Abstract: It is a well-known result that the only spheres which can admit almost complex structures occur in dimensions two and six. We show that if one instead considers rational homology spheres (manifolds with the same rational homology as a sphere), then the same conclusion holds: they can only admit almost complex structures in dimensions two and six. However, there are infinitely many six-dimensional rational homology spheres and not all of them admit almost complex structures. We will explain how to construct infinitely many examples which do admit almost complex structures and infinitely many which do not. This is joint work with Aleksandar Milivojevic.
Abstract: I will explain the role of stacks and descent in the theory of bundle gerbes, and talk about various examples and features. I will then describe a new application of non-abelian gerbes to T-duality, in which their stacky nature is of crucial importance.
Abstract: In 2004 Fornaess gave the the first example of a “short C^2”: a domain of C^2 which is an increasing union of (biholomorphic images of) balls and whose Kobayashi metric vanishes, but which is not biholomorphic to C^2 since it admits a nonconstant bounded plurisubharmonic function. Such a domain appears as the basin of attraction at the origin of a nonautonomous sequence of holomorphic automorphisms of C^2. In this talk I will explain how to construct a short C^2 as a Fatou component of a single holomorphic automorphism (a transcendental Hénon map). By the classical Rosay-Rudin result it is clear that such a Fatou component cannot be the basin of attraction of a point, and indeed it follows from the construction that the Fatou component is wandering and oscillating. This is a joint work with Luka Boc Thaler and Han Peters.
Abstract: In this somewhat coals-to-Newcastle exercise I will describe joint work with Chris Kottke in which we introduce the notion of a bigerbe. Michael Murray's bundle gerbes are smoother versions of the Čech objects due to Brylinski and Hitchin, all giving realizations of integral 3-classes in cohomology. Our bigerbes are special 2-gerbes in the sense of Stevenson, so realizing integral 4-classes, but more constrained and are direct generalizations of bundle gerbes to a bisimplicial setting. Motivating examples include a `Brylinski-McLauglin' bigerbe arising from the transgression of a principal G-bundle.
Abstract: Periodic Floer Homology(PFH) is a Gromov type invariant for fibered 3-manifold, which is defined by counting of J-holomorphic curves. Given a symplectic 4-manifold whose boundaries are fibered 3-manifolds, it is expected that this 4-manifold induces a homomorphism (cobordism map) between the periodic Floer homologies of its boundaries. Due to certain technical difficulties, the cobordism map haven't been defined completely yet. Motivated by defining the cobordism map, I will discuss a concrete example that the 4-manifold is a elementary Lefschetz fibration. I will start with a brief introduction of PFH and ECH index.
Abstract: I will report on new joint work with Franc Forstnerič about introducing ideas and methods from Oka theory into complex contact geometry. This is a new research area in which only a few papers have been published so far (by Alarcón, Forstnerič, López, and myself). More specifically, we study holomorphic Legendrian curves in the projectivised cotangent bundle X=PT*Z of a complex manifold Z of dimension at least 2. Such a manifold X carries a natural complex contact structure. We prove several approximation and general position theorems, as well as some h-principles, for holomorphic Legendrian curves in X. I will start with a very brief introduction to complex contact geometry.
Abstract: Motivating in constructing conformal field theories Jones recently discovered a very general process that produces actions of the Thompson groups F,T and V such as unitary representations or actions on C*-algebras. I will give a general panorama of this construction along with many examples and present various applications regarding analytical properties of groups and, if time permits, in lattice theory (e.g. quantum field theory).
Abstract: A bounded linear operator F between Hilbert spaces V and W is Fredholm if it is invertible modulo compact operators. Then ker(F) and W/im(F) are finite-dimensional, and the difference of their dimensions is the index of F. This applies for example to elliptic differential operators on compact manifolds, viewed as bounded operators between suitable Sobolev spaces. A more general and abstract view of index theory is that such an operator F is A-Fredholm, for some subalgebra A of the bounded operators on the direct sum of V and W, if it is invertible modulo A. Then it has an index in the K-theory of A. If A is the algebra of compact operators, this reduces to the classical index just mentioned. I will give an introduction to this approach to index theory, assuming no knowledge of K-theory or C*-algebras. Then I will mention some work in this context with Hao Guo and Mathai Varghese, where the algebra A has its origins in coarse geometry, and its K-theory equals the K-theory of a group C*-algebra in cases we are interested in. An application is recent work with Bai-Ling Wang and Hang Wang on an equivariant Atiyah-Patodi-Singer type index theorem for proper actions by locally compact groups on possibly noncompact manifolds with boundary.
