Das allgemeine Kolloquium des mathematischen Instituts findet während der Vorlesungszeit donnerstags um 17:15 Uhr im Raum 05-426 statt. Ab 16:45 Uhr gibt es im Hilbertraum (05-432) Kaffee und Kuchen.
(Teilnahme ggf. auch online möglich, Zugangsdaten über den Koll.beauftragten.)
Programm Sommersemester 2023
27.4. Prof. Dr. Hans Jockers (JGU Mainz)
The Quest of Hyperbolic 3-Manifolds in Mirror Symmetry
Abstract:
Mirror symmetry predicts the “classical" algebraic and transcendental invariants of degenerate Calabi-Yau threefold to match with the symplectic “quantum” invariants of the mirror Calabi-Yau manifold. An instance of this correspondence arises from open-string mirror symmetry, in which algebraic cycles of the degenerate Calabi-Yau threefold correspond to Lagrangian submanifolds of the mirror manifold. I discuss this open-string mirror symmetry correspondence, and I illustrate how to calculate invariants in this context. These results propose a connection to hyperbolic 3-manifolds.
4.5. Prof. Dr. Felix Finster (Univ. Regensburg)
An introduction to causal fermion systems and the causal action principle
Abstract: The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. Moreover, causal fermion systems provide a general framework for modelling and analyzing non-smooth spacetime structures. The dynamics of a causal fermion system is described by a nonlinear variational principle, the causal action principle. In the talk I will give an introduction from the point of view of geometry and the calculus of variations.
11.5. Prof. Dr. Stefan Schröer (Univ. Düsseldorf)
Algebraic surfaces over the integers
Abstract: In this talk I give a gentle introduction to a general problem in arithmetic algebraic geometry:
What geometric objects can be defined by polynomials with integral coefficients such that no singularities arise over any prime field? After discussing the theorems of Minkowski, Tate, Ogg, Fontaine and Abrashkin I will explain some recent results on Enriques surfaces.
25.5. Prof. Alexander Kurganov (Southern University of Science and Technology, Shenzhen, China)
Low-Dissipation Central-Upwind Schemes
Abstract: The talk will be focused on central-upwind schemes, which are simple, efficient, highly accurate and robust Godunov-type finite-volume methods for hyperbolic systems of conservation and balance laws. I will first briefly go over the main three steps in the derivation of central-upwind schemes. First, we assume that the computed solution is realized in terms of its cell averages, which are used to construct a global in space piecewise polynomial interpolant. We then evolve the computed solution according to the integral form of the studied hyperbolic system. The evolution is performed using a nonsymmetric set of control volumes, whose size is proportional to the local speeds of propagation: this allow one to avoid solving any (generalized) Riemann problems. Once the solution is evolved, it must be projected back onto the original grid as otherwise the number of evolved cell averages would double every time step and the scheme would become impractical. The projection should be carried out in a very careful manner as the projection step may bring an excessive amount of numerical dissipation into the resulting scheme as was the case in previous versions of the central-upwind schemes.
In order to more accurately project the solution, we have recently introduced a new way of making the projection. A major novelty of the new approach is that we use a subcell resolution and reconstruct the solution at each cell interface using two linear pieces. This allows us to perform the projection in the way, which would be extremely accurate in the vicinities of linearly degenerate contact waves. This leads to the new second-order semi-discrete low-dissipation central-upwind schemes, which clearly outperform their existing counterparts as confirmed by a number of numerical experiments conducted for both the 1-D and 2-D Euler equations of gas dynamics in both single- and multifluid settings.
The accuracy of the low-dissipation central-upwind schemes can be further increased in two ways. First, we develop a scheme adaption strategy: we automatically detect "rough" parts of the computed solution and apply an overcompessive slope limiter in these areas at the piecewise linear reconstruction step. The adaptive low-dissipation central-upwind schemes achieve a superb resolution in a variety of challenging numerical examples. Second, we utilize the new low-dissipation central-upwind numerical fluxes to construct new fifth-order finite-difference A-WENO schemes, which outperfom their existing A-WENO counterparts based on less accurate central-upwind numerical fluxes.
1.6. Prof. Dr. Thomas Schick (Göttingen):
Rigidity of scalar curvature
Abstract: The round metric on the n-dimensional sphere is very special.
By a celebrated theorem of Llarull, it has e.g. the property to be extremal
among metrics whose scalar curvature is nowhere smaller than the one of the sphere
(in the sense: one has to shrink the metric somewhere to increase the scalar curvature).
Indeed, this even holds if one allows to change the topology.
If n=2, this can be derived from the Gauss-Bonnet theorem, in higher dimensions one uses
the spectral theory of the Dirac operator.
We discuss these classical results and recent improvements (jointly obtained with Cecchini and Hanke)
which allow for metrics and comparison maps of low regularity.
15.6. Festkolloquium Manfred Lehn:
Coffee at 3pm in Hilbertraum (Details and program under this link)
Lectures by:
Prof. Dr. Dmitry Kaledin (HSE Univ. Moscow)
Geometry and topology of symplectic resolutions
Prof. Dr. Christoph Sorger (Nantes Univ.)
The topology of Hilbert schemes
22.6. Dr. Hans Fischer (Kath. Univ. Eichstätt)
Real Analysis 1830–1850: Propagation and Further Development of Cauchy's Basic Concepts by Peter Gustav Lejeune Dirichlet
Abstract: Peter Gustav Lejeune Dirichlet (1801–859) is considered as one of the most significant promulgators of rigorous analytic standards in the pre-Weierstrass era. Through his studies in Paris (1822–1826) he became especially influenced by Cauchy's "new" analysis, and he adopted and modified its most important concepts, as one can see from some of his papers and in particular from lecture notes. In this talk I will explain in which way Dirichlet adapted or specified Cauchy's notions of function, continuity including uniform continuity, and definite integrals in one and two dimensions. Finally, the question will be briefly discussed what influence Dirichlet actually had on the development of "epsilontic" analysis.
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