Self-Adjoint Fredholm Operators in K-Theory
Montag, 3.7.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk we introduce certain ways to think about the group K^1(X):\nFor a compact space X, the group K^1(X) is the Grothendieck group of the monoid of fnite rank complex vector bundles over the suspension SX. \nThere are several classifying spaces for the gorup K^1(X). We study here two of them: the infinite dimensional unitary group and the space of selfadjoint fredholm operator. Both of them are connected via the Cayley transform. \nFinally we will consider the kernel dimension of self-adjoint fredholm operators. Using the Fredholm operators as classifying space of K^1 we get an obstruction for high kernel dimension.
Asymptotic behavior of discrete Laplacian
Dienstag, 4.7.17, 10:30-11:30, Raum 226, Hermann-Herder-Str. 10
This thesis is motivated by the problem - how to define a quantum\nfield theory rigorously. As one of the most successful theories in 20th\ncentury, quantum field theory agrees with experiments to a very high\nprecision. However, how to make this theory mathematically complete\nis still an open question. One approach is to consider discrete versions\nof the theory and then study the limit as the lattice spacing approaches\nto zero. In this thesis, we take such an approach for free scalar field\ntheory. In particular, we study the partition function by analyzing the\nasymptotic behavior of discrete partition function. We show that the\nlogarithm of the zeta regularized determinant of continuous Laplacian,\nwhich can be interpreted as the logarithm of the partition function of\nthe continuous theory, is contained in the asymptotic expansion of log-determinant\nof the discrete Laplacian.
Globale Existenz schwacher Lösungen für die Interaktion eines Newtonschen Fluides mit einer linearen, transversalen Koiter-Schale unter natürlichen Randbedingungen
Dienstag, 4.7.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Bio-inspired materials research: Tapping the wondrous world of plant structures and functions
Donnerstag, 6.7.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Modules as exact functors
Donnerstag, 6.7.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
A module - a linear representation of a ring (or other object) - can\noccur as a representation of many different rings (under Morita, or more\ngenerally tilting, equivalence, for example). This can be seen as\nchoosing a different generator for an abelian category canonically\nassociated to the module. From the point of view of model theory it is\nchoosing a different home sort in the associated category of\nimaginaries. Through this we are led to an alternative view of what a\nmodule is, which I will illustrate with some examples and applications.\n
A decomposition theorem for the pushforwards of pluricanonical bundles to abelian varieties
Freitag, 7.7.17, 10:15-11:15, Raum 404, Eckerstr. 1
I will describe a direct-sum decomposition in pull-backs of ample sheaves for the pushforwards of pluricanonical bundles via morphisms from smooth projective varieties to an abelian varieties. The techniques to proving this decomposition rely on generic vanishing theory, and the use of semipositive singular hermitian metrics. Time permitting, I will provide an application of the above decomposition towards the global generation and very ampleness properties of pluricanonical divisors defined on singular varieties of general type. The talk is based on a recent joint work with M. Popa and C. Schnell.
Compactification of a moduli space of lattice polarized K3 surfaces
Montag, 10.7.17, 16:15-17:15, Raum 404, Eckerstr. 1
After a short introduction to hermitian symmetric spaces, I will explain the classical statement that the moduli space of complex elliptic curves is isomorphic to the Siegel modular variety of genus 1. In analogy Clingher and Doran proved that the moduli space of certain lattice polarized K3 surfaces is isomorphic to the Siegel modular variety of genus 2. Finally I will introduce a compactification for this moduli space and show that it is isomorphic to the Baily-Borel compactification of the Siegel modular \nvariety of genus 2.
Ancient pancakes
Dienstag, 11.7.17, 16:00-17:00, Raum 404, Eckerstr. 1
We will show how to construct a compact, convex ancient solution of mean curvature flow which lies in a slab region of \(\bmathbb{R}^3\) (of width \(\bpi\)) and prove unique asymptotics for such solutions: The maximum of the mean curvature is close to one and the `edge' regions are close to grim planes (of width \(\bpi\)) when \(t\) is close to minus infinity. This is joint work with Theodora Bourni (FU Berlin) and Giuseppe Tinaglia (King's College London).
