For Jan 30--Feb 1:
Title: Monodromy of the Hitchin fibration
Abstract: The moduli spaces of Higgs bundles on a Riemann surface are of considerable interest due to their relation to the geometric Langlands program and to moduli spaces of flat connections via the non abelian Hodge correspondence. Langlands duality of these moduli spaces is realised by mirror symmetry along a torus fibration, known as the Hitchin vibration. In this talk I will describe the monodromy of the SL(n) Hitchin fibration. The monodromy is given by a structure known as a skew-symmetric vanishing lattice, a skew-symmetric analogue of a root system. If time permits I will describe an application to counting components of character varieties.
Title: K-theory, gapped quantum system and C-symmetric indefinite metric space
Abstract: Topological K-theory is a useful tool for classifications of gapped quantum systems such as topological insulators. To describe a quantum system, we consider a Hilbert space of `states' or `wave functions', whose inner product is usually positive definite. However, there is a system described by a Hilbert space with a possibly indefinite inner product. Toward classifications of such systems, a K-theory is considered in my joint work with Giuseppe De Nitts. This K-theory will be the theme of my talk.
Title: Higher degree of the covering monopole map
Abstract: I will introduce a monopole map over universal covering spaces of compact four manifolds. In particular we can formulate higher degree of the covering monopole map when the linearized maps are isomorphic, as an element in the equivariant E theory. It induces a homomorphism between K theory of group C^* algebras.
As an application we propose an aspherical inequality on compact aspherical four manifolds. This presents a stronger version to 10/8 inequality by Furuta, in the aspherical class of four manifolds. This holds for many cases which include some complex surfaces of general type. Technically the construction of the covering monopole map requires non linear estimates in Sobolev spaces and will motivate L^p analysis on non compact manifolds.
Title: Relative Bauer-Furuta invariants for 4-manifolds with boundary
Abstract: In this talk, I will describe construction of unfolded version of the relative Bauer-Furuta invariants for general 4-manifolds with boundary. I will explain main ingredients such as the double Coulomb slice and finite dimensional approximation. Some applications will be discussed if time permitted.
Title: Inner amenable groups, stable actions, and central extensions
Abstract: Inner amenable groups arise from a probability-measure-preserving action whose orbit equivalence relation absorbs the hyperfinite equivalence relation under direct product. I will discuss recent progress around the structure of linear inner amenable groups (due to Tucker-Drob) and connection between inner amenability and existence of group-actions having the absorption property, especially focusing on central group-extensions.
Title: Representation varieties detect splittings of 3-manifolds
Abstract: Culler and Shalen established a method to construct nontrivial actions of a group on trees from ideal points of its SL_2-character variety. The method, in particular, gives essential surfaces in a 3-manifold corresponding to such actions of its fundamental group. Essential surfaces in some 3-manifold are known to be not detected in the classical SL_2-theory. We show that every essential surface in a 3-manifold is given by an ideal point of the SL_n-character variety for some n. The talk is partially based on joint works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.
Title: Symmetry and dynamics of random walks on the mapping class groups
Abstract: We discuss random walks on the mapping class groups of surfaces of negative Euler characteristics. Symmetry of random elements and dynamical properties of random walks will be mainly considered in this talk.
Title: Geometry of Pseudodifferential operator algebra bundles
Abstract: I will motivate the construction of pseudodifferential algebra bundles arising in index theory, and also outline the construction of the most general pseudodifferential algebra bundles (and the associated twisted sphere bundles), showing that there are many that are purely infinite dimensional that do not arise from standard constructions in index theory. I will also discuss characteristic classes of such bundles. This is joint work with Richard Melrose.
Title: Scalar curvature and the twisted Seiberg-Witten equations
Abstract: We compute the Yamabe invariants for a new infinite class of closed four-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations.
Title: Unfolded Seiberg-Witten-Floer spectra
Abstract: Manolescu defined an invariant, called Seiberg-Witten-Floer stable homotopy type, for rational homology 3-spheres. This invariant is an refinement of Seiberg-Witten-Floer homology defined by Kronheimer-Mrowka and has produced applications to topology. I will talk about two versions of generalization of the invariant to 3-manifolds with positive first Betti number. Joint work with T. Khandhawit, J. Lin.
