Kreck-Stolz-Invarianten der Grove-Wilking-Ziller-Familie N
Monday, 1.12.14, 16:15-17:15, Raum 404, Eckerstr. 1
The Existence of Hermitian-Yang-Mills connection over compact Kähler manifold
Tuesday, 2.12.14, 16:00-17:00, Raum 404, Eckerstr. 1
Silver trees and Cohen reals
Wednesday, 3.12.14, 16:30-17:30, Raum 404, Eckerstr. 1
I will sketch the main ideas of my recent result that the\nmeager ideal is Tukey reducible to the Mycielski ideal. The latter one is\nthe ideal associated with Silver forcing. This implies that every\nreasonable amoeba forcing for Silver adds a Cohen real. This has been open\nfor some years.\n
On Shoenfield's Absoluteness Theorem
Thursday, 4.12.14, 17:00-18:00, Hörsaal II, Albertstr. 23b
In 1961, Joseph R. Shoenfield (1927 - 2000) proved the following theorem, that later has been called Shoenfield's Absoluteness Theorem:\n\nEvery Sigma^12(a) relation and every Pi^12(a) relation is absolute for all inner models M of the Zermelo-Fraenkel axioms and dependent choice that contain the real number a as an element.\n\nThe notions will be explained. On our way towards a sketch of proof we will encounter computable reals (same as in computer science and in numerical mathematics), arithmetical properties (same as in algebra and in number theory), provability (same all over classical mathematics), and Borel sets (same as in measure theory and in probability theory). Absoluteness of a relation, and in particular absoluteness of truth of a statement, is a useful property. The axiomatic background, Zermelo-Fraenkel and dependent choice, is much weaker than the axiomatic basis, e.g., for Linear Algebra 1.\n
Conjugacy classes of \(n\)-tuples in semi-simple Jordan algebras
Friday, 5.12.14, 10:15-11:15, Raum 404, Eckerstr. 1
Let \(J\) be a (complex) semi-simple Jordan algebra, and consider the action of the automorphism group acts\non the \(n\)-fold product of \(J\) via the diagonal action. In the talk, geometric properties of this action\nare studied. In particular, a characterization of the closed orbits is given.\n\nIn the case of a complex reductive linear algebraic group and the adjoint action on the \(n\)-fold product\nof its Lie algebra, a result of R.W. Richardson characterizes the closed orbits. A similar condition can\nbe found in the case of Jordan algebras. It turns out that the orbit through an \(n\)-tuple \(x=(x_1,\bldots, x_n)\)\nis closed if and only if the Jordan subalgebra generated by \(x_1,\bldots, x_n\) is semi-simple.\n\nFor the proof, the existence of certain one-parameter subgroups of the automorphism group is important. Those\none-parameter subgroups have special properties with respect to a given subalgebra of the Jordan algebra \(J\).
Risk sensitive utility indifference pricing of perpetual American options under fixed transaction costs
Friday, 5.12.14, 11:30-12:30, Raum 404, Eckerstr. 1
The problems of risk sensitive portfolio optimisation under transaction costs have taken a considerable attention in the recent literature on mathematical finance. We study the associated problems of risk sensitive utility indifference pricing for perpetual American options with fixed transaction costs in the classical model of financial market with two tradable assets. Assume that the investors trading in the market must pay transaction costs equal to a fixed fraction of the entire portfolio wealth each time they trade. The objective is to maximise the asymptotic (risk null and risk adjusted) exponential growth rates based on the expected logarithmic or power utility of the difference between the terminal portfolio wealth and a certain amount of the option payoffs. It is shown that the optimal trading policy keeps the number of shares held in the assets unchanged between the transactions. In order to determine the optimal trading times and sizes, we reduce the initial problems to the appropriate (discounted) time-inhomogeneous optimal stopping problems for a one-dimensional diffusion process representing the fraction of the portfolio wealth held by the investor in the risky asset. The optimal trading and exercise times are proved to be the first times at which the risky fraction process exits certain regions restricted by two time-dependent boundaries. Then, certain amounts of assets should be bought or sold or the options should be exercised whenever the risky fraction process hits either the lower or the upper time-dependent curve. The latter are characterised as unique solutions of the associated parabolic-type free-boundary problems for the value functions satisfying the smooth-fit conditions at the curved boundaries. The optimal asymptotic growth rates and trading sizes are specified as parameters maximising the value functions of the resulting optimal stopping problems. We illustrate these results on the examples of the perpetual American call and put as well as the asset-or-nothing options, for which we obtain the utility indifference prices as well as the optimal trading and exercise boundaries in a closed form.
