Institutsöffentliche Vorstellungsvorträge Didaktik
Dienstag, 10.10.17, 10:00-11:00, Raum 404, Eckerstr. 1
Die Vorträge finden zu folgenden Zeiten statt:\n\n10:00-10:30 Uhr \n11:15-11:45 Uhr \n12:30-13:00 Uhr
Algebraic trees versus metric trees as states of stochastic processes
Donnerstag, 19.10.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
In this talk we are interested in limit objects of graph-theoretic trees\nas the number of vertices goes to infinity. Depending on which notion of\nconvergence we choose different objects are obtained.\n\nOne notion of convergence with several applications in different areas is\nbased on encoding trees as metric measure spaces and then using the\nGromov-weak topology. Apparently this notion is problematic in the\nconstruction of scaling limits of tree-valued Markov chains whenever the\nmetric and the measure have a different scaling regime. We therefore\nintroduce the notion of algebraic measure trees which capture only the tree\nstructure but not the metric distances.\nConvergence of algebraic measure trees will then rely on weak convergence\nof the random shape of a subtree spanned a sample of finite size.\nWe will be particularly interested in binary algebraic measure trees which\ncan be encoded by triangulations of the circle. We will show that in the\nsubspace of binary algebraic measure trees sample shape convergence is\nequivalent to Gromov-weak convergence when we equip the algebraic measure\ntree with an intrinsic metric coming from the branch point distribution.\nWe will illustrate this with the example of a Markov chain arising in\nphylogeny whose mixing behavior was studied in detail by Aldous (2000) and\nSchweinsberg (2001).\n\n (based on joint work with Wolfgang Löhr and Leonid Mytnik)\n
Classification of principal bundles via motivic homotopy theory
Freitag, 20.10.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
I will talk about recent joint work with Aravind Asok and Marc Hoyois on classification results for principal bundles over smooth affine varieties. The main point will be to explain how the general homotopical results boil down to very concrete examples of interesting octonion algebras.
Flat bundles, \(\bmathbb{R/Z}\)-K-theory and rho invariants
Montag, 23.10.17, 16:15-17:15, Raum 404, Eckerstr. 1
Atiyah, Patodi and Singer constructed the relative K-theory class \([\balpha]\) associated with a flat unitary vector bundle over a closed manifold. \nThis class is related to the spectral invariant rho of a Dirac operator by the so called index theorem for flat bundles, which computes the pairing between \([\balpha]\) and the K-homology class \([D]\) of the Dirac operator.\n\nIn this talk, after introducing the context and the needed tools, we show that \([\balpha]\) admits a canonical construction, using von Neumann algebras and that, as a secondary class, it results from Atiyah's \(L^2\)-index theorem for covering.\n\nTaking an operator algebraic point of view, we show that Atiyah's property can be encoded using KK-theory with real coefficients (which will be introduced). This permits to generalise the constructions of secondary classes of rho-type in the noncommutative setting of a discrete group \(\bGamma\) suitably acting on a \(C^*\)-algebra \(A\).\nBased on joint work with Paolo Antonini and Georges Skandalis.
Transport of a two-phase flow with sharp interface in three dimensions
Dienstag, 24.10.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Eigenschaften der J-Hierarchie und L[E]
Mittwoch, 25.10.17, 16:15-17:15, Raum 404, Eckerstr. 1
Die J-Hierarchie ermöglicht eine alternative Konstruktion des\nkonstruktiblen Universums L und erlaubt es, für jede Klasse E ein Universum\nL[E] zu konstruieren, das - wie L - ein Modell von ZFC ist. In diesem\nVortrag werden wir L[E] und die J-Hierarchie untersuchen. Dabei werden wir\nauch sehen, dass L[E] mit der Forcingerweiterung von L bezüglich E\nübereinstimmt, falls die Menge E ein generischer Filter ist.\n\n
Donnerstag, 26.10.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
K-theory of locally compact modules
Freitag, 27.10.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
We generalize a recent result of Clausen: For a number field with integers O, we compute the K-theory of locally compact O-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite different: Instead of a homotopy coherent cone construction in infinity categories, we rely on calculus of fraction type results in the style of Schlichting. This produces concrete exact category models for certain quotients, a fact which might be of independent interest. As in Clausen's work, our computation works for all localizing invariants, not just K-theory.
Trajectorial Models based on Operational Assumptions
Montag, 30.10.17, 14:15-15:15, Raum 125, Eckerstr. 1
We illustrate by example the construction of\none-dimensional models for\noption pricing based on operational and observable features of a\nsingle class of investors and a\nrisky asset. Market models are defined based on a class of investors\ncharacterized by how they operate on financial data leading to\npotential portfolio re-balances.\nOnce observable variables are selected for modeling, necessary conditions\nconstraining these variables and resulting from the operational setup are\nderived. Future uncertainty is then reflected in the construction of\ncombinatorial trajectory spaces satisfying such constraints. In the absence\nof probability assumptions, a minmax methodology is available to price option\ncontracts; numerical results are presented based on worst case estimation of\nparameters.
Donnerstag, 2.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Walks, Difference Equations and Elliptic Curves
Freitag, 3.11.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
In the recent years, the nature of the generating series of walks in\nthe quarter plane has attracted the attention of many authors in\ncombinatorics and probability. The main questions are: are they\nalgebraic, holonomic (solutions of linear differential equations) or at\nleast hyperalgebraic (solutions of algebraic differential equations)? In\nthis talk, we will show how the nature of the generating function can\nbe approached via the study of a discrete functional equation over a\ncurve E, of genus zero or one. In the first case, the functional\nequation corresponds to a so called q-difference equation and all the\nrelated generating series are differentially transcendental. For the\ngenus one case, the dynamic of the functional equation corresponds to\nthe addition by a given point P of the elliptic curve E. In that\nsituation, one can relate the nature of the generating series to the\nfact that the point P is of torsion or not.\n This is a collaboration with T. Dreyfus (Irma, Strasbourg), J. Roques\n(Institut Fourier, Grenoble) and M.F. Singer (NCSU, Raleigh).
Vorträge zum 30-jährigen Bestehen des FDM-Seminars
Freitag, 3.11.17, 13:30-14:30, Raum 404, Eckerstraße 1, Freiburg i. Br.