On a theorem of Campana and Paun
Freitag, 14.7.17, 10:15-11:15, Raum 404, Eckerstr. 1
Let X be a smooth projective variety over the complex numbers, D a divisor with normal crossings, and consider the bundle of log one-forms on (X, D). I will explain a slightly simplified proof for the following theorem by Campana and Paun: If some tensor power of the bundle of log one-forms on (X, D) contains a subsheaf with big determinant, then (X, D) is of log general type. This result is a key step in the proof of Viehweg's hyperbolicity conjecture.\n\n
Forensic DNA Phenotyping
Freitag, 14.7.17, 12:00-13:00, Raum 404, Eckerstr. 1
Forensic DNA Phenotyping (FDP) is a relatively new development in the field of forensic genetics. It aims at predicting selected so-called externally visible characteristics (EVCs) of a trace donor from their DNA as left behind at the crime scene. The best results for FDP were achieved for eye colour where the IrisPlex DNA test system was developed (Walsh et al. 2011), which includes six SNPs in six different genes, and was found to obtain relatively high levels of prediction. The second best predictable EVC after eye colour is hair colour.\n In the first part of this talk, results of a study investigating the prediction of the pigmentation phenotypes eye, hair and skin colour in a Northern German population will be presented (Caliebe et al. 2016). With this study, we aimed at answering the following research questions: (1) do existing models allow good prediction of high-quality phenotypes in a genetically similar albeit more homogeneous population? (2) Would a model specifically set up for the more homogeneous population perform notably better than existing models? (3) Can the number of markers included in existing models be reduced without compromising their predictive capability in the more homogenous population?\nIn the second part of the talk we differentiate FDP from trace donor identification problems. In the latter, it has become widely accepted in forensics that, owing to a lack of sensible priors, the evidential value of matching DNA profiles is most sensibly communicated in the form of a likelihood ratio (LR). This agreement is not in contradiction to the fact that the posterior odds (PO) would be the preferred basis for returning a verdict. A completely different situation holds for FDP. The statistical models underlying FDP typically yield PO for an individual possessing a certain EVC. This apparent discrepancy has led to confusion as to when LR or PO is the appropriate outcome of forensic DNA analysis to be communicated. We thus set out to clarify the distinction between LR and PO in the context of forensic DNA profiling and FDP from a statistical point of view (Caliebe et al. 2017). \nCaliebe, A., M. Harder, R. Schuett, M. Krawczak, A. Nebel and N. von Wurmb-Schwark, 2016. The more the merrier? How a few SNPs predict pigmentation phenotypes in the Northern German population. Eur. J. Hum. Genet. 24: 739-747.\nCaliebe, A., S. Walsh, F. Liu, M. Kayser and M. Krawczak, 2017. Likelihood ratio and posterior odds in forensic genetics: Two sides of the same coin. Forensic Sci Int Genet 28: 203-210.\nWalsh, S., F. Liu, K. N. Ballantyne, M. van Oven, O. Lao and M. Kayser, 2011. IrisPlex: a sensitive DNA tool for accurate prediction of blue and brown eye colour in the absence of ancestry information. Forensic Sci Int Genet 5: 170-180.
Time-delay reservoir computers: nonlinear stability of functional differential systems and optimal nonlinear information processing capacity. Applications to stochastic nonlinear time series forecasting.