Quantum cohomology of affine flag manifolds and periodic Toda lattices
Monday, 8.12.14, 16:15-17:15, Raum 404, Eckerstr. 1
A theorem of Bumsig Kim (1999) says that the quantum cohomology ring of a full flag manifold (i.e. generic adjoint orbit of a compact Lie group) is determined by a certain integrable system, the open Toda lattice\n, which is canonically associated to the Lie group.\nIn my talk I will present this result in some more detail and then I will explain how one can extend it to the context of affine Kac-Moody flag manifolds. The quantum cohomology ring is this time determined by\nanother integrable system, the periodic Toda lattice. This has been observed by Martin Guest and Takashi Otofuji (2001) for some particular flag manifolds.\nExtensions of their result have been obtained recently by Leonardo Mihalcea and myself in a joint work.\nThey will be outlined in the talk. \n\n
Schanuel's Conjecture and Exponential Fields
Wednesday, 10.12.14, 16:30-17:30, Raum 404, Eckerstr. 1
Schanuel's Conjecture states that for a collection of n complex\nnumbers z1, ..., zn, linearly independent over the field of\nrational numbers, the transcendence degree of z1, ..., zn,\nexp(z1), ..., exp(zn) is at least n.\n\nZilber constructs in [Zilber, Pseudo-exponentiation on\nalgebraically closed fields of characteristic zero] a sentence\nwhose models are structures called strongly\nexponentially-algebraically closed fields with\npseudo-exponentiation, which are unique in every uncountable\ncardinality. One of their main properties is that Schanuel's\nConjecture holds in those fields.\n\nFirstly, I will outline the properties of Zilber's fields.\nSecondly, I will sketch the proof given in [Marker,\nA Remark on Zilber's Pseudoexponentiation] showing that, if one\nassumes Schanuel's Conjecture, the simplest case of one of the\naxioms of Zilber's fields holds in the complex exponential field.\n
Thursday, 11.12.14, 17:00-18:00, Hörsaal II, Albertstr. 23b
On automorphic forms for Calabi-Yau threefolds
Friday, 12.12.14, 10:15-11:15, Raum 404, Eckerstr. 1
I will present a novel approach to relate Hodge theory of elliptic curves to quasimodular forms. Then we consider its generalization to the Hodge theory of Calabi-Yau threefolds, leading to the appearance of a new family of Lie algebras.
Eta-forms for fibrewise Dirac operators with kernel over a hypersurface
Monday, 15.12.14, 16:15-17:15, Raum 404, Eckerstr. 1
Amoeba and tree ideals
Wednesday, 17.12.14, 16:30-17:30, Raum 404, Eckerstr. 1
I will talk about what I asked to Spinas in the end of his talk, i.e., whether an\namoeba for Silver might add Cohen reals. Two weeks ago he proved that add(J(Silver)) is at most\nadd(M). However this is not strictly sufficient to infer that any proper amoeba for Silver does\nadd Cohen reals, but only that it does not have the Laver property. I will clarify this\nissue. If there will be any time left I will also present some results about other tree ideals,\nwhich are part of a joint work, still in preparation, with Yurii Khomskii and Wolfgang\nWohofsky.\n
Khovanov homology and the geometry of Springer fibers
Thursday, 18.12.14, 17:00-18:00, Hörsaal II, Albertstr. 23b
Walled Brauer algebra and higher Schur-Weyl duality
Friday, 19.12.14, 10:15-11:15, Raum 404, Eckerstr. 1
Thursday, 25.12.14, 17:00-18:00, Hörsaal II, Albertstr. 23b