We show that for a special class of geometric quantizations with "small" quantum errors, the quantum classical correspondence gives rise to an approximate projective representation of the group of Hamiltonian diffeomorphisms. As an application, we get an obstruction to Hamiltonian actions of finitely presented groups. Joint work with Laurent Charles.

We show that correlation matrices with particular average and variance of the correlation coefficients have a notably restricted spectral structure. Applying geometric methods, we derive lower bounds for the largest eigenvalue and the alignment of the corresponding eigenvector. We explain how and to which extent, a distinctly large eigenvalue and an approximately diagonal eigenvector generically occur for specific correlation matrices independently of the correlation matrix dimension.

We present two results about subelliptic PDEs. The first one (joint with Chenmin Sun) concerns observability for a family of subelliptic Schrodinger equations: using tools of semiclassical analysis, we prove a resolvent estimate which shows that energy propagates slowly in directions which need brackets to be generated. The second one (joint with Yves Colin de Verdiere) is a general result about the propagation of singularities for subelliptic PDEs, formulated with the help of sub-Riemannian geometry.

In this talk, we will start with a conjecture posed by Cheng, which states that the Bergman metric of a bounded, strongly pseudoconvex domain in $\mathbb{C}^n$ with smooth boundary is Kahler-Einstein if and only if the domain is biholomorphic to the unit ball $\mathbb{B}^n$. Then we will discuss the recent developments on solving and generalizing Cheng's conjecture. The talk is based on a joint paper with Huang, and a recent preprint with Ebenfelt and Xu.

Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semiclassical analysis. In this talk I will describe how to associate classes of semiclassical oscillating functions to isotropic submanifolds in phase space, and show that these classes are invariant under the action of arbitrary Fourier integral operators (modulo the usual clean intersection condition). Some special classes and applications will also be discussed. This is based on joint works with V. Guillemin (MIT) and A. Uribe (U. Michigan).

Let $X$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature, which admits a connected compact Lie group $G$ action. In this talk, we will discuss a Boutet de Monvel-Sjostrand type theorem for $G$-equivariant Szego kernels $S_k^{(q)}$. When $X$ admits also a transversal CR $S^1$ action, we study the asymptotics of Fourier components of $S_k^{(q)}$. If time permits, we will show the coefficients of lower order terms in the asymptotic expansion when $X$ is strongly pseudoconvex.

In this talk, we discuss the classification problem of Levi flat hypersurfaces in complex surfaces by restricting ourselves to the case that the Levi foliation is transversely affine. After presenting known examples, we give a proof for the non-existence of real analytic Levi flat hypersurface whose complement is 1-convex and Levi foliation is transversely affine in a compact Kaehler surface. This is a joint work with Severine Biard (arXiv:2011.06379).

In this talk, I will give a discussion when a Stein space with compact strongly pseudoconvex boundary is biholomorphic to a ball under the assumption the Bergman metric is K ahler-Einstein. This talk is based on a joint work with Professor Xiaojun Huang.

A compact subset of $\mathbb{C^n}$ is polynomially convex if it is defined by a family of polynomial inequalities. In this talk, we will elaborate on this definition and discuss some questions and recent results regarding the minimum embedding (complex) dimension of abstract compact (real) manifolds subject to this convexity constraint. The primary challenge arises from the CR-singularities of a generic embedding. We will explain why this is the case, and discuss the main techniques that have been used in this problem so far. This is joint work with R. Shafikov.

We consider the CR equivalence problem of 5-dimensional real hypersurfaces in C3 that are Levi degenerate of constant rank 1, and are 2 non-degenerate in the sense of Freeman; and we will review some of the progress made in this area. A well-known homogeneous model is the tube over the future light cone, whose Lie algebra of infinitesimal CR automorphism is isomorphic to so(2,3). In recent joint works with Zhangchi Chen, Joel Merker, and The-Anh Ta, we study such manifolds in the rigid setting, giving rise to 7-dimensional Lie sub-algebra of so(2,3), with 2 primary invariants.

Firstly, we recall Siu and Demailly's proof of Grauert-Riemenschneider conjecture, and Berndtsson's refined estimates on the cohomology of semipositive line bundles over compact complex manifolds. Secondly, on possibly non-compact complex manifolds, we present the refined estimate for the dimension of the space of harmonic (0,q)-forms with values in high tensor powers of semipositive line bundle when the fundamental estimate holds. In the end, we discuss two questions about nef and pseudoeffective line bundles given by Demailly and Matsumura analogue to Berndtsson's estimates. Both the background and the recent progresses of this topic would be introduced.