Programm:\n\n13:30 Einführung mit Beiträgen von Rektor Prof. Dr. Hans-Jochen Schiewer und dem Mit-Gründer des FDM, Prof. Dr. Josef Honerkamp\n\n14:00 Prof. Dr. Leonhard Held (University of Zurich): Building a Statistical Model: The Endemic-Epidemic Modelling Framework\n\n15:00 Kaffee\n\n15:30 Prof. Dr. Rainer Dahlhaus (Heidelberg University): Cointegration and Phase Synchronization: Bridging Two Theories\n\n16:30 Prof. Dr. Josef Teichmann (ETH Zürich): Affine processes in mathematical Finance\n\n17:30 Schluss\n\n
Deformation problems on Calabi-Yau manifolds, Picard-Fuchs equations and potential functions
Montag, 6.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
In my talk I will explain various results concerning Calabi-Yau threefolds and geometric objects on the Calabi-Yau variety, e. g. a divisor, a curve or a coherent sheaf. I will discuss their deformation theory and the connection to Picard-Fuchs equations and potential functions. These are special holomorphic functions describing the obstructions of a deformation problem.
Bifurcation of the compressible Taylor vortex
Dienstag, 7.11.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The inverse mean curvature flow and the Riemannian Penrose inequality
Dienstag, 7.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
\(\bkappa\)-trees and the sup game
Mittwoch, 8.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
We use the so-called sup game to obtain some results about\nthe relationships between dominating \(\bkappa\)-sequences and Cohen\n\(\bkappa\)-sequences. In particular, we improve a couple of results\nconcerning the amoeba for \(\bkappa\)-Sacks forcing and \(\bkappa\)-Laver forcing\nthat I presented in some previous seminar-talk.\n
The algebraic topology of p-adic Lie groups
Donnerstag, 9.11.17, 10:15-11:15, Raum 403, Eckerstr. 1
Lazard proved that the cohomology of compact p-adic Lie groups satisfies a version of Poincare duality. This indicates an analogy between p-adic Lie groups and real manifolds. I will explain some results which develop this analogy further.
Donnerstag, 9.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
From elementary number theory to string theory and back again
Freitag, 10.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
I will describe some surprising interactions between number theory, algebraic geometry and mirror symmetry that have appeared in my recent work with Mircea Mustata and Chenyang Xu and that have led to a solution of Veys' 1999 conjecture on poles of maximal order of Igusa zeta functions. The talk will be aimed at a general audience and will emphasize some key ideas from each of the fields involved rather than the technical aspects of the proof.
The extended ν invariant of Joyce manifolds
Montag, 13.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
The Crowley-Nordström ν invariant and its extension ν¯ is an invariant of compact G2 manifolds. It has been computed for several kinds of connected sums by Crowley, Nordström and Goette (see next talk), in this talk we discuss strategies of computations for Joyce manifolds.
Contact and Adhesion of Fractal Interfaces
Dienstag, 14.11.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The inverse mean curvature flow and the Riemannian Penrose inequality
Dienstag, 14.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
Modular representations of sl_n with two-row nilpotent p-character
Mittwoch, 15.11.17, 10:45-11:45, Hörsaal FRIAS
We will study the category of modular representations of the\nspecial linear Lie algebra with a central p-character given by a\nnilpotent whose Jordan type is a two-row partition. Building on work of\nCautis and Kamnitzer, we construct a categorification of the affine\ntangle calculus using these categories; the main technical tool is a\ngeometric localization-type result of Bezrukavnikov, Mirkovic and\nRumynin. Using this, we give combinatorial dimension formulae for the\nirreducible modules, composition multiplicities of the simples in the\nbaby Vermas, and a description of the Ext spaces. This Ext algebra is an\n"annular" analogues of Khovanov's arc algebra, and can be used to give\nan extension of Khovanov homology to links in the annulus. This is joint\nwith Rina Anno and David Yang.
Praesentation der Masterarbeit
Mittwoch, 15.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
Lokal modulare additive Redukte von den komplexen Zahlen
Mittwoch, 15.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
Vektorräume sind modulare Strukturen, denn es gilt die Dimensionsformel. Diesen Begriff der Modularität können wir auf streng minimale Strukturen verallgemeinern, indem wir mit dem algebraischen Abschluss und dem dadurch gegebenen Dimensionsbegriff arbeiten. Eine streng minimale Struktur ist lokal modular (oder monobasiert), wenn die Dimensionsformel für alle endlich erzeugten algebraisch abgeschlossenen Mengen gilt, deren Durchschnitt positive Dimension hat.\nMarker und Pillay zeigten im Jahre 1990, dass in jedem nicht-monobasierten additiven Redukt der komplexen Zahlen die Muliplikation definierbar ist. Der Beweis erfolgt in zwei Teilen. Im ersten Teil wird mit Hilfe der Gruppenkonfiguration von Hrushovski ein unendlicher Körper im Redukt interpretiert. Wie man dann die ursprüngliche Multiplikation des Körpers im Redukt definieren kann, wird im zweiten Teil gezeigt. Im Vortrag wird der erste Teil des Beweises präsentiert und die Idee des zweiten Teils erklärt.
Donnerstag, 16.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Functorial test modules
Freitag, 17.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
In my talk I will report on joint work with Manuel Blickle. I will explain how one can generalize the definition of test ideals \btau to so-called Cartier modules in a functorial way. We obtain several transformation rules with respect to f^! and f_* for various classes of morphisms f: X \bto Y, e.g. for f smooth one has an isomorphism f^! \btau = \btau f^!. Part of the reason for working in this generality is that one has an equivalence with constructible etale p-torsion sheaves up to nilpotence of Cartier modules and these results further support the idea that the test module construction relates to etale nearby cycles similarly to the complex situation where multiplier ideals relate to complex nearby cycles.
Mini-Workshop on "Singular Variational Problems"
Sonntag, 19.11.17, 10:00-11:00, Raum 226, Hermann-Herder-Str. 10
Luigi Berselli (U Pisa): Time averages and Reynolds equations for dissipative equations; \nGiuseppe Buttazzo (U Pisa): Shape optimization under uncertainty;\nPatrick Dondl (U Freiburg): A phase field model for Willmore’s energy with topological constraint;\nMichael Ruzicka (U Freiburg): On a Clement type operator; \nSoeren Bartels (U Freiburg): Semi-implicit time stepping for p-Laplace equations\n
The extended \(\bnu\) invariant of exotic extra twisted connected sums
Montag, 20.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
There is nowadays a large supply of compact manifolds of special holonomy \(G_2\). The Crowley-Nordström \(\bnu\) invariant and its extension \(\bbar\bnu\) help to distinguish different connected components of the \(G_2\)-moduli space on a given manifold \(M\). We already know some manifolds \(M\) whose \(G_2\)-moduli space has at least 7 different connected components.\nHowever all examples of \(G_2\)-manifolds investigated so far turn out to be topologically \(G_2\)-nullbordant, that is, their \(\bnu\)-invariant is divisible by 3.\n\nIn this talk, we will consider Nordström's extra twisted connected sum construction with at least one \(\bmathbb Z/3\)- or \(\bmathbb Z/4\)-block. Although \(\bbar\bnu(M)\) is defined in terms of \(\beta\)-invariants, we will show that in the end it can be computed explicitly for these examples using elementary hyperbolic geometry.
