Number Theory Day
Freitag, 1.6.18, 00:00-01:00, Basel
Isolated singularities of conformal flat metrics
Freitag, 1.6.18, 14:15-15:15, Raum 127, Ernst-Zermelo-Str. 1
I will talk about isolated singularities of conformal flat (i.e., Gauss curvature is zero) metrics on Riemann surfaces. Under the condition that the area grows at most polynomially near the singularities, there are two possible types of singularities one of which is conical. I will also present a local expression of the metric near a singularity in suitable coordinates.
A general mirror symmetry construction
Montag, 4.6.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
I will talk about joint work with Bernd Siebert, which aims to give a general construction of mirrors to either log Calabi-Yau manifolds or maximal degenerations of Calabi-Yau manifolds. The construction goes by way of building the coordinate ring of the mirror as an abstract ring whose multiplication law is governed by counting curves on the original (log) Calabi-Yau.
TBA
Donnerstag, 7.6.18, 10:00-11:00, Raum 404, Ernst-Zermelo-Str. 1
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Freitag, 8.6.18, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
Volatility estimation for stochastic PDEs using high-frequency observations
Freitag, 8.6.18, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
Motivated by random phenomena in natural science as well as by mathematical finance, stochastic partial differential equations (SPDEs) have been intensively studied during the last fifty years with a main focus on theoretical analytic and probabilistic aspects. Thanks to the exploding number of available data and the fast progress in information technology, SPDE models become nowadays increasingly popular for practitioners, for instance, to model neuronal systems or interest rate fluctuations to give only two examples. Consequently, statistical methods are required to calibrate this class of complex models.\nWe study the parameter estimation for parabolic, linear, second order SPDEs observing a mild solution on a discrete grid in time and space. A high-frequency regime is considered where the mesh of the grid in the time variable goes to zero. Focusing on volatility estimation, we provide an explicit and easy to implement method of moments estimator based on the squared increments of the process. The estimator is consistent and admits a central limit theorem. This is established moreover for the estimation of the integrated volatility in a semi-parametric framework. Starting from a representation of the solution as an infinite factor model and exploiting mixing properties of Gaussian time series, the theory considerably differs from the statistics for semi-martingales literature. The performance of the method is illustrated in a simulation study.\nThis is joint work with Markus Bibinger.
TBA
Freitag, 8.6.18, 14:00-15:00, Hörsaal II, Albertstr. 23b
Quantised dihedral angles and quantum dilogarithms
Montag, 11.6.18, 13:15-14:15, Hörsaal FRIAS, Albertstr. 19
I will describe a relation between quantum dilogarithms and\n3-dimensional hyperbolic geometry obtained by quantising the dihedral\nangles of an ideal hyperbolic tetrahedron with respect to the\nNeumann—Zagier symplectic structure. In this way, one constructs a\n(metaplectic) quantum operator \(Q\) realising the 3-3 Pachner move for\n4-dimensional triangulations. This realisation admits a natural\ngeneralisation to any self-dual locally compact abelian group, together\nwith a fixed gaussian exponential. The 5-term operator identity,\nsatisfied by a quantum dilogarithm over such a group, is equivalent to\nan integral identity involving the operator kernel of \(Q\).
Moduli Spaces of Nonnegatively Curved Riemannian Metrics
Montag, 11.6.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
I will report on general results and questions about spaces and moduli spaces of Riemannian metrics \nwith non-negative Ricci or non-negative sectional curvature on closed and open manifolds,\nand present recent joint work with Michael Wiemeler. In particular, \nwe construct the first classes of manifolds for which these spaces \nhave non-trivial rational homotopy, homology and cohomology groups.
Norm-Resolvent Convergence in Perforated Domains
Dienstag, 12.6.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
TBA
Dienstag, 12.6.18, 16:00-17:00, Hörsaal II, Albertstr. 23b
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Donnerstag, 14.6.18, 10:00-11:00, Raum 119, Ernst-Zermelo-Str. 1
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Freitag, 15.6.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
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Freitag, 15.6.18, 14:00-15:00, Hörsaal II, Albertstr. 23b
Wave equations with initial data on compact Cauchy horizons
Montag, 18.6.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Wave equations are usually studied in globally hyperbolic regions of \nspacetimes. However, to approach the famous strong cosmic censorship \nconjecture, this is not sufficient. One needs to understand the behavior \nof waves close to the boundary of the globally hyperbolic region, the \nCauchy horizon. The purpose of this talk is to discuss the \ncharacteristic Cauchy problem with initial data on a compact Cauchy \nhorizon. We prove an energy estimate close to compact non-degenerate \nCauchy horizons which implies existence and uniqueness results for wave \nequations. In particular, we overcome the essential remaining difficulty \nin proving the Moncrief-Isenberg conjecture in the non-degenerate case. \nThis can be seen as a special case of the strong cosmic censorship \nconjecture.