We obtain a localization formula in differential K-theory for S1-action. Then by combining an extension of Goette's result on the comparison of two types of equivariant $\eta$ invariants, we establish a version of localization formula for equivariant $\eta$ invariants. An important step of our approach is to construct a pre-$\lambda$-ring structure in differential K-theory which is interesting in its own right.

The study of the Szego kernel is a classical subject in several complex variables and Cauchy--Riemann geometry. For example, when $X$ is the boundary of a strongly pseudoconvex domain, Boutet de Monvel and Sjostrand proved that locally the Szego kernel $\Pi^{(0)}(x,y)$ on $(0,0)$-forms is a Fourier integral operator with complex-valued phase function. This kind of description of kernel function has profound impact in many aspects, such as spectral theory for Toeplitz operator, geometric quantization and Kahler geometry. In this talk, we let $(X,T^{1,0}X)$ be a compact CR manifold of real dimension $2n+1$, $n\geq 2$, admitting a $S^1\times T^d$ action, $d\geq 1$. We consider a lattice point $(p_1,\cdots,p_d)\in Z^d$, where $(-p_1,\cdots,-p_d)$ is a regular value of the associate CR moment map $\mu$. Our goal is to study the asymptotic expansion of the corresponding torus equivariant Szego kernel $\Pi^{(0)}_{m,mp_1,\cdots,mp_d}(x,y)$ as $m\to\infty$ under certain assumptions on $Y:=\mu^{-1}(-p_1,\cdots,-p_d)$. As a corollary, we find a condition when the space of R-equivariant CR functions on an irregular Sasakian manifold is non-trivial in semi-classical limit.

We introduce a method of geometric quantization for of non-compact symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We then apply it to a class of compact manifolds with singular symplectic structure, called b-symplectic manifolds. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg ``quantization commutes with reduction'' property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper. (joint with Yiannis Loizides and Yanli Song)

The frame flow on hyperbolic 3-manifolds is an example of a flow that is only partially hyperbolic (Anosov) due to the presence of an additional neutral direction besides the flow direction. I will present some very recent results, obtained together with Colin Guillarmou, on the existence of a spectral gap in the resonance spectrum of the generator of the frame flow on hyperbolic 3-manifolds, based on Fourier analysis and a semiclassical calculus for powers of line bundles.

In this talk, we discuss a notion of semi-isometric immersions of strictly pseudoconvex CR manifolds into a Kahler manifold. This is applied in particular to estimate the first positive eigenvalue of the Kohn Laplacian on compact manifolds, to determine the CR umbilical points, and to characterize the totally umbilic pseudohermitian submanifolds of the complex euclidean space.

We construct a pointwise parametrix of Boutet de Monvel- Sjostrand type for weakly pseudoconvex three dimensional CR manifolds of finite type assuming the range of the tangential CR operator to be closed. This improves the earlier analysis of Christ and consequently generalizes an embedding theorem of Lempert. Joint work with C-Y. Hsiao.

Recently Dyatlov and Zworski proved that the order of vanishing of the Ruelle zeta function at zero, for the geodesic flow of a negatively curved surface, is equal to the negative Euler characteristic. They more generally considered contact Anosov flows on 3-manifolds. In this talk, I will discuss an extension of this result to volume-preserving Anosov flows, where new features appear: the winding cycle and the helicity of a vector field. A key question is the (non-)existence of Jordan blocks for one forms and I will give an example where Jordan blocks do appear, as well as describe a resonance splitting phenomenon near contact flows. This is joint work with Gabriel Paternain.

Let $M$ be a complex manifold of dimension $n$ with smooth connected boundary $X$. Assume that $\overline{M}$ admits a holomorphic $S^1$-action preserving the boundary $X$ and the $S^1$-action is transversal and CR on $X$. We show that the $m$-th Fourier component of the $q$-th Dolbeault cohomology group $H^q_m(\overline{M})$ is finite dimensional, for every $m\in\mathbb{Z}$ and every $q=0,1,\ldots,n$. This enables us to define$\sum^{n}_{j=0}(-1)^j{\rm dim \,} H^j_m(\overline{M})$ the $m$-th Fourier component of the Euler characteristic on $M$ and to study large $m$-behavior of $H^q_m(\overline{M})$.In this talk, I will present an indexformula for $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^j_m(\overline{M})$ and Morse inequalities for $H^q_m(\overline{M})$. This is based on a joint work with Chin-Yu Hsiao, Xiaoshan Li and Guokuan Shao.

