ABSTRACT

Shizuo Kaji    A secondary equivariant string product

A product on homology of the space of free loops LM over a closed manifold M is first defined by Chas and Sullivan and various similar constructions have been discovered since then. Among them is Chataur-Menichi’s product on homology of LBG where BG is the classifying space of a finite group G. We give a common generalisation to both by defining a product on homology of L(M_G), where M_G is the Borel construction of a compact (not necessarily connected) Lie group action on M. We also discuss a secondary product and show it is related to the cup product in negative Tate cohomology of G.

This is joint work with Haggai Tene.


Takahito Naito    Sullivan's coproduct on the reduced loop homology

In the theory of string topology initiated by Chas and Sullivan, the homology of free loop spaces of manifolds (called the loop homology) has very rich algebraic structures. The loop product is a multiplication on the loop homology and it is the most basic operation in string topology. Cohen and Godin discovered a 2-dimensional TQFT structure on the homology. In this talk, I will discuss a coproduct on the reduced loop homology which is introduced by Sullivan. The coproduct and the loop product give the loop homology an infinitesimal bialgebra structure. I will explain how to construct Sullivan's coproduct and give some computational examples.


Matthew Burfitt    Free loop cohomology of homogenous spaces

A complete flag manifold is the quotient of a Lie group by its maximal torus and is one of the nicer examples of homogeneous spaces. Related objects are studied in different areas of mathematics and mathematical physics. In topology, the study of free loop spaces on manifolds has two folded motivation. First there is a relation between geometrically distinct periodic geodesics on a manifold and their free loop spaces, originally studied by Gromoll and Meyer in their 1969 paper. More recently the study of string topology has been an active area of research with connection to interesting areas in algebraic topology including topological quantum field theory, operads and topological cyclic homology. In this talk I will discuss my work on the cohomology of the free loop space of complete flag manifolds. In particular I will discuss the cohomology algebras of some such space in low rank cases.


Ingrid Membrillo-Solis    Homotopy types of gauge groups related to certain 7-manifolds

Let X be a path-connected pointed topological space and let G be a topological group. Given a principal G-bundle over X, P→X, the gauge group is the group of G-equivariant automorphisms of P that fix X. The study of the topology of gauge groups when X is a low dimensional manifold has played a prominent role in mathematics and mathematical physics over the last thirty years. In 2011, however, Donaldson and Segal established the mathematical set-up to construct gauge theories using principal G-bundles over high dimensional manifolds. In this talk I will present some results on the homotopy theory of gauge groups when X is a certain 2-connected 7-manifold and G is a simply connected simple compact Lie group.


Tse Leung So    Homotopy types of gauge groups over 4-manifolds

Gauge groups originate from physics and they have many applications in physics and mathematics. Given a Lie group G, a gauge group is defined to be the group of G-equivariant automorphisms of a principal G-bundle which fix its base manifold. In general gauge groups are difficult to compute. In this talk, I will give a homotopy decomposition method for gauge groups over certain non-simply-connected 4-manifolds and discuss their homotopy types.


Mitsunobu Tsutaya    Homotopy theoretic classifications of gauge groups

We will give a survey about various classification results of gauge groups. In particular, we will present the relation between the classification of gauge groups and homotopy commutativity of the structure group.


Shitaro Kuroki    On flagged Bott manifolds

A Bott manifold (also called a Bott tower) is a special type of iterated CP¹ bundles which was introduced in the work of Grossberg-Karshon. Grossberg-Karshon show that Bott manifolds are toric varieties (toric manifolds) and some of them are obtained by a deformation of complex structures of Bott-Samelson varieties. After the work of Grossberg-Karshon, this class of manifolds has been often appeared in several areas of mathematics. In particular, toric topologists generalize this class of manifolds to iterated complex projective bundles (called a generalized Bott manifold). Their motivation is to verify the cohomological rigidity of special class of toric manifolds (generalized Bott manifolds are toric manifolds). Though this generalization is one of the natural generalizations of Bott manifolds, it seems to be difficult to find the counterpart of Bott-Samelson varieties unlike Bott manifolds. In this talk, we generalize Bott manifolds from the different point of view. Namely we regard CP¹ as the flag manifold and introduce the special type of iterated flag manifold bundles, called a flagged Bott manifold, and introduce some basic properties about this class of manifolds.　

This is the progress work with Eunjeong Lee, Jongbaek Song and Dong Youp Suh.


Tatsuya Horiguchi    Hessenberg varieties and hyperplane arrangements

Hessenberg varieties are subvarieties of a flag variety. The study of topology of Hessenberg varieties makes connection with many research areas such as: geometric representation theory, quantum cohomology of a flag variety, graph theory, and hyperplane arrangements. In this talk, I will talk about the connection between Hessenberg varieties and hyperplane arrangements. In particular, from this connection we obtain that the cohomology ring of a regular nilpotent Hessenberg variety is isomorphic (as rings) to the invariant subring of the cohomology ring of a regular semisimple Hessenberg variety under the Weyl group action.

This is joint work with Takuro Abe, Mikiya Masuda, Satoshi Murai, and Takashi Sato.


Hiraku Abe    Extensions of pavings by affines of Peterson varieties

The Peterson variety is defined to be a subvariety of the full flag variety. In Lie type A, it is known that the restriction map from the cohomology of the full flag variety to the cohomology of Peterson variety is surjective. In this talk, I will give a simple and topological explanation of this fact.


Takahiro Matsushita    Box complexes and model structures on the category of graphs

An n-coloring of a simple graph is a map from the vertex set V(G) to the n-point set so that adjacent vertices have different values. The chromatic number χ(G) of G is the smallest numver n such that G has an n-coloring. The graph coloring problem, which is one of the most classical problems in graph theory, is to compute chromatic numbers. The neighborhood complex N(G) of a graph was introduced by Lovász in the context of the graph coloring problem. He showed that the connectivity of the neighborhood complex N(G) gives a non-trivial lower bound for the chromatic number. The box complex B(G) is a Z/2-space homotopy equivalent to N(G), but its Z/2-homotopy invariant give more precise information about the chromatic number. In this talk, I introduce a model structure on the category of graphs related to box complexes. A weak equivalence is a graph homomorphism f which induces a Z/2-homotopy equivalence between box complexes. Moreover, the model structure is Quillen equivalent to the category of Z/2-spaces.


Seonjeong Park    Smooth toric varieties over a cube with one vertex cut

We say that a smooth toric variety is over a polytope P if its quotient by the compact torus is homeomorphic to P as a manifold with corners. Every smooth toric variety over a cube Iⁿ is projective, and it is known as a Bott manifold. Then blowing up a Bott manifold at a fixed point produces a projective smooth toric variety over vc(Iⁿ), a cube with one vertex cut. But not every smooth toric variety over vc(Iⁿ) is projective. Oda's 3-fold is a non-projective smooth toric variety over vc(I³). In this talk, we discuss the classification of smooth toric varieties over vc(Iⁿ) as smooth manifolds and also as varieties.

This is joint work with S. Hasui, M. Masuda and H. Kuwata.