Heuristic Solvers for Edge Clique Cover Graph Problems Based on Deep Neural Networks
Dienstag, 19.6.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Combinatorial optimization over graphs often presents NP-hard problems, which require considerable manual effort for deriving problem-specific heuristic solvers. Recent research suggests learning of such solvers with deep neural networks and yields high experimental performance on various problems. Yet a theoretical discussion on learnability from a statistical perspective is omitted. We propose traditional feedforward neural networks (FNN) and recurrent neural networks (RNN) that adapt to graph size, in order to learn heuristic solvers for the NP-hard edge clique cover number (ECCN) problem. Both types of architectures are examined in the framework of statistical learning theory, which allows derivation of problem-independent sample complexity bounds for the respective networks. We find that, whereas iterating through all graphs with n vertices takes O(2^{n^2}), FNN-based solvers require at most O(n^3 ln(n)), and RNN-based solvers at most O(n^6) samples to provide reliable heuristic solvers. Experimental evaluation with random graphs on the ECCN problem con firrms a high solution quality, especially of RNNs. On dense graphs, an accuracy of 82.7% is reached, which outperforms the state-of-the-art heuristic from the operations research literature by 15.6 percentage points.
Anwendungen von Typen in der Theorie reeller Körper
Mittwoch, 20.6.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Ein angeordneter Körper heißt archimedisch, falls die Menge der ganzen Zahlen in ihm unbeschränkt ist. Es ist leicht zu sehen, dass diese Eigenschaft nicht erststufig axiomatisierbar ist, weshalb es sich anbietet, schwächere Bedingungen zu studieren. Dies führt auf sehr natürliche Weise zur Sprache von Typen im Sinne der Modelltheorie, und einigen interessanten (und recht subtilen) Fragen zu deren Realisierbarkeit.\n
TBA
Donnerstag, 21.6.18, 10:00-11:00, Raum 404, Ernst-Zermelo-Str. 1
Multi-monopole invariants and Vertex Algebras
Donnerstag, 21.6.18, 14:15-15:15, Raum 414, Ernst-Zermelo-Str. 1
Donnerstag, 21.6.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Dynamical Systems, Topology, and Modular Forms
Donnerstag, 21.6.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
What connects these three subjects? The goal of the talk will be to introduce a function that answers a question in topology, can be computed via methods more common in the theory of dynamical systems, and in the end turns out to enjoy beautiful modular properties of the type first observed by Ramanujan.
Moduli spaces of sheaves on K3 surfaces and irreducible symplectic varieties
Freitag, 22.6.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Irreducible symplectic manifolds are one of the three building blocks of compact K\b"ahler manifolds with numerically trivial canonicl bundle (together with abelian varieties and Calabi-Yau manifolds), thanks to the Beauville-Bogomolov decomposition theorem. A recent result of A. H\b"oring and T. Peternell has completed the extension of this decomposition theorem to singular projective varieties: irreducible symplectic varieties are the singular analogue of irreducible symplectic manifolds, and they are one of the building blocks of normal, projective varieties having canonical singularities and numerically trivial canonical bundle. In a recent joint work with A. Rapagnetta we prove that all moduli spaces of semistable sheaves over projective K3 surfaces (with respect to a generic polarization) are irreducible symplectic varieties, with the only excption of those isomorphic to symmetric products of K3 surfaces, and compute their Beauville form and Fujiki constant. Similar results are shown to hold for the Albanese fiber of moduli spaces of sheaves over Abelian surfaces.
TBA
Freitag, 22.6.18, 14:00-15:00, Hörsaal II, Albertstr. 23b
Equivalence of field theories in the BV-BFV formalism. Insights from General Relativity
Montag, 25.6.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The standard notion of equivalence of field theories roughly requires the Euler-Lagrange loci of the two associated variational problems to be diffeomorphic, possibly modulo the action of the respective symmetry distributions, in some appropriate framework.\nThis can be made stated more precisely in the Batalin-Fradkin-Vilkovisky setting, where some cohomological presentation of said locus is constructed.\nI will discuss a series of examples, all related to General Relativity in different space-time dimensions, that suggest that higher codimension data should play a role in defining (and refining) equivalence between classical theories, and raise the question of whether (and how) this picture carries over to quantisation.
TBA
Dienstag, 26.6.18, 16:00-17:00, Hörsaal II, Albertstr. 23b
A variety that cannot be dominated by one that lifts
Mittwoch, 27.6.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
In the sixties, Serre constructed a smooth projective variety in\ncharacteristic p that cannot be lifted to characteristic 0. If a variety\ndoesn't lift, a natural question is whether some variety related to it does\nlift. We construct an example of a smooth projective variety that cannot be\nrationally dominated by a smooth projective variety that lifts.\n
TBA
Donnerstag, 28.6.18, 10:00-11:00, Raum 404, Ernst-Zermelo-Str. 1
Learning Genetic Architecture of Complex Traits Across Populations
Freitag, 29.6.18, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
Genome-wide association studies (GWAS) have become a standard approach for identifying loci influencing complex traits. However, GWAS in non-European populations are hampered by limited sample sizes and are thus underpowered. We introduce an empirical Bayes approach, which improves the power of mapping trait loci relevant in minority populations through adaptively leveraging multi-ethnic evidence. Likewise, trans-ethnic information can improve genetic risk prediction of traits and diseases. I will discuss how these statistical approaches can be extended to integrate other types of biological knowledge.