Let $B$ be a compact integral affine manifold. If there is an atlas whose coordinate changes are not only affine but also preserve the lattice $\mathbb{Z}^n$, then there is a well-defined notion of "integral points" in $B$, and we call $B$ an \emph{integral integral affine manifold.} I will discuss the relation of integral integral affine structures to quantization as well as some associated results, in particular the fact that for a regular Lagrangian fibration $M \to B$, the Riemann-Roch number of $M$ is equal to the number of "integral points" in $B$. Along the way we encounter the fact that the volume of $B$ is equal to the number of integral points, a simple claim from "integral integral affine geometry" whose proof turns out to be surprisingly tricky. This is joint work with Yael Karshon and Takahiko Yoshida.

An important comparison theorem in global analysis is the comparison of analytic and topological torsion for smooth compact manifolds equipped with a unitary flat vector bundle. It has been conjectured by Ray and Singer and has been independently proved by Cheeger and Mueller in the 70ies. Bismut and Zhang combined the Witten deformation and local index techniques to generalise the result of Cheeger and Mueller to arbitrary flat vector bundles with arbitrary Hermitian metrics. The aim of this talk is to present an extension of the Cheeger-Mueller theorem to spaces with isolated conical singularities by generalising the proof of Bismut and Zhang to the singular setting.

In this talk, I will first explain how one can reformulate the known semiclassical limit results for the heat trace of Schrodinger operators on Riemannian manifolds in a form which makes sense for abstract Schrodinger type operators on locally compact spaces. These are results of the form ``the quantum partition function converges to the classical partition function as the Planck parameter tends to zero''. Then I will give a probabilistic proof of this reformulation in case the ``free operator'' stems from a regular Dirichlet form which satisfies a principle of not feeling the boundary. This abstract result leads to completely new results for Schrodinger operators on arbitrary Riemannian manifolds, and simultaniously allows to recover also some results for weighted infinite graphs.

Almost periodic chains (models for incommensurate phases) and quasiperiodic chains (models for quasicrystals) have a richer topological phase structure than periodic chains, due to the existence of phasons. This effects the bulk boundary correspondence for these models. We investigate in particular quasiperiodic chains where the phason degree of freedom lives in a totally disconnected space. We show how we can nevertheless define a winding number for the boundary resonances and thus obtain an equation which relates the integrated density of states at a gap to a winding number which can be interpreted as the work the phason motion exhibits on the edge states of the system.

We generalize Riemann-Roch-Grothendieck theorem on the level of differential forms for families of Riemann surfaces with hyperbolic cusps. The study of the spectral properties of the Kodaira Laplacian lies in the heart of our approach. When applied directly to the moduli space of punctured stable curves, our main result gives a formula for the Weil-Petersson form in terms of the first Chern form of the Hodge line bundle, which generalizes the result of Takhtajan-Zograf. Our result gives also some non-trivial consequences on the growth of the Weil-Petersson form near the compactifying divisor of the moduli space, which permits us to give a new approach to some well-known results of Wolpert on the Weil-Petersson geometry of the moduli space of curves.

Abstract: Let $f:\mathbb{C}\rightarrow\mathbb{P}^2(\mathbb{C})$ be an entire holomorphic curve and let $\{L_i\}_{1\leq i\leq q}$ be a family of $q\geq 4 $ lines in general position in projective plane. If $f$ is linearly nondegenerate, i.e. its image is not contained in any line, then the classical Second Main Theorem of Cartan states that the following inequality holds true outside a subset of $(0,\infty)$ of finite Lebesgue measure: \[ (q-3)\,T_f(r) \leq \sum_{i=1}^qN_f^{[2]}(r,L_i) + o(T_f(r)). \] Here $T_f(r)$ and $N_f^{[2]}(r,L_i)$ stand for the characteristic function and the $2$-truncated counting functions in Nevanlinna theory. It is conjectured that in the above estimate, the truncation level of the counting functions can be decreased to $1$, provided that that $f$ is algebraically nondegenerate (i.e. its image is not contained in any algebraic curve). In this talk, we will provide a partial answer to this conjecture in the special case where $f$ clusters to some algebraic curve. We also want to propose a strategy to achieve the full proof.