The inverse mean curvature flow and the Riemannian Penrose inequality from Husiken and Imanen.
Dienstag, 21.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
\(\bkappa\)-trees and the sup game, part 2
Mittwoch, 22.11.17, 16:00-17:00, Raum 404, Eckerstr. 1
This is the continuation of the talk of November 8, 2017.
Donnerstag, 23.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Local systems in motivic homotopy theory
Freitag, 24.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
Local systems in motivic homotopy theory
Freitag, 24.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
This talk is first devoted to explain how Voevodsky's theory of motivic complexes is built on, and provides, a natural notion of local system. Two key players involved here are homotopy sheaves and cycle modules, which are respectively the theoretical and concrete side of the same notion. The first part of the talk will recall these two notions and their fundamental link.\n The second part of the talk will move on relative motivic complexes. I will explain the construction of a natural t-structure, analogous to the perverse t-structure (but not realized to it!), obtained in collaboration with Bondarko along ideas of Ayoub. Besides describing some of its good properties, I will explain a program to compute its heart, in terms of cycle modules.\n If time allows, I will wander a little bit on the aimed tool motivating these technicalities: the delta-homotopy Leray spectral sequence.\n
On detecting changes in the jumps of arbitrary size of a time-continuous stochastic process
Freitag, 24.11.17, 12:00-13:00, Raum 404, Eckerstr. 1
An Ito semimartingale is a superposition of a roughly fluctuating Brownian part and a pure jump process. Therefore, it is a very challenging task to disentangle the small jumps of the process from increments of the continuous part. We solve this problem by deriving a statistical procedure for inference on the general jump behaviour of an Ito semimartingale. Finally, we apply this technique to detect abrupt and gradual changes in the jumps of the underlying process using bootstrap tests, where we also allow for local alternatives.
Detection and Estimation of Local Signals
Dienstag, 28.11.17, 12:00-13:00, Raum 404, Eckerstr. 1
I will discuss a general framework for detection of local signals, primarily defined\nby change-points, in sequences of data. Changes can occur continuously, e.g., a\nchange in the slope of a regression line, or discontinuously, e.g., a jump in the\nlevel of a process. I will focus on the problem of segmentation of independent\nnormal observations according to changes in the mean. Results will be illustrated\nby simulations and applications to copy number changes and to historical weather\npatterns. Confidence regions for the change- points and some difficulties associated\nwith dependent observations will also be discussed. Aspects of this research involve\ncollaboration with Fang Xiao, Li Jian, Liu Yi, Nancy Zhang, Benjamin Yakir and\nLi (Charlie) Xia.
Numerical homogenization by localized orthogonal decomposition and connections to the mathematical theory of homogenization
Dienstag, 28.11.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The inverse mean curvature flow and the Riemannian Penrose inequality from Husiken and Imanen.
Dienstag, 28.11.17, 16:15-17:15, Raum 404, Eckerstr. 1
Präsentation der Masterarbeit: Grunerts gerichteter Pseudoraum
Mittwoch, 29.11.17, 16:30-17:30, Raum 404, Eckerstr. 1
Grunerts gerichteter Pseudoraum
Mittwoch, 29.11.17, 16:30-17:30, Raum 404, Eckerstr. 1
In dem Vortrag werden wir \ndie Axiome des N-Pseudoraums und einige seiner Eigenschaften besprechen, \ndie Axiomatisierung des gerichteten Pseudoraums angeben, \nbesprechen, welche Eigenschaften einen 1-Typ über der leeren Menge eindeutig\nbestimmen, \nund die Unabhängigkeitsrelation besprechen.\n\n
Donnerstag, 30.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Hodge theory and formality
Freitag, 1.12.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
Given a differential graded algebra or any algebraic structure\nin chain complexes, one may ask if it is quasi-isomorphic to its\nhomology equipped with the zero differential. This property is called\nformality and has important consequences in algebraic topology. For\ninstance, if the de Rham algebra of a manifold is formal, then certain\nhigher operations in cohomology, called Massey products, are known to\nvanish. In this talk, I will first discuss the notion of formality and\nits consequences in different algebraic and topological contexts. Then,\nI will explain how mixed Hodge theory and Galois actions can be used to\nprove formality, for algebraic structures arising from the category of\ncomplex algebraic varieties.
Understanding Biological Processes using Stochastic Modelling
Freitag, 1.12.17, 12:00-13:00, Raum 404, Eckerstr. 1
The molecular biology of life seems inaccessibly complex, and gene expression is an essential part of it. It is subject to random variation and not exactly predictable. Still, mathematical models and statistical inference pave the way towards the identification of underlying gene regulatory processes. In contrast to deterministic models, stochastic processes capture the randomness of natural phenomena and result in more reliable predictions of cellular dynamics. Stochastic models and their parameter estimation have to take into account the nature of molecular-biological data, including experimental techniques and measurement error.\n \nThis talk presents according modelling and estimation techniques and their applications: the derivation of mRNA contents in single cells; the identification of differently regulated cells from heterogeneous populations using mixed models; and parameter estimation for stochastic differential equations using computer-intensive Markov chain Monte Carlo techniques.
Generalized Seiberg-Witten equations and almost-Hermitian geometry
Montag, 4.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
I will talk about a generalisation of the Seiberg-Witten equations introduced by Taubes and Pidstrygach, in dimension 3 and 4 respectively, where the spinor representation is replaced by a hyperKahler manifold admitting certain symmetries. I will discuss the 4-dimensional equations and their relation with the almost-Kahler geometry of the underlying 4-manifold. In particular, I will show that the equations can be interpreted in terms of a PDE for an almost-complex structure on 4-manifold. This generalises a result of Donaldson.
