Donnerstag, 3.1.13, 17:00-18:00, Hörsaal II, Albertstr. 23b
Laminations of solutions in the Frenkel-Kontorova model with weak coupling
Montag, 7.1.13, 16:15-17:15, Raum 404, Eckerstr. 1
Perfect subsets of generalized Baire spaces and Banach-Mazur games
Mittwoch, 9.1.13, 16:30-17:30, Raum 404, Eckerstr. 1
Let \(\bkappa\) be an uncountable cardinal with \(\bkappa^{< \bkappa} = \bkappa\). We consider the generalized Baire space of functions \(f : \bkappa \bto \bkappa\) with basic open sets \(U_s = \b{f \bin \bkappa^\bkappa \bmid s \bsubseteq f \b}\) for \(s \bin {}^{< \bkappa} \bkappa\). A subset of \(\bkappa^\bkappa\) is perfect if it is the set of branches of a \(< \bkappa\)-closed subtree of \({}^{< \bkappa} \bkappa\) which splits above every node. We prove that after an inaccessible \(\blambda > \bkappa\) is collapsed to \(\bkappa^+\), every set \(A \bsubseteq \bkappa^\bkappa\) definable from ordinals and subsets of \(\bkappa\) either has size \(\bleq \bkappa\) or a perfect subset, and that the Banach-Mazur game for \(A\) is determined.
Donnerstag, 10.1.13, 17:00-18:00, Hörsaal II, Albertstr. 23b
The Hele Shaw Flow and the Moduli of Holomorphic Discs
Freitag, 11.1.13, 10:00-11:00, Raum 404, Eckerstr. 1
The Hele-Shaw Flow is a model for describing the propagation of\nfluid in a cell consisting of two parallel places separated by a small\ngap. This model has been intensely studied for over a century, and is a\nparadigm for understanding more complicated systems such as the flow of\nwater in porous media, melting of ice and models of tumor growth.\n\nIn this talk I will discuss how this flow fits into the more general\nframework of "inverse potential theory" through the idea of complex\nmoments. I will then discuss joint work with David Witt Nystrom that\nconnects to the moduli space of holomorphic discs with boundary in a\ntotally real manifold. Using this we prove a number of short time\nexistence/uniqueness results for the flow, including the case of the Hele\nShaw flow with varying permeability starting from a smooth Jordan domain,\nand for the Hele Shaw flow starting from a single point.
Talk 1: A sharp quantitative isoperimetric inequality in higher codimension
Mittwoch, 16.1.13, 16:15-17:15, Hörsaal II, Albertstr. 23b
The Pila-Zannier proof of the Manin-Mumford conjecture
Mittwoch, 16.1.13, 16:30-17:30, Raum 404, Eckerstr. 1
Talk 2: A sharp quantitative isoperimetric inequality in higher codimension
Mittwoch, 16.1.13, 17:30-18:30, Hörsaal II, Albertstr. 23b
Donnerstag, 17.1.13, 17:00-18:00, Hörsaal II, Albertstr. 23b
Modified surgery theory - Application to Bott manifolds
Freitag, 18.1.13, 10:00-11:00, Raum 404, Eckerstr. 1
One of the fundamental questions in geometric topology is the question whether two manifolds are diffeomorphic. Surgery theory translated that question as follows: Assuming they are simply connected it now reads: Are our manifolds h-cobordant? But for certain settings yet another transformation of our question is useful leading to modified surgery theory. The key idea here is to compare controlled bordism classes of manifolds.\n\nIn my talk I will explain how one can apply this theory to problems related to Bott manifolds. Bott manifolds are a very nice and explicite class of manifolds given as iterated CP^1-fiber bundles. They are of special interest since, conjecturally, they are diffeomorphic if and only if their cohomology rings are isomorphic.
Adiabatic limit of the Ray-Singer torsion
Montag, 21.1.13, 16:15-17:15, Raum 404, Eckerstr. 1
Amoeba of Sacks forcing without Cohen reals
Mittwoch, 23.1.13, 16:30-17:30, Raum 404, Eckerstr. 1
Limitations of the Euler method
Donnerstag, 24.1.13, 17:00-18:00, Hörsaal II, Albertstr. 23b
Our goal is to calculate moments of stochastic differential equations (SDEs)\n(e.g. the stochastic Langevin equation, the stochastic Duffing-van der Pol\nequation, the stochastic Lotka-Volterra model, the Lewis-3/2-volatility\nmodel, the stochastic Lorenz equation). In the global Lipschitz case, the\napproximation method with optimal rate of convergence is the multilevel\nMonte Carlo Euler method. Most of the applied SDEs, however, have\nsuper-linearly growing coefficients. We show for this case that the\nmultilevel Monte Carlo Euler method does not even converge in general. The\nreason herefore is strong divergence of Euler's method. For this reason, we\nrecommend to be careful with Euler's method in applications. Instead, we\npropose a strongly converging numerical method with optimal rate of\nconvergence and provide its short Matlab code.\n
Canonical degree of curves on varieties of general type
Freitag, 25.1.13, 10:00-11:00, Raum 404, Eckerstr. 1
Analysis on crepant resolutions of Calabi-Yau orbifolds
Montag, 28.1.13, 16:15-17:15, Raum 404, Eckerstr. 1
A Calabi-Yau orbifold is locally modeled on C^n/G with G a finite subgroup of SU(n). If the singularity is isolated, then the crepant resolution (if it exists) is an ALE manifold, for which index-type results are well known. However, most of the time the singularity is not isolated, and for the corresponding crepant resolution there is no index theorem so far. In this talk, I present the first step towards obtaining such a result: I will introduce the class of iterated cone-edge singular manifolds and the corresponding quasi-asymptotically conical spaces (for which orbifolds and their crepant resolutions are an example), and build-up the general set-up for studying Fredholm properties of geometrical elliptic operators on these spaces. This is joint work with Rafe Mazzeo.\n
On rings of continuous p-adic valued functions
Dienstag, 29.1.13, 11:30-12:30, Raum 318, Eckerstr. 1
The Galois closure of ring extensions and a discriminant theorem of Stickelberger
Donnerstag, 31.1.13, 15:15-16:15, Raum 404, Eckerstr. 1
A classical result of algebraic number theory by L. Stickelberger states that the discriminant of a number field is an integer congruent to 0 or 1 modulo 4. We generalize this theorem to n-ic extensions of arbitrary base rings. To achieve this, we use the notion of Galois closures of arbitrary ring extensions, as introduced by M. Bhargava and M. Satriano, combined with new results from invariant theory generalizing the fundamental theorem of symmetric polynomials. In the end, we will also state a conjecture strengthening the given results.
Donnerstag, 31.1.13, 17:00-18:00, Hörsaal II, Albertstr. 23b