Abstract: Given a d-dimensional manifold M with a spin structure, we can consider the looped spin frame bundle LSpin(M), which is a principal bundle for the loop group of spin, LSpin(d), over the loop space LM. Supposing that this bundle lifts to a principal bundle for the basic central extension of LSpin(d), I outline the construction of an infinite rank vector bundle F over the loop space LM. This bundle is moreover equipped with "Fusion over compatible loops", more precisely, if x and y are loops such that the second half of x agrees with the first have of y, there is a natural isomorphism F_x \otimes F_y -> F_(x*y), where the loop x*y is the loop obtained by deleting the middle segment, and the tensor product is the Connes fusion product over a suitable von Neumann algebra.
Abstract: In this talk I will discuss the gauging of two-dimensional sigma models with respect to a non-isometric Lie algebroid symmetry, and its application to T-duality. In particular I will discuss the potential relevance to Poisson-Lie duality.
Abstract: In this talk we will discuss new non-existence results on non-compact homogeneous Einstein manifolds. After introducing the problems in the area and explaining the main results, we will explain the key idea behind its proof: to consider non-transitive group actions on these spaces (more precisely, actions with cohomogeneity one), and to find geometric monotone quantities for the ODE that results from writing the Einstein equation in such a setting. As an application, we will show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds. This is joint work with C. Böhm.
Abstract: It is a famous result (obtained by combining work of Lichnerowicz, Hitchin, Gromov-Lawson, and Stolz) that if M^n is a simply connected closed spin manifold, vanishing of the "alpha-invariant" in KO_n (up to torsion this is just the A-hat-genus) is necessary for M to admit a Riemannian metric of positive scalar curvature, and if n > 4, this condition is also sufficient. I will explain some joint work with Boris Botvinnik and Paolo Piazza in which we extend this theorem to certain manifolds with singularities
Abstract: Nakajima's work in Geometric Representation Theory uses quiver varieties to construct representations of certain quantum algebras. One of the first examples was the Heisenberg algebra, where the quiver is the ADHM quiver, and the generators of the algebra are constructed via push-pull maps using the Hecke correspondence for quivers. In this talk I will explain how to use Morse theory on singular spaces to carry out this construction, where the Hecke correspondence appears as a space of flow lines, and the push-pull maps appear naturally as the cup product on the Morse complex.
Abstract: In the 1980s, Julg and Valette provided a beautiful geometric construction that was at the foundation of their proof of the K-amenability of groups acting on trees. Motivated by the construction, in collaboration with Guentner and Higson we have extended this construction to the CAT(0) cubical spaces and demonstrated that it can be used to prove the K-amenability for groups acting on CAT(0)-cubical spaces. More recently, using the so called gamma-property invented by Nishikawa, we have used these very geometric ideas to prove the Baum-Connes conjecture with coefficients for these groups. In my talk I will present the main geometric and analytic ideas that go into the proof.
Abstract: Cuntz-Pimsner algebras of tautological line bundles over noncommutative spaces — obtained out of a Fock-space construction of creation and annihilation operators — are thought of as total spaces algebras of principal circle bundles. A Gysin-like sequence in KK-theory can be used to compute KK classes of the total space algebras or as a way to consider T-duality for noncommutative line bundles. Examples include the Irrational Rotation Algebra for quadratic irrationals and quantum lens spaces out of line bundles over weighted quantum projective spaces.
Abstract: It is a classical result of Lichnerowicz that the index of the Dirac operator on a closed, spin manifold vanishes in the presence of positive scalar curvature. When the manifold is non-compact but has bounded geometry, this operator still has an index that lives in the K-theory of the maximal Roe algebra. The aim of this talk is to outline a proof of the fact that, if the manifold has uniformly positive scalar curvature, then this index vanishes. The argument relies on a subtle distinction between the usual maximal Roe algebra and a uniform version of it; the latter allows one to avoid some of the analytical difficulties encountered when working with the former, in particular allowing one to establish a functional calculus for the Dirac operator, from which one obtains the desired result. This work is joint with Zhizhang Xie and Guoliang Yu.