Additive manufacturing of scaffolds for bone regeneration
Dienstag, 5.12.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Inverse mean curvature flow in complex hyperbolic space
Dienstag, 5.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
Abstract: We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub- Riemannian metric on the sphere. Finally we show that there exists a family of examples such that the Webster curvature of this sub-Riemannian limit is not constant.
tba
Mittwoch, 6.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
Infinite Populations, Choice and Determinacy
Mittwoch, 6.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
This talk criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow's theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the "Hildenbrand criterion" and argue that results based on unrestricted Axiom of Choice (AC) do not meet this criterion. The technically novel part is a proposal to use a set-theoretic principle known as the Axiom of Determinacy (AD). A particularly appealing aspect of AD from the point of view of the research area in question is its game-theoretic flavor.\n
Donnerstag, 7.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Recent results and open problems about Oeljeklaus-Toma manifolds
Freitag, 8.12.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
Oeljeklaus - Toma manifolds are compact complex manifolds associated to number fields with at least one real and and at least one complex place. The construction is similar to tori, but it involves not only the lattice of integers but also a suitable group of units. We investigate how the number-theoretic properties influence their geometric properties: existence of special metrics, existence of closed subvarieties, etc. The talk is based mainly on joint work with L.Ornea and M. Verbitsky.
Borcherds-Kac-Moody Algebras in Conformal Field Theory
Montag, 11.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
Borcherds-Kac-Moody algebras are a generalization of finite dimensional semisimple Lie algebras obtained by weakening the requirements on Cartan matrices. In his proof of the Moonshine Conjectures, Borcherds related them to the vertex operator algebras from conformal field theory. At the same time, there appears to be a connection to automorphic forms via denominator functions. This can hopefully be leveraged, in particular, to investigate so-called Bogomol'nyi-Prasad-Sommerfield states in field theories with supersymmetry.
tba
Mittwoch, 13.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
Forcing over ord-transitive models
Mittwoch, 13.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
Usually forcing is performed over transitive countable ground models.\nHowever, there are technical means to waive transitivity. In this talk we\nshall focus on the algebraic features of suitable ground models. We explain\nord-transitive models, labelled models, the ord-collapse, and\ntheir relations to the Mostowski collapse.
1-Motives
Donnerstag, 14.12.17, 11:15-12:15, Hörsaal FRIAS, Albertstr. 19
In this expository talk we want to present Deligne's category of 1-motives and its realisations. This ties up\nwith the talk of Wüstholz on Friday, but the two talks\nwill be independent of each other.
Donnerstag, 14.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Period domains
Freitag, 15.12.17, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Variational formulas for the Selberg zeta function and applications to curvature asymptotics
Montag, 18.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk, we will study the Selberg zeta function and its relatives. We will recall the celebrated Selberg trace formula, and the geometric setting of our work, the Teichmüller space of Riemann surfaces of genus, g. As shown by Zograf and Takhtajan, the Selberg trace formula connects the Ricci curvature of the Hodge bundle \(H^0 (K^m)\) over Teichmüller space together with the second variation of the Selberg zeta function at integer points. We will briefly explain this connection and the role of the Selberg trace formula in its derivation. \n \nFurther, we will investigate the behavior of the Selberg zeta function, \(Z(s)\), as a function on Teichmüller space. We will deduce an explicit formula for the second variation of \(\blog( Z(s) )\) via a certain infinite sum involving lengths of closed geodesics of the underlying surface and their variations. We will then utilize this formula to study the asymptotics of the second variation of \(\blog( Z(s) )\) as \(s \bto \binfty\). We shall see that the most prominent role is played by the systole geodesics. Moreover, the dimension of the kernel of the first variation of the latter appears in the signature of the Hessian of \(\blog Z(s)\) for large \(s\). In conclusion, we will show how our variational formula and its asymptotics have interesting implications for the curvature of the Hodge bundle and its relationship to the Quillen curvature. \n\nThis is a joint work with Julie Rowlett and Genkai Zhang.
Dienstag, 19.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
The inverse mean curvature flow and the Riemannian Penrose inequality from Husiken and Imanen.
Dienstag, 19.12.17, 17:00-18:00, Raum 404, Eckerstr. 1
Separabel abgeschlossene Körper sind äquational.
Mittwoch, 20.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
Der Imperfektionsgrad eines Körpers \(K\) positiver Charakteristik \(p\) ist im Grunde die linear Dimension von \(K\) als \(K^p\)-Vektorraum. Der Körper \(K\) ist separabel abgeschlossen, falls er keine echte separable algebraische Erweiterung besitzt, wobei ein algebraisches Element \(\balpha\) über \(K\) separabel ist, wenn sein minimal Polynom keine doppelten Nullstellen (im algebraischen Abschluss) hat. \n\nDie Theorie separabel abgeschlossener Körper der Charakteristik \(p>0\) ist axiomatisierbar, und ihre Vervollständigungen werden durch den Imperfektionsgrad bestimmt. Insbesondere ist die Theorie separabel abgeschlossener Körper der Charakteristik \(p>0\) und unendlichen Imperfektionsgrades vollständig und stabil. Diese Theorie hat keine Elimination von Imaginären in der Ringsprache. \n\nIn Zusammenarbeit mit Martin Ziegler werden wir zeigen, dass diese Theorie äquational ist. Äquationalität ist eine Art lokaler Noetherianität und impliziert eine relative Elimination von Imaginären. Wir werden zeigen, dass gewisse Formeln Gleichungen in einem geeigneten Modell sind, nämlich in einem differentiell abgeschlossenen Körper der Charakteristik \(p\), dessen modelltheoretische Eigenschaften von Carol Wood beschrieben wurden.
Complete convex surfaces: intrinsic and extrinsic properties
Donnerstag, 21.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
We will examine convex surfaces which divide the ambient Euclidean 3-space into two parts. By the Theorem of Gauss-Bonnet such surfaces need to be of the intrinsic type of a cylinder, plane, or sphere. We will then discuss the extrinsic symmetry property of rotational invariance, and its infinitesimal version at a point of the surface. We outline the proof of a global extrinsic conjecture of Victor Andreevich Toponogov : "Any convex plane admits (at least) one point of infinitesimal symmetry, possibly at infinity". The proof, in collaboration with Brendan Guilfoyle, uses complex analysis and a parabolic curvature flow in the space of lines of Euclidean 3-space.