Title: The differential topology of semimetals
Abstract: The "Weyl fermion" was discovered in a topological semimetal in 2015. The mathematical theory of these new topological phases of matter turns out to involve deep and subtle results in differential topology. I will outline this theory, and explain some connections to Euler structures, torsion of manifolds, and Seiberg-Witten invariants. I also propose an interesting generalisation with torsion topological charges, based on Kervaire semi-characteristics and the Atiyah-Dupoint-Thomas theory of singular vector fields. This is joint work with V. Mathai.
Title: Index Theory and Character Formulas
Abstract: Weyl character formula describes characters of irreducible representations of compact Lie groups. This formula can be obtained using geometric method, for example, from the Atiyah-Bott fixed-point theorem. Harish-Chandra character formula, the noncompact analogue of the Weyl character formula, can also be studied from the point of view of index theory. We apply orbital integrals on K-theory of Harish-Chandra Schwartz algebra of a semisimple Lie group G, and then use geometric method to deduce Harish-Chandra character formulas for discrete series representations of G. This joint work with Peter Hochs.
For Feb 2--Feb 4:
Title: Classifications of some twisted equivariant bundles
Abstract: Some nonsymmorphic topological crystalline insulators are classified by Z/2 subgroups in twisted equivariant K-theory. To understand the corresponding Z/2 indices, I will present some results about the classifications of twisted equivariant vector bundles.
Title: Family of monopole maps and its application
Abstract: Furuta’s 10/8 inequality presents a strong constrain to existence of smooth four manifolds satisfying topological condition. This is obtained by computing the degree of the monopole map. We will present a partial formula of its family version for families of monopole maps. We will apply it to the moduli space of complex surfaces and discuss on constrain to existence of different smooth families of complex surfaces, particularly K3 case. This is joint work with Nakamura and Kawaguchi.
Title: Symmetry and dynamics of random walks on the mapping class groups II
Abstract: In the second talk, applications of the results in the first talk will be discussed. Especially, we consider mapping tori of random mapping classes and discuss their properties.
Title: Perturbation of the SW equations on symplectic 4-manifolds revisited
Abstract: We introduced a new class of perturbations of the SW equations via the Weitzenbock formulae of harmonic spinors and ASD 2-forms. By using these perturbations, we will give yet another proof of non-triviality of the SW invariant of symplectic 4-manifolds.
Title: Spectral gap-labelling conjecture for magnetic Schrodinger operators
Abstract: With the help of an earlier result, joint with Quillen, I will formulate a spectral gap-labelling conjecture for magnetic Schrodinger operators with smooth aperiodic potentials on Euclidean space. Results in low dimensions (2 and 3) as well as in the case of periodic potentials will be given. This is joint work with Moulay Benameur.
Title: Gluing formula for the stable homotopy Seiberg-Witten invariants
Abstract: Bauer and Furuta showed that finite dimensional approximations of the Seiberg-Witten equations on a closed spin-c 4-manifold define a stable homotopy class of a map from the Thom space of a vector bundle on the Picard torus of the 4-manifold to a sphere. This stable homotopy class is an invariant of the spin-c 4-manifold and is a refinement of the Seiberg-Witten invariant in the sense that the Seiberg-Witten invariant is a variant of the mapping degree of the map. Suppose that the 4-manifold splits into two 4-manifolds with boundary along a 3-manifold. The unfolded Seiberg-Witten-Floer spectra of the 3-manifold enable us to define relative stable homotopy Seiberg-Witten invariants for the 4-manifolds with boundary. We will discuss how to calculate the invariant of the closed 4-manifold from the relative invariants. Joint work with T. Khandhawit and J. Lin.
Title: Twisted Donaldson Invariants
Abstract: Fundamental groups and "higher theory" are useful tools in the study of geometry and topology of compact Riemannian manifolds. They give rise to refined invariants in K-theory of a group C*-algebra, and can be calculated by pairing their Chern character in cyclic homology with group coccyges. Motivating examples include the Connes-Moscovici localised index formulas and Lott's construction of higher eta-invariants in cyclic homology. The former are used to prove the Novikov conjecture for hyperbolic groups, and the latter give rise to refined spectral invariants for compact manifolds with boundary. Donaldson invariants, an important concept in gauge theory of closed 4-manifolds, also have a counterpart in the higher theory. We will show that they can be defined and calculated using cyclic homology. This is work in progress with T. Kato and H. Sasahira.
Last updated on January 29, 2017.