Abstract: Let $S_1,...,S_N$ be a collection of codimension 2 smooth submanifolds, of arbitrarily complicated topology, in the round sphere $\mathbb{S}^n$, $n \geq 3$, ($N$ being the complex dimension of the spinor bundle). In this talk I will show that there is always an eigenfunction $\psi:=(\psi_1,..., \psi_N)$ of the Dirac operator for which each submanifold $S_j$ is (modulo ambient diffeomorphism) a structurally stable nodal set of the spinor component $\psi_j$. These structures appear at small scales and sufficiently high energies. The result holds for any choice of trivialization of the spinor bundle.

Abstract: We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.

Abstract: This talk is devoted to the spectrum of the electro-magnetic Laplacian $(-ih\nabla-A)^2-hB$ on a bounded, regular, and simply-connected open set of the plane. Here $B$ is the magnetic field associated with $A$. We will see that, when the magnetic field is positive (and under generic conditions), we can accurately describe the low-lying eigenvalues in the semiclassical limit $h\to 0$. We will show the crucial role of the magnetic Cauchy-Riemann operators (and of their ellipticity), of the Bergman-Hardy spaces, and of the Riemann mapping theorem in the description of the spectrum. This is a joint work with Jean-Marie Barbaroux, Loïc Le Treust, and Edgardo Stockmeyer.

Abstract: In many cases, the size of torsion subgroups in the cohomology of a closed manifold can be studied by computing the Ray-Singer analytic torsion, which can be related to the topological torsion by the Cheeger-Müller theorem. Müller initiated the study of asymptotic analytic torsions associated with a family of flat vector bundles on compact locally symmetric spaces. Bismut, Ma and Zhang considered the analytic torsion forms in the more general context and they introduced the W-invariants to denote the leading terms of asymptotic analytic torsion forms. Here we consider the leading term of the asymptotics of equivariant analytic torsions on compact locally symmetric spaces, which suggests an extension of Bismut-Ma-Zhang's results to the equivariant case.

Abstract: The vortex equations provide an equivariant generalization of Gromov-Witten theory for Kähler manifolds X equipped with a holomorphic Hamiltonian action of a compact Lie group. Their moduli spaces support Kähler structures which are invaluable to understand certain gauge theories (for example gauged sigma-models, but not only) at both classical and quantum level. In my talk, I shall describe the geometric quantization of the moduli spaces of vortices in line bundles (i.e when X=C with usual circle action) on a compact Riemann surface \Sigma with fixed compatible area form \omega_\Sigma. As complex manifolds, the moduli spaces identify with symmetric powers of \Sigma. A crucial ingredient of our construction is the Deligne pairing of line bundles over a familiy of curves, which carries a metric defined in terms of Quillen's metric on a determinant of cohomology. In a natural complex polarization, the resulting quantum Hilbert spaces are finite-dimensional, and they can be interpreted as spaces of multi-spinors on \Sigma valued in a prequantization of an integral rescaling of \omega_\Sigma. I will also address the issue of relating Hilbert spaces corresponding to different quantization data geometrically. Joint work with Dennis Eriksson.

Abstract: Let F be a compact with piecewise smooth boundary in n dimensional Euclidean space and N_k the dimension of the vector space of the restrictions of real polynomials of n variables to K. A Fekete point of order k is a point in F^{N_k} maximising the determinant of a certain matrix of Vandermonde type. These points are important in the interpolation problem of continuous functions on F by polynomials. By using deep tools from complex geometry and pluripotential theory, we prove that Fekete points of large order are equidistributed toward a canonical measure associated to F with an explicit speed of convergence. Such equidistribution is crucial for approximation of Fekete points in practice.

Abstract: There are several well-known embedding theorems for real and complex manifolds, the most prominent one arguably being the embedding theorem of Whitney. If one adds an additional structure to the manifold in the form of a Lie-group action, one may ask about the existence of embeddings that respect the group action, which are called equivariant embeddings. I will give an introduction into this topic and talk about my results for non-proper group actions. I will also give a small introduction into the theory of moment maps and their connection to CR manifolds.