Donnerstag, 28.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 4.1.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Riemannian shape analysis
Montag, 8.1.18, 16:15-17:15, Raum 404, Eckerstr. 1
Shape analysis aims at a mathematical description and\nanalysis of geometric data such as e.g. curves or surfaces. The key\nparadigm is to view these data as elements of an infinite-dimensional\nRiemannian manifold, which is called shape space. I will give an\nintroductory talk to shape spaces and Riemannian metrics thereon. Some\nmain results to be covered are (non-)degeneracy of the Riemannian path\nlength functional and wellposedness of the geodesic equation.
Kac-Moody algebras and derived algebraic geometry
Dienstag, 9.1.18, 09:15-10:15, SR 119, Eckerstr. 1
Kac-Moody algebras are usually used in physics to describe\nsome 2-dimensional conformal field theories. In this talk, we will introduce a new version of Kac-Moody algebras supposed to describe some physical phenomenons in higher dimensions.\n\nThose new algebras are in fact Lie algebras up to homotopy and we will study them using tools from algebraic topology and derived algebraic geometry.
Ein Färbungssatz
Mittwoch, 10.1.18, 16:30-17:30, Raum 404, Eckerstr. 1
Sei \(k \bgeq 1\) eine natürliche Zahl. Gowers' Satz über eine Partition der Menge der \(k\)-wertigen Blöcke in endlich viele Teile sagt, dass in einem Teil der Partition eine gegen die Tetrisoperation abgeschlossene Unterhalbgruppe liegt. Die partiell definierte Gruppenoperation auf den Blöcken ist die stellenweise Addition, die auf hintereinanderliegenden Blöcken mit der Konkatenation übereinstimmt. Wir verallgemeinern Gowers' Satz, indem wir den Grundraum auf Blocksequenzen, deren Projektionen auf \(\bomega\) aus bestimmten selektiven Koidealen über \(\bomega\) stammen, einschränken. Diese neue Variante führt dazu, dass es in Forcingserweiterungen durch Gowers-Matet-Forcing erweiterte Ramseyräume gibt. Der Vortag wird sich auf die Beweisschritte ohne Forcing konzentrieren. \n
Adelic formal neighborhoods and chiral modules with support
Donnerstag, 11.1.18, 09:15-10:15, SR 119, Eckerstr. 1
In this talk I will explain how one can use Ind-Pro objects\nto consider the restriction to formal and punctured formal\nneighborhoods of a subvariety Z in a variety X. This restriction functor can then be applied to give a description of modules for a chiral algebra over X supported at Z in terms of modules for an associative algebra object over Z. This is work in progress.\n\n\n(no worries, the speaker will explain what a "chiral algebra" is...)
Donnerstag, 11.1.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Complete intersections in varieties with finite-dimensional motive
Freitag, 12.1.18, 09:30-10:30, Hörsaal II, Albertstr. 23b
We consider complete intersections inside a variety with finite-dimensional motive for which the Lefschetz standard conjecture B holds. We show how conditions on the (modified) niveau filtration on homology influence the Chow groups. This leads to a generalization of results of Voisin and Vial on injectivity of cycle class and Abel-Jacobi maps. Using a variant involving group actions, we obtain several new examples of complete intersections with finite-dimensional motive. This is joint work with Robert Laterveer and Chris Peters.
The generalized Franchetta conjecture for hyperkaehler varieties
Freitag, 12.1.18, 11:15-12:15, Hörsaal II, Albertstr. 23b
The generalized Franchetta conjecture as formulated by O’Grady is about algebraic cycles on the universal K3 surface. It is natural to consider a similar conjecture for algebraic cycles on universal families of hyperkaehler varieties. This has close ties to Beauville’s conjectural ``splitting property’’, and the Beauville-Voisin conjecture (stating that the Chow ring of a hyperkaehler variety has a certain subring injecting into cohomology). I will attempt to give an overview of these conjectures, and present some cases where they can be proven. This is joint work with Lie Fu, Mingmin Shen and Charles Vial.
Derivator Six Functor Formalisms
Freitag, 12.1.18, 14:45-15:45, Hörsaal II, Albertstr. 23b
Grothendieck, Verdier, and Deligne in the 60's observed that classical duality theorems like Poincaré, or Serre duality can be most elegantly expressed, and vastly generalized, by a formalism of the six functors. This makes essential use of derived categories. The latter are, however, not sufficient for the purpose of descent. Descent is essential to define equivariant (co)homology and for equivariant duality theorems, and more generally to extend six-functor-formalisms to stacks, which is very important in applications. The problem with (co)homological descent is that the ``glueing data'' has a higher-categorical nature. In this talk we explain how our theory of fibered derivators, based on the idea of derivator due to Grothendieck and Heller (will be explained as well !), solves the problem of (higher-categorical) descent in a way closely related to the classical theory of cohomological descent (due to Deligne in SGA4). However, it is, in contrast, completely self-dual, making it very suitable for the descent of six-functor-formalisms.
Stability of Ricci flow on singular spaces
Montag, 15.1.18, 16:15-17:15, Raum 404, Eckerstr. 1
We discuss recent results on the Ricci flow for spaces with incomplete edge singularities. In the special case of isolated cones we establish stability of the flow near Ricci flat metrics.
Dynamics of fronts in some singularly coupled Allen-Cahn equations
Dienstag, 16.1.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
When coupling the scalar Allen-Cahn model for phase separation with large scale linear fields, the dynamics of interfaces can be rather intriguing. We consider the one-dimensional situation and apply methods from spatial dynamics to identify and unfold singularities that complex motion even for a single interface. Specifically, we can imbed scalar singularities such as a butterfly catastrophe that yield accelerated and direction reversing fronts. In recent work we unfold a degenerate Takens-Bodganov point with additional oscillatory dynamics.
Fortsetzung des Vortrags vom 10.1.2018 über Färbungen
Mittwoch, 17.1.18, 16:30-17:30, Raum 404, Eckerstr. 1
INVERSE CURVATURE FLOWS AND GEOMETRIC INEQUALITIES
Donnerstag, 18.1.18, 16:00-17:00, Hörsaal II, Albertstr. 23b
In recent years curvature flows have played a crucial role in proving important geometric theorems. For instance, the Ricci flow lead to a proof of the Poincare conjecture, and the inverse mean curvature flow (IMCF) was crucial in the proof of the Riemannian Penrose inequality. In this talk we present further applications of the IMCF. First we review, how classical geometric inequalities, such as the Minkowski inequality for closed convex hypersurfaces, can be generalised to a wider class of hypersurfaces using curvature flows. Secondly, we present new estimates for a Willmore-type energy of hypersurfaces with boundary, satisfying a perpendicular Neumann-type condition on the unit sphere. The crucial ingredient is the IMCF with boundary conditions.
