Vorsicht! Zeit- und Raumänderung: Mittwoch, 16:00 Uhr im Hörsaal des MI

Abstract: In light of the discoveries that the universe went through a phase of accelerated expansion and is in a state with extremely small vacuum energy, physicists are struggling to produce theoretical models that are consistent with these observations. I will discuss the potential implications of the recent detection of B mode polarization by the BICEP2 experiment. I will discuss two theoretical approaches giving rise to (1) large field inflation and (2) small field inflation. The first realizes N-flation via kinematic alignment in the axion kinetic term while the second approach aims towards an understanding inflation within random supergravity theories. Both approaches crucially rely on a detailed understanding of the geometry on which the underlying string theory is compactified.

Abstract: Let K be a field of characteristic zero. Consider the polynomial ring S=K[X_{i,j}]_{1\le i\le m,1\le j\le n} on the entries of a generic m\times n matrix X=(X_{i,j}). Let I_p be the ideal in S generated by p\times p minors of X. I explain how to calculate completely the local cohomology modules H^i_{I_p}(S). I will also explain why the problem is interesting. It turns put the result allows to classify the maximal Cohen-Macaulay modules of covariants for the action of SL(n) on the set of m n-vectors. It also allows to describe the equivariant simple D-modules, where D is the Weyl algebra of differential operators on the space of m\times n matrices. This is a joint work with Claudiu Raicu and Emily Witt. The relevant references are arXiv 1305.1719 and arXiv 1309.0617.

Abstract: Field theories, as axiomatised by Atiyah and Segal, have a natural 'multiplication' coming from the monoidal structure on the source and target categories. We show that when the target category is permutative (roughly meaning that it has two monoidal structures that interact as the addition and multiplication on a ring), that the field theories may also be 'added', and that the 'multiplication' distributes over the addition.

Abstract: The talk will discuss a random-matrix approach to quantum transport in chaotic quantum dots with one non-ideal lead and Dyson's symmetry parameter 1, 2 and 4. The reflection eigenvalues (the fundamental quantities of the theory) are shown to form a novel probability ensemble, described in terms of Jack polynomials, which are objects appearing in various settings in mathematics and physics; an introduction to this subject is given. This ensemble reveals links to various challenging mathematical questions.

Abstract: Atiyah's Reelle K-Theorie, oder KR-Theorie, ist eine Verallgemeinerung topologischer K-Theorie auf Räume mit Involution, die reelle und komplexe topologische K-Theorie als Spezialfälle enthält. Nach Einführung der grundlegenden Definitionen, werde ich ein Modell für das konnektive Cover kr als C_2-Spektrum vorstellen, welches auf Segal und Suslin zurückgeht. Anschließend werde ich, hierbei Suslin folgend, zeigen, wie aus Segal's Arbeit zu Gamma-Räumen eine Version der (1, 1)-Periodizität für KR-Theorie folgt.

Abstract: Random matrix theory has recently raised a lot of interest both in mathematics and physics. It is one of the few solvable models, said to be integrable, which at the same time can be explicitly solved and has many applications ranging from biology to high-energy physics. This makes it not only a formidable toy model but also an important tool for studying modern complex systems as well as dualities in high-energy physics and mathematics. In particular, it was recently understood that, in some regime, a large class of matrix models can be solved by a universal inductive method called topological recursion. In this elementary talk, I will review some of the main applications of this new method in problems such as statistical physics on a random lattice, combinatorics, Gromov-Witten theories, Givental theory, knot theory or integrable systems in a larger sense. I will show how combinatorics allows very often to fill the gap between a solvable system and its solution.

Vorsicht! Zeit- und Raumänderung: Mittwoch, 10:15 Uhr im Seminarraum B der Chemie

Abstract: Let G be a complex reductive group and V a representation space of G. Then there is a quotient space Z and a canonical map pi: V-->Z. The quotient space Z has a natural stratification which reflects properties of the G-action on V. Let phi: V-->Z be an automorphism. Then one can ask the following questions. (1) Does phi automatically preserve the stratification? (2) Is there an automorphism Phi: V-->Z which lifts phi? This is, can we have pi(Phi(v))=phi(pi(v)) for all v in V. If so, can we choose Phi to be equivariant, i.e., can we have that Phi(gv)=g Phi(v) for all v in V and g in G? We give conditions for positive responses to these questions, expanding upon work of Kuttler and Reichstein.

Abstract: We propose a new approach to define theories of random metrics in two and higher dimensions, based on recent methods in Kahler geometry. The main idea is to use finite dimensional spaces of Bergman metrics, parameterized by large N hermitian matrices, as an approximation to the full space of Kahler metrics. This approach suggests the relevance of a new type of gravitational effective actions, corresponding to the energy functionals in Kahler geometry. These actions appear when a non-conformal field theory is coupled to gravity, and generalize the standard Liouville model in two dimensions.

Abstract: Spherical representations are interesting because their matrix coefficients are eigenfunctions of invariant differential operators on symmetric spaces. A classical theorem proven by Helgason in his '84 book classifies all spherical representations of a non-compact semisimple Lie group with Iwasawa decomposition G=KAN. It states that a given finite dimensional irreducible representation (which is necessarily a highest weight representation) contains a non-zero K-invariant vector if and only if the highest weight vector is M-invariant, where M is the centralizer of A in K. This immediately translates into a condition on the highest weight. For the simplest example, the spherical representations of su(2) containing a u(1) invariant vector are exactly those with even highest weight. (For physicists: The multiplets with integer total spin are exactly those containing a state with vanishing magnetic quantum number.) In my Diploma thesis we could fully generalize this results to the case of gl(q|r+1) (for r>q or high enough highest weight). A necessary condition for a representation to be spherical is given for any gl(q|r+s) and in fact also for any strongly reductive symmetric pair. The method of prove is similar to the one used by Schlichtkrull ('84).