Abstract: A recent breakthrough in condensed matter physics was the discovery of so-called topological insulators. These are materials for which a topological non-triviality in their mathematical description forces them to behave "non-local“ in a certain sense. We model this by a Riemannian manifold carrying a cocompact action of a discrete symmetry group G, together with a G-invariant Hamiltonian operator. The question is then whether a certain spectral subspace of $L^2(X)$ has a G-basis of rapidly decaying functions, called "Wannier functions“. We show that this is equivalent to the (non-)triviality of the spectral subspace, when considered as a Hilbert module over the group C*-algebra $C^*_r(G)$. This is joint work with Guo Chuan Thiang.
Abstract: What is a restriction problem? Why is it important? This talk is intended to be a gentle introduction to Fourier and eigenfunction restriction, a central theme in harmonic analysis. We will review some of the classical results in the area, leading up to problems of recent vintage.
Abstract: This is a report on joint work with Ahmad Reza Haj Saeedi Sadegh. An Euler-like vector field associated to the embedding of a smooth manifold M into a smooth manifold V is a vector field on V that identifies near M with the Euler vector field on the normal bundle of M in V under some tubular neighborhood embedding. The deformation to the normal cone (DNC) construction is a smooth manifold that is functorially associated to the embedding of M into V, and that fibers over the real line, with fibers equal to V away from zero, and equal to the normal bundle at zero. It can serve as a sort of alternative tubular neighborhood construction (it was originally introduced in other categories, e.g. complex manifolds, where tubular neighborhood embeddings do not always exist). Bursztyn, Lima and Meinrenken showed that Euler-like vector fields actually determine their associated tubular neighborhood embeddings. I shall explain this result from the point of view of the DNC construction. One reason for doing so is that the DNC construction extends readily to other contexts of current interest in index theory, and I shall discuss this too.
Abstract: In this talk, I will describe how the geometry of an arbitrary parabolic second order equation governs the existence of its conservation laws, and conversely, how the existence of even a single conservation law puts strong geometric restrictions on a parabolic equation. In particular, I will describe the strong connection between conservation laws and parabolic Monge-Ampere equations.
Abstract: One of the most important challenges of Riemannian geometry is to understand the Ricci curvature. A problem that is still open is: determine all possible signatures of the Ricci curvature of all Riemannian metrics on a given manifold. The aim of this talk is to present the problem in the setting of nilpotent Lie groups with left-invariant metrics, and give an answer in the case that the nilpotent Lie group admits a nice basis. This talk is based on work in progress with Ramiro Lafuente (The University of Queensland).
Abstract: In the mid 70s Goresky and MacPherson introduced intersection cohomology. Their idea was to limit the extent to which cycles could intersect the singularities of the space. This was recast in sheaf theoretic language by Deligne and Goresky-MacPherson. Intersection cohomology has provided a powerful tool in topology, algebraic geometry, representation theory and combinatorics. It was pointed out in the first paper of Goresky and MacPherson that intersection cohomology over the integers does not posess a non-degenerate intersection form, in contrast to cohomology of smooth spaces. I will describe a solution to this issue, if one works over p-adic rings, and allows ramified extensions. I have no idea whether this story is useful, but it does seem aesthetically appealing, and appears to solve a foundational issue.
Abstract: This talk is about applications of elliptic cohomology theory in representation theory. I will begin with some basic notions and examples of elliptic cohomology and symplectic resolutions. Then, I will focus on two examples in representation theory where elliptic cohomology of symplectic resolutions occur. In the first example, the symplectic resolution is taken to be the Springer resolution. The resulting algebra has irreducible representations parametrized by certain nilpotent Higgs bundles on the elliptic curve, which can be viewed as the Deligne-Langlands correspondence in the elliptic setting. In the second example, the symplectic resolution is the quiver variety. The resulting algebra is the elliptic quantum group, which I will describe in detail. I will also mention recent work in progress on representation theory obtained beyond the elliptic cohomology theory.
Abstract: In the last decade, an exciting program has emerged connecting the arithmetic of elliptic curves to classical questions in complex algebraic dynamics; that is, the study of iteration of maps on complex algebraic varieties. We will discuss this program and the fruit it has yielded, providing a new and surprising approach to fundamental questions about the interaction between geometry and arithmetic of elliptic curves.