Homologie linearer Gruppen und die Vermutung von Quillen
Donnerstag, 18.1.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Unlikely intersections between isogeny orbits and curves
Freitag, 19.1.18, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
In the spirit of the Mordell-Lang conjecture, we consider the intersection of a curve in a family of abelian varieties with the images of a finite-rank subgroup of a fixed abelian variety A0 under all isogenies between A0 and some member of the family. After excluding certain degenerate cases, we can prove that this intersection is finite. This proves the so-called André-Pink-Zannier conjecture in the case of curves. We can even allow translates of the finite-rank subgroup by abelian subvarieties of controlled dimension if we strengthen the degeneracy hypotheses suitably. In my talk, I will try to explain the motivation for this problem and give an outline of the proof, which follows a strategy due to Pila-Zannier.
Chronological Age Determination for Forensic Applications using Random Forest Regression and DNA Methylation Analysis
Freitag, 19.1.18, 12:00-13:00, Raum 404, Eckerstr. 1
Over the last few years it became clear that additional information is hidden wi-\nthin epigenetic modifications of the DNA, and that especially DNA methylation\n(DNAm) could provide useful evidence to the criminal justice system. Within this\nproject, specific changes in DNAm levels upon age progression at selected loci were\nused to develop an objective scientific tool to determine the chronological age of\nan (unknown) individual. This information can be used to narrow down the list\nof suspects during criminal investigations or to determine the age of a person in\nother legal contexts such as human trafficking. A model for age prediction based\non whole blood samples, 13 selected age-dependent DNAm markers, and a ran-\ndom forest regression (RFR) approach was developed. The analysis of the DNAm\nwas performed using amplicon based massive parallel sequencing (MPS) and the\nRFR model created with the R package RandomForest. The performance of the\nmodel was evaluated using cross-validation for the training set and by indepen-\ndent analysis of an additional test set. Within the seminar, a short introduction\ninto the field of forensic (epi-)genetics, the marker selection and development of\nthe DNA methylation tool based on RFR and MPS as well as the results of the\nage-determination tool will be presented. Furthermore, the potential and (current)\nlimitations of the experimental and machine learning approach in respect to the\nimplementation into forensic investigations will be discussed. The here presented\nproject of the University of Amsterdam in cooperation with the Netherlands Fo-\nrensic Institute was funded by the NCTV grant of the Dutch Ministry of Security\nand Justice.
Crystal graphs and semicanonical functions for symmetrizable Cartan matrices
Freitag, 19.1.18, 14:00-15:00, Raum 404, Eckerstr. 1
In joint work with B. Leclerc and J. Schröer we propose a 1-Gorenstein algebra H, defined over an arbitrary field K, associated to the datum of a symmetrizable Cartan Matrix C, a symmetrizer D of C and an orientation Ω. The H-modules of finite projective dimension behave in many aspects like the modules over a hereditary algebra, and we can associate to H a kind of preprojective algebra Π. If we look, for K algebraically closed, at the varieties of representations of Π which admit a filtration by generalized simples, we find that the components of maximal dimension provide a realization of the crystal B(-∞) corresponding to C. For K being the complex numbers we can construct, following ideas of Lusztig, an algebra of constructible functions which contains a family of "semicanonical functions", which are naturally parametrized by the above mentioned components of maximal dimensions. Modulo a conjecture about the support of the functions in the "Serre ideal" those functions yield a semicanonical basis of the enveloping algebra U(n) of the positive part of the Kac-Moody Lie algebra g(C).
BPS-states and automorphic representations of exceptional groups
Freitag, 19.1.18, 15:45-16:45, Raum 404, Eckerstr. 1
Automorphic forms on exceptional Lie groups appear naturally in string theory compactifications. They manifest themselves as couplings in higher derivative corrections and in terms of generating functions of BPS-states. I will explain how to treat automorphic forms in the modern theory of automorphic representations, which can be directly connected to BPS-states in string theory. Various recent results, conjectures and open problems are outlined.
Twisted intertwining operators and nonabelian orbifold theories
Freitag, 19.1.18, 17:00-18:00, Raum 404, Eckerstr. 1
The study of two-dimensional conformal field theories can in fact be reduced to the study of intertwining operators. In particular, the study of orbifold conformal field theories corresponds to the study of twisted intertwining operators (intertwining operators among twisted modules). However, for more than twenty years, there was even no mathematical definition of twisted intertwining operators when the twisted modules involved are associated to non-commuting automorphisms of the vertex operator algebra. In this talk, I will discuss a recently introduced notion of twisted intertwining operator and the basic properties of such operators. I will also present the main conjecture on nonabelian orbifold theories in terms of these operators and discuss the applications.
On the algebraic approach to QFT
Montag, 22.1.18, 16:15-17:15, Raum 404, Eckerstr. 1
Residual-type a posteriori estimator for a quasi-static contact problem.
Dienstag, 23.1.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
tba
Mittwoch, 24.1.18, 16:30-17:30, Raum 404, Eckerstr. 1
The dp-rank of an abelian group
Mittwoch, 24.1.18, 16:30-17:30, Raum 404, Eckerstr. 1
Abstract: Abelian groups form an archetypical example of stable groups. Their model theory is well-understood and in fact, distinct degrees of stability can be easily described for abelian groups in terms of the lattice of definable subgroups. For instance, an abelian group is omega-stable if and only if it satisfies the descending chain condition on definable subgroups.\n\nIn this talk, I will characterise the notion of dp-rank, which originates in Shelah's work on NIP theories, for abelian groups. Furthermore, I will explain how to compute it explicitly. This is joint work with Yatir Halevi.\n\n
On the motivic Tamagawa number of number fields
Donnerstag, 25.1.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
The Tamagawa number of a linear algebraic group is a classical arithmetic invariant.\nIt was computed and related to special values of L-functions in the second half of the 20th century.\nIn 1990 Bloch and Kato proposed a Tamagawa number of a (certain kind of) motive, related it to special L-values of motives and stated a conjecture on their value in the spirit of the classical case. We will discuss motivic Tamagawa numbers for the (seemingly easiest) unknown case, namely the twisted motive of a number field. I will present precise formulas relating these two notions of Tamagawa numbers to one another and to Borel regulators.