Abstract: An unexpected outcome of the recent proof of the 30 year old conjecture that disks and annuli are the only domains where analytic content - the uniform distance from z bar to analytic functions - achieves its lower bound, is a new insight into projective connections and the classification of quadratic differential spaces. In particular, we reveal a new relation between the symmetry constraints characterizing extremal domains (in the approximation theory sense) and invariance groups for projective connections in the case of finite-genus Riemann surfaces.

Abstract: We show that the (graded) spectral flow of a family of Toeplitz operators on a complete Riemannian manifold is equal to the index of a certain Callias-type operator. When the dimension of the manifold is even this leads to a cohomological formula for the spectral flow. As an application, we compute the spectral flow of a family of Toeplitz operators on a strongly pseudoconvex domain. This result is similar to the Boutet de Monvel's computation of the index of a single Toeplitz operator on a strongly pseudoconvex domain. Finally, we show that the bulk-boundary correspondence in the Graf-Porta model of topological insulators is a special case of our result.

Abstract: Let $f: X\to P^1$ be a meromorphic function of degree N with simple poles and simple critical points on a compact Riemann surface $X$ of genus $g$ and let $m$ be the standard round metric of curvature $1$ on the Riemann sphere $P^1$. Then the pullback $f^*m$ of $m$ under $f$ is a metric of curvature $1$ with conical singularities of conical angles $4\pi$ at the critical points of $f$. We study the $\zeta$-regularized determinant of the Laplace operator on $X$ corresponding to the metric $f^*m$ as a functional on the moduli space of the pairs $(X,f)$ (i.e. on the Hurwitz space $H_{g,N}(1,...,1)$) and derive an explicit formula for the functional. Using closely related methods, we find an explicit expression for the determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature 1 having a single conical singularity of angle $4\pi$. The talk is based on the joint works with V. Kalvin (Concordia University).

Abstract: I will first review the structure of the large N expansion in 1d log-gases/matrix models, which is for a large class of models governed by a "topological recursion". Then, I will present a refinement of the topological recursion, called "geometric recursion", which is a general construction of functorial assignments for surfaces, by means of successive excisions of embedded pairs of pants. This leads to some (very preliminary) thoughts about the relevance of the possible relevance of the geometric recursion in the study of matrix models. Based on joint works with Guionnet-Kozlowski, and Andersen-Orantin.

Abstract: We prove a probabilistic refinement of the Gauss-Bonnet-Chern theorem at the level of differential forms in the spirit of local index theory. Namely, for a real oriented vector bundle with metric connection, we show that its Euler form may be identified with the expectation of the current defined by the zero-locus of an appropriate random section of the bundle.

Abstract: A quantization is a process which, given a classical dynamical system, produces the underlying quantum dynamics. In the case of Berezin-Toeplitz quantization, to a symplectic manifold with some additional structures, we associate a sequence of Hilbert spaces parametrized by the integers. Asymptotic results when this parameter tends to infinity are supposed to describe the so-called semi-classical limit, when the scale gets so large that we recover the laws of classical mechanics. In the case of geometric quantization associated to a real polarization, quantum states are represented by Lagrangian submanifolds satisfying the so-called Bohr-Sommerfeld condition. In this talk, I will construct these "Lagrangian states" in the framework of Berezin-Toeplitz quantization and study their semi-classical properties. I will then give an application to the problem of relative Poincaré series in the theory of automorphic forms, and if time allows, I will present some links with geometric quantization of Chern-Simons theory.

Abstract: We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of "quantum chaotic dynamics". If the coefficients are i.i.d. Gaussian random variables, then the roots tend to concentrate near the unit circle in the complex plane. In contrast to this singular distribution, the zeros of SU(2) polynomials spread uniformly over the Riemann sphere. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity. They admit higher-dimensional generalizations and form a field called "stochastic Kähler geometry".

Abstract: We will introduce the Gaussian free field and the model of random interlacements as prototypical examples for percolation models with long-range correlations. After a review some of the developments in these fields during the last decades, a recently established isomorphism theorem will be introduced that leads to a deeper understanding of the connection between these two models. In particular, we will then outline how this isomorphism theorem can be used in order to infer new interesting properties from one of the models via known properties of the other.

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).