Freitag, 14.7.17, 13:00-14:00, Raum 404, Eckerstr. 1
Reservoir computing is a recently introduced brain-inspired\nmachine learning paradigm capable of excellent performances in the processing of empirical data. We focus on a particular kind of time-delay based reservoir computers that have been physically implemented using optical and electronic systems and have shown unprecedented data processing rates. Reservoir computing is well-known for the ease of the associated training scheme but also for the problematic sensitivity of its performance to architecture parameters.\nThis talk addresses the reservoir design problem, which remains the biggest challenge in the applicability of this information processing scheme. More specifically, we use the information available regarding the optimal reservoir working regimes to construct a functional link between the reservoir parameters and its performance. This function is\nused to explore various properties of the device and to choose the optimal reservoir architecture, thus replacing the tedious and time consuming parameter scanning used so far in the literature.\n
One can hear the corners of a drum
Montag, 17.7.17, 16:15-17:15, Raum 404, Eckerstr. 1
Analytically computing the spectrum of the Laplacian is impossible\nfor all but a handful of classical examples. Consequently, it can be tricky\nbusiness to determine which geometric features are spectrally determined; such\nfeatures are known as geometric spectral invariants. Weyl demonstrated in 1912\nthat the area of a planar domain is a geometric spectral invariant. In the\n1950s, Pleijel proved that the perimeter is also a spectral invariant. Kac,\nand McKean & Singer independently proved in the 1960s that the Euler\ncharacteristic is a geometric spectral invariant for smoothly bounded domains. \nAt the same time, Kac popularized the isospectral problem for planar domains in\nhis article, "Can one hear the shape of a drum?'' Colloquially, one says that\none can "hear'' spectral invariants. Hence the title of this talk in which we\nwill show that the presence, or lack, of corners is spectrally determined. \nThis talk is based on joint work with Zhiqin Lu. \n
Introduction to Motives I
Dienstag, 18.7.17, 14:15-15:15, Raum 414, Eckerstr. 1
Introduction to Motives II
Mittwoch, 19.7.17, 10:15-11:15, Raum 414, Eckerstr. 1
Interpretable Fields in algebraically closed fields
Mittwoch, 19.7.17, 16:30-17:30, Raum 404, Eckerstr. 1
Abstract: D. Marker and A. Pillay proved that in a reduct of an algebraically closed field F, which is non-locally modular and expanding the additive structure, an infinite field is interpretable and then the multiplication on F is definable in this reduct. In their work, they use a result of B. Poizat, which states an infinite field K which is definable in the pure algebraically closed field F is definably\nisomorphic to F. I will present this result and its proof.\n
Representation theory in stable derivators and tilting bimodules
Donnerstag, 20.7.17, 14:15-15:15, Raum 403, Eckerstr. 1
I will discuss some classical concepts and results from the representation theory of finite dimensional hereditary algebras (reflection functors, Coxeter functors, Serre duality) and their incarnation in an arbitrary stable derivator. I will also show how to represent such functors and relations among them in terms of spectral tilting bimodules (these are rather small diagrams of spectra, in the sense of topology, with very favorable properties).\n\n
Singularity Formation in Geometric Flows
Donnerstag, 20.7.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
An Extension Theorem for differential forms on 4-dimensional GIT-quotients
Freitag, 21.7.17, 10:15-11:15, Raum 404, Eckerstr. 1
Introduction to Motives III
Freitag, 21.7.17, 14:15-15:15, Raum 414, Eckerstr. 1
The large scale geometry of the Higgs bundle moduli space
Montag, 24.7.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiß and Frederik Witt on the asymptotics of the natural L2-metric on the moduli space of rank-2 Higgs bundles over a Riemann surface as given by the set of solutions to the so-called self-duality equations \nfor a unitary connection and a Higgs field. \n\nExtended abstract (including formulas) can be found using the link above.