Invariance of closed convex cones for stochastic partial differential equations
Donnerstag, 25.1.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
The goal of this talk is to clarify when a closed convex cone is invariant for a stochastic partial differential equation (SPDE) driven by a Wiener process and a Poisson random measure, and to provide conditions on the parameters of the SPDE, which are necessary and sufficient. As a particular example, we will show how the Heath-Jarrow-Morton-Musiela (HJMM) equation from Financial Mathematics, which models the evolution of interest rate curves, fits into the present SPDE setting. Moreover, we will apply our result about the invariance of closed convex cones in order to investigate when the HJMM equation produces nonnegative interest rate curves.\n\n
Invariance of closed convex cones for stochastic partial differential equations
Donnerstag, 25.1.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
The goal of this talk is to clarify when a closed convex cone is\ninvariant for a stochastic partial differential equation (SPDE) driven\nby a Wiener process and a Poisson random measure, and to provide\nconditions on the parameters of the SPDE, which are necessary and\nsufficient. As a particular example, we will show how the\nHeath-Jarrow-Morton-Musiela (HJMM) equation from Financial Mathematics,\nwhich models the evolution of interest rate curves, fits into the\npresent SPDE setting. Moreover, we will apply our result about the\ninvariance of closed convex cones in order to investigate when the HJMM\nequation produces nonnegative interest rate curves.
The Koszul duality between D-modules and Omega-modules
Freitag, 26.1.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Koszul duality between D-modules and Omega-modules (dg modules over the algebraic de Rham complex) has first been studied by Kapranov, and subsequently by various authors including Beilinson-Drinfeld and Positselski. In this talk, I will report on work in progress (joint with Dmitri Pavlov) on a refinement of Koszul duality. An application of this refinement is a presentation of the derived pushforward and pullback functors for D-modules which avoids the usual mixing of right and left adjoints.
Higher Order Elicitability
Freitag, 26.1.18, 12:00-13:00, Raum 404, Eckerstr. 1
Elicitability of a statistical functional means that it can be obtained as the minimizer of an expected loss function. Such a loss function leads to a natural way of forecast comparison or model selection, and allows for M-estimation and generalized regression.\n\nPrime examples of elicitable functionals are the mean or quantiles of a random variable. Independently, Weber (2006, Mathematical Finance) and Gneiting (2011, JASA) have shown that expected shortfall (ES), an important risk measure in banking and finance, is not elicitable. However, it turns out that ES is jointly elicitable with a certain quantile, that is, it is elicitable of second order.\n\nIn this talk, we present our results on higher order elicitability of ES and some other functionals, and we provide characterizations of the associated classes of consistent scoring functions. We illustrate the usefulness of scoring functions for forecast comparison.
Pollicott-Ruelle-Resonanzen
Montag, 29.1.18, 14:00-15:00, Bibliothek Angewandte Mathematik, R216, RZ, Hermann-Herder-Strasse 10
Pollicott-Ruelle-Resonanzen werden auch klassische Resonanzen genannt. Sie lassen sich definieren als Eigenwerte des erzeugenden Vektorfelds des geodätischen Flusses auf dem (ko)-Sphärenbündel einer geeigneten Riemannschen Mannigfaltigkeit, wobei das Vektorfeld als Operator auf einem sogenannten anisotropen Sobolevraum aufgefasst wird. Das Gegenstück zu klassischen Resonanzen sind Quantenresonanzen, d.h. die Eigenwerte des Laplace-Beltrami-Operators. Wir betrachten, zunächst anhand eines einfachen Beispiels, Resultate und Forschungsfragen zu der Beziehung zwischen klassischen und Quanten-Resonanzen und den zugehörigen Resonanzzuständen.
Hadamard states for quantum Abelian duality
Montag, 29.1.18, 16:15-17:15, Raum 404, Eckerstr. 1
Cohomological properties of OT manifolds
Dienstag, 30.1.18, 11:00-12:00, SR 119
Conical spherical metrics: Lecture I
Dienstag, 30.1.18, 13:00-14:00, SR 119, Eckerstr. 1
Cone spherical, flat and hyperbolic metrics are conformal metrics with constant curvature +1, 0 and −1, respectively, and with finitely many conical singularities on compact Riemann surfaces. The Gauss-Bonnet formula gives a natural necessary condition for the existence of such three kinds of metrics with prescribed conical singularities on compact Riemann surfaces. The condition is also sufficient for both flat and hyperbolic metrics. However, it is not the case for cone spherical metrics, whose existence has been an open problem over twenty years.
Gradient Flow for a phase Field Model of the Willmore Energy
Dienstag, 30.1.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\nWe consider a phase field model for Willmore’s energy originally proposed by de Giorgi. In essence, the energy of this phase field model is given by taking the first variation of the well known Modica-Mortola energy and integrating the square of this variation. A Gamma-convergence result was proved in 2006 by Röger and Schätzle. In this presentation, we examine the viscous gradient flow of de Giorgi’s energy and prove existence of weak solutions using a Galerkin approximation. Finally, we give an outlook to the addition of further constraints for the energy, for example using a term to control certain topological properties of the phase field.
Conical spherical metrics: Lecture II
Mittwoch, 31.1.18, 10:15-11:15, SR 119, Eckerstr. 1
Conical spherical metrics: Lecture III
Donnerstag, 1.2.18, 13:00-14:00, SR 403, Eckerstr. 1
A multi-scale approach to reaction-diffusion processes in domains with microstructure
Donnerstag, 1.2.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Reaction-diffusion processes occur in many materials with microstructure such as biological cells, steel or concrete. The main difficulty in modelling and simulating accurately such processes is to account for the fine microstructure of the material. One method of upscaling multi-scale problems, which has proven reliable for obtaining feasible macroscopic models rigorously, is the method of periodic homogenisation. \n\nThe talk will give an introduction to multi-scale modelling of physico-chemical mechanisms in domains with microstructure as well as to the method of periodic homogenisation. Moreover, certain aspects particularly relevant in upscaling reaction-diffusion processes in biological cells will be discussed together with their applications.
