz (Cinvestav, MX) | Oct 22, 2014 | 5 PM s.t. | Title: The cohomology ring away from 2 of F(RP^n,k). Abstract: In this talk we present a computation of the cohomology ring of the configuration space of k ordered points in the n-th dimensional real projective space. The strategy used for this computation allows us to compute the cohomology ring of ordered configurations in the punctured n-th dimensional real projective space. |
(U Bonn) | Oct 29, 2014 | 5:15 PM | Title: On the homology of Sullivan Diagrams. Abstract: In string topology one studies the algebraic structures of the chains of the free loop space of a manifold by defining operations on them. Recent results show that these operations are parametrized by certain graph complexes that compute the homology of compatifications of the Moduli space of Riemann surfaces. Finding non-trivial homology classes of these compactifications is related to finding non-trivial string operations. However, the homology of these complexes is largely unknown. In this talk I will describe one of these complexes: the chain complex of Sullivan diagrams. In the genus zero case, I'll give a reinterpretation of it in terms of weighted partitions, give some computational results, and if time permits, I'll describe work in progress that suggests that these are highly connected. This talk is based on joint work with F. Lutz. |
(IITP Moscow) | Nov 05, 2014 | 5PM s.t. | Title: Flexible polyhedra and their volumes. Abstract: Consider a closed polyhedral surface in the Euclidean three-space with rigid faces and with hinges at edges. If this polyhedral surface admits a deformation (a flexion) that is not induced by an ambient rotation of the space, then it is called a flexible polyhedron. A definition of a flexible polyhedron in an n-dimensional Euclidean space is completely similar. A problem on existence of flexible polyhedra turned out to be rather non-trivial. First examples of flexible polyhedra are Bricard's self-intersected flexible octahedra (1897). However, the first example of an embedded (i.e., non-self-intersected) flexible polyhedron was constructed by Connelly only in 1977. One of the most amazing facts in the theory of flexible polyhedra is Sabitov's theorem claiming that the volume of an arbitrary flexible polyhedron in the three-dimensional Euclidean space is constant during the flexion. (Earlier this assertion was known as the Bellows Conjecture.) The talk will contain a survey of recent results by the speaker concerning flexible polyhedra, and the main ideas behind these results. This will include: (1) The constructions of self-intersected flexible cross-polytopes in Euclidean and Lobachevsky spaces of all dimensions, and of embedded flexible cross-polytopes in spheres of all dimensions. In dimensions 5 and higher these are the first known examples of flexible polyhedra and in dimensions 4 and higher these are the first known examples of embedded flexible polyhedra. (2) The proof of the analogue of the Bellows Conjecture in the Euclidean spaces of arbitrary dimensions. (3) The proof of the analogue of the Bellows Conjecture in odd-dimensional Lobachevsky spaces, and the counterexamples to the Bellows Conjecture in spheres of all dimensions. (4) Results on the polyhedral relations among the entries of the Gram matrix of the period vectors for a doubly-periodic two-dimensional polyhedral surface in the three-dimensional Euclidean space. |
| -- | Nov 12, 2014 | (no talk) | |
(FU Berlin) | Nov 19, 2014 | 5PM s.t. | Title: Obstacles for splitting multidimensional necklaces Abstract: "The well-known “necklace splitting theorem” of Alon asserts that every k-colored necklace can be fairly split into q parts using at most t cuts, provided k(q − 1) ≤ t. In a joint paper with Alon et al. we studied a kind of opposite question. Namely, for which values of k and t there is ameasurable k-coloring of the real line such that no interval has a fair splitting into 2 parts with at most t cuts? We proved that k > t + 2 is a sufficientcondition (while k > t is necessary).We generalize this result to Euclidean spaces of arbitrary dimension d, and to arbitrary number of parts q. We prove that if k(q − 1) > t + d + q − 1,then there is a measurable k-coloring of R^d such that no axis-aligned cube has a fair q-splitting using at most t axis-aligned hyperplane cuts. Our bound is of the same order as a necessary condition k(q − 1) > t. Moreover, for d = 1, q = 2 we get exactly the previous result. Additionally, we prove that if a stronger inequality k(q−1) > dt+d+q−1 is satisfied, then there is a measurable k-coloring of R^d with no axis-alignedcube having a fair q-splitting using at most t arbitrary hyperplane cuts. The proofs are based on the topological Baire category theorem and use algebraic independence over suitable chosen fields." |
(TU Berlin) | Nov 26, 2014 | 5PM s.t. | Title: Constraining with equivariant maps: Barycenters of polytope skeleta Abstract: By the van Kampen−Flores theorem the d-skeleton of the (2d+2)-simplex is not embeddable into Euclidean space of dimension 2d. The closely related Conway−Gordon−Sachs theorem states that for any embedding of the complete graph on six vertices in 3-space there are two vertex-disjoint linked triangles. Recently, Dobbins proved that any point in an nd-polytope is the barycenter of n points in the d-faces of the polytope. We will give simple, mostly combinatorial proofs of these three results that all build on the same idea: constraining points in a cell complex to an appropriate subcomplex by exploiting symmetry and the pigeonhole principle. Moreover, we will remark on generalizations. This is joint work with Pavle V. M. Blagojević and Günter M. Ziegler. |
(HU Berlin) | Dec 17, 2014 | 5PM s.t. | Title: Wonderful Renormalization Abstract: This talk is about how so-called wonderful compactifications can be used to solve an extension problem for distributions appearing in quantum field theory. This extension problem is one variant of what physicists call renormalization, a collective term for various ways of extracting physical sensible quantities out of a priori ill-defined integrals arising in perturbative calculations. Roughly speaking, physics associates to a given graph G (representing an element in the perturbative expansion of some physical quantity) a distribution that is defined only outside of a subspace arrangement determined by G; renormalization then amounts to extending this distribution onto this arrangement - this is where wonderful compactifications enter the game as a way to systematically reduce the problem to a toy model case. These compactifications were first introducedby DeConcini and Procesi in the case of linear arrangements, based on ideas from Fulton and MacPherson's famous article "A Compactification of Configuration Spaces". What makes them so well-suited for this extension problem is the fact that both the wonderful construction as well as renormalization in general, are governed by the underlying combinatorial structure. This structure is encoded in a certain subset of the poset of all subgraphs of G and allows to describe the problem's solution in purely combinatorial terms (once some initial data is fixed). I will quickly sketch how the problem emerges in physics and describe its solution. Then I show how this geometric/combinatorial approach allows us to study the ambiguity of extensions obtained in this way. This leads to the renormalization group, a powerful tool that even allows for statements beyond perturbation theory (i.e. about the "real world"). |
Organizers
| Contact | |
| Pavle Blagojević | blagojevic(at)math.fu-berlin.de |
| Holger Reich | holger.reich(at)fu-berlin.de |
| Elmar Vogt | vogt(at)math.fu-berlin.de |
| Günter M. Ziegler | ziegler(at)math.fu-berlin.de |