Introduction to Motives IV
Dienstag, 25.7.17, 14:15-15:15, Raum 414, Eckerstr. 1
Depinning as a coagulation process
Dienstag, 25.7.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Phase-Field Models for Thin Elastic Structures
Dienstag, 25.7.17, 16:45-17:45, Raum 226, Hermann-Herder-Str. 10
We will discuss phase-field approximations of a\ngeometric energy functional defined on surfaces embedded\ninto a small container. The novelty in our work is the\ncontrol of the connectedness of limiting surfaces by a\npenalty on the diffuse interface level. This is achieved by\npenalising a double integral of a suitable geodesic\ndistance function. We will also show that no finer\ntopological control can be achieved and present numerical\nevidence of the effectiveness of our method.\n
Some maximum principles on complete manifolds and their applications in geometry and analysis
Dienstag, 25.7.17, 17:00-18:00, Raum 404, Eckerstr. 1
The Hopf maximum principle is a fundamental tool for geometry and analysis on compact manifolds. For non-compact complete manifolds, in 1960-70's, H. Omori, S. T. Yau, and S. T. Yau-S. Y. Cheng respectively established maximum principles with the sectional/Ricci curvature bounded below by a constant. This kind of results are called Omori-Yau\nmaximum principles, they provide a powerful tool in the geometry and analysis on non-compact complete manifolds. In this talk, we will present some new Omori-Yau maximum principles and give their applications in submanifold geometry, harmonic maps and holomorphic maps.\n
Introduction to Motives V
Mittwoch, 26.7.17, 10:15-11:15, Raum 414, Eckerstr. 1
Some consequences from Hodge theory in representation theory
Mittwoch, 26.7.17, 12:00-13:00, Raum 119, Eckerstraße 1
Essentially Different Functions
Mittwoch, 26.7.17, 16:30-17:30, Raum 404, Eckerstr. 1
The terminology "Wesentlich verschiedene Abbildungen" (which means "essentially different functions") is taken from Hausdorff's work "Über zwei Sätze von Fichtenholz\nund Kantorovich'' (1935).\n\nWe will follow Hausdorff's proof of the existence of continuum many essentially different functions: i.e. there is some \(H \bsubseteq {^\bomega \bomega}\) of size continuum\nsuch that for every finitely many \(f_0, \bdots, f_i \bin F\) there is a level \(x \bin \bomega\) such that \(f_l(x) \bneq f_j(x)\) for \(l<j \bleq i\).\n\nWe will then see how to generalize the result to find a family of size continuum of "independent functions" using a construction with trees. If the audience is\ninterested, we could also compare it with some other well known (but less pictorial) proofs.\nIf time remains, we will show how the existence of continuum many independent functions applies to prove that a finite support iteration of σ-centred forcing notions is\nagain σ-centred (this is a question asked by Goldstern and answered by Blass in Mathoverflow).\n\n
Donnerstag, 27.7.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Motivic Hodge modules and the Decomposition Theorem
Freitag, 28.7.17, 10:15-11:15, Raum 404, Eckerstr. 1
Let \(p: X \bto S\) be a proper morphism of complex varieties. If we regard the higher direct image \(\boperatorname{R}^ip_{\bast} \bmathbf{Q}\) of the constant analytic sheaf \(\bmathbf{Q}\) as an \(S\)-parametrized family of mixed Hodge structures via the identifications \((\boperatorname{R}^ip_{\bast} \bmathbf{Q})_s = \boperatorname{H}^i(X_s(\bmathbf{C}),\bmathbf{Q})\), then M. Saito's theory of Hodge modules provides a categorical framework for studying such families and their functoriality. In this talk, we will explore an alternative framework for this inspired by \(\bmathbf{A}^1\)-homotopy theory. As an application, we sketch a proof of the Decomposition Theorem that avoids many of the subtleties of Saito's proof.
Linking differential equation modeling to population statistics in metabolomics - insights from general population data for statistical analyses and study design of metabolome wide association analyses
Freitag, 28.7.17, 12:00-13:00, Raum 404, Eckerstr. 1
Metabolomics has developed fast in the last decade, presenting promising results, both in terms of improving the understanding of physiological and pathophysiological processes and in terms of predictive and diagnostic models aiming at personalized medicine. However, statistical modeling has been relying almost exclusively on linear models like partial least squares or ordinary least squares regression analyses, despite them being physiologically implausible in a wide range of scenarios. Here, by using data from the large general population cohort Study of Health in Pomerania (SHIP, n=4068), we show that differential equation modeling can be utilized to inform and refine statistical regression models on the population level, describing successfully important features of one-time metabolome measures. As shown, the information derived from differential equation modeling can then be used to modify and optimize several steps of metabolome wide association analyses from data sampling (e.g. which factors should be sampled or controlled) and data preparation (e.g normalization of urine data) to model specification (e.g. correct adjustment for important confounder) and data interpretation (e.g. metabolite-phenotype interactions). In conclusion, we demonstrate that metabolome data contain more information than usually extracted and that theoretical modelling via differential equations can be helpful in understanding attributes of one-time metabolomic measurements, paving the way for better applications of metabolomics in the clinical sciences.\n
Introduction to Motives VI
Freitag, 28.7.17, 14:15-15:15, Raum 414, Eckerstr. 1