Conical spherical metrics: Lecture IV
Freitag, 2.2.18, 10:15-11:15, Raum 404, Eckerstr. 1
Probing Solar and Stellar Physics by Helio- and Asteroseismology
Freitag, 2.2.18, 14:15-15:15, Hörsaal II, Albertstr. 23b
The Sun and the stars are subject to sound waves that probe their interiors. Observations of these stellar oscillations have emerged as a powerful tool to gain information on the processes inside the Sun and the stars.\n\nThrough helio- and asteroseismology detailed inferences of the stellar internal structure and of the physical processes inside stars can be obtained. In particular, helioseismology allows studying sunspots and other magnetic active areas on the Sun, which have an important impact on our technological society through potentially harmful solar eruptions.\n\nHowever, a complete understanding of the Sun, and in particular of its magnetism, can only be obtained by understanding the internal structure and properties of the stars in general. Asteroseismology offers solving this problem.
Various flavours of Chern classes
Montag, 5.2.18, 16:15-17:15, Raum 404, Eckerstr. 1
Characteristic classes of vector bundles provide an important tool to study these geometric objects using techniques from algebraic topology, i.e. cohomology. In my talk I will give an introduction to Chern classes, which are characteristic classes of complex vector bundles. I will present several points of view onto this topic, each emphasising a certain aspect of Chern classes. This will help to understand the significance of this machinery.
Hyperbolic AMD mass
Dienstag, 6.2.18, 16:15-17:15, Raum 404, Eckerstr. 1
This talk is about the hyperboic AMD mass.\n1. The background\n2. Its definition\n3. Its well-definedness and its invariance.\n4. A generalization.\n
Compactness and reflection in mathematics
Mittwoch, 7.2.18, 16:30-17:30, Raum 404, Eckerstr. 1
Abstract: One of the most fruitful research area in set theory is the study of the so-called reflections principles'. Roughly speaking, by reflection principle we mean a combinatorial statement of the following form: given a structure S (e.g. a stationary set, a tree, a graph, a groups ...) and a property P of the structure, the principle establishes that there exists a smaller substructure of S that satisfies the same property P. Compactness is dual to reflection, namely bycompactness property' we mean roughly a statement of the following form: given a structure S and a property P in the language of the structure, if every smaller substructure has the property P, then S satisfies P as well. \n\nMany interesting mathematical problems can be formulated as compactness problems; for instance, there is an extensive literature on the compactness problem for the property of being a free group: given a group G, suppose that every small subgroup (i.e. of smaller size) is free, is G itself free? This problem is independent from ZFC and the answer depends on the cardinality of the group. \n\nStrong forms of reflection are typically associated with large cardinals axioms, which therefore imply interesting compactness results. There is a tension between large cardinals axioms and the axiom of constructibility V=L at the level of reflection: on the one hand, large cardinals typically imply reflection properties, on the other hand L satisfies the square principles which are anti-reflection properties. Two particular cases of reflection received special attention, the reflection of stationary sets and the tree property. We will discuss the interactions between these principles and a version of the square due to Todorcevic. This is a joint work with Menachem Magidor and Yair Hayut. \n\n
tba
Mittwoch, 7.2.18, 16:30-17:30, Raum 404, Eckerstr. 1
Compactness and reflection
Mittwoch, 7.2.18, 17:30-18:30, Raum 404, Eckerstr. 1
tba
Donnerstag, 8.2.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Resolution of singularities of the cotangent sheaf of a singular variety
Freitag, 9.2.18, 10:30-11:30, Hörsaal FRIAS, Albertsstr. 19
The subject of the talk is resolution of singularities of differential forms on an algebraic or analytic variety. We address the problem of finding a resolution of singularities \(\bsigma : X \bto X_0 \) of a singular algebraic or analytic variety \(X_0\) such that the pulled back cotangent sheaf of \(X_0\) (i.e., the pull-back of the Kahler differential forms defined in \(X_0\)) is given, locally in \(X\), by monomial differential forms (with respect to a suitable coordinate system). This problem is related with monomialization of maps, the \(L^2\) cohomology of singular varieties and reduction of singularities of vector-fields. In a work in collaboration with Bierstone, Grandjean and Milman, we give a positive answer to the problem when \(dim\b, X_0 \bleq 3\).
On the topology of smooth hypersurfaces
Freitag, 9.2.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
To what extent the Chern class of a divisor (in singular\ncohomology) determines its topology?\nDiscussion of a conjecture by Totaro concerning the topology\nof smooth hypersurfaces on projective manifolds.
Algebraic curves and modular forms of low degree
Freitag, 23.2.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
For genus 2 and 3 modular forms are intimately connected with\nthe moduli of curves of genus 2 and 3. We give an explicit way to\ndescribe such modular forms for genus 2 and 3\nusing invariant theory and give some applications.\nThis is based on joint work with Fabien Clery and Carel Faber.\n
The McKay correspondence and the eta-invariant
Montag, 26.2.18, 14:00-15:00, Raum 404, Eckerstr. 1
GPSD 2018
Dienstag, 27.2.18, 09:00-10:00, Institusviertel
GPSD 2018
Mittwoch, 28.2.18, 09:00-10:00, Institusviertel
GPSD 2018
Donnerstag, 1.3.18, 09:00-10:00, Institusviertel
Beilinson conjectures for curves
Donnerstag, 1.3.18, 15:00-16:00, Raum 404, Eckerstr. 1
GPSD 2018
Freitag, 2.3.18, 09:00-10:00, Institutsviertel
Inference of biogeographical ancestry from SNP data -- an evaluation of selection methods and classifiers on a variety of SNP data sets
Mittwoch, 7.3.18, 14:30-15:30, Raum 404, Eckerstr. 1
Single nucleotide polymorphism in DNA have proven to be suitable for inferring biogeographical ancestry of human individuals. Various methods have been developed and recent articles in this field focus on their advantages and evaluate their qualities in a variety of settings and under different aspects, such as the ability to predict admixture rates, the dependency on assumptions or handling different rates of missing data. This thesis includes three aspects that are heavily linked:\nFirst, we test forward selection algorithms to select a minimal sufficient or maybe even best subset of SNPs for ancestry prediction from a given set of SNPs that may not be preselected. We compare the quality of predictions on SNP sets chosen by forward selection with different methods with those on a SNP set selected using a procedure that is based on FST values. Secondly we introduce a novel version of a naive Bayesian classifier.Different versions of Bayesian classifiers have been developed for this purpose and they show good performances. Finally we use SNP data simulation software to systematically test our methods and compare the Bayesian classifier with logistic regression, which is an established method in eye color prediction from SNP data. We investigate the impact of parameters, such as migration and the number of islands, on the prediction performances and conclude the analysis with comparing the results to those on real data sets.
siehe Webseite
Dienstag, 13.3.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19