Let a compact torus $T$ of dimension $k$ act effectively on a smooth closed manifold $X$ of dimension $2n$, and assume that the action has nonempty finite set of fixed points. The difference $n-k$ is called the complexity of the action. An action is called equivariantly formal if odd-degree cohomology modules of $X$ vanish.

When the action has complexity 0, the orbit space is typically a manifold with corners. Masuda and Panov proved in 2006 that the action of complexity 0 on $X$ is equivariantly formal if and only if all faces of its orbit space are acyclic. In this case, equivariant and ordinary cohomology of $X$ can be described using face rings of simplicial posets. This result generalizes the result of Davis and Januszkiewicz on quasitoric manifolds.

We study actions of complexity one, in particular, equivariantly formal actions. The action of complexity one is said to be in general position, if any $n-1$ of the $n$ tangent weights at each fixed point are linearly independent. In the joint work with Masuda we prove that the orbit space of an equivariantly formal action in general position is a homology sphere. This result recovers several previously known particular results if one applies Poincare conjecture in topological category. Moreover, we give a criterion for equivariant formality of the action in terms of its orbit space' structure.

On the other hand, dropping the assumption of general position of tangent weights, the orbit space of equivariantly formal action is not always a sphere. In the joint work with Cherepanov we proved that a triple suspension over any finite simplicial complex can be realized, up to homotopy, as the orbit space of an equivariantly formal (even Hamiltonian) action of complexity one.

Differential graded algebras (DGAs) are one of the most important objects of study in homological algebra. These are chain complexes with an associative and unital multiplication. Examples of DGAs include cochain complexes of topological spaces equipped with the cup product.

In fact, there are interesting interactions between DGAs and stable homotopy theory. I will present these connections, and show how they can be used to classify DGAs with polynomial homology.

The talk will be accessible to a general audience.

Let G be a compact connected Lie group and K a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of G and K is invertible in a given principal ideal domain R. It has been known since around 1970 that in this case the cohomology of the homogeneous space G/K with coefficients in R and the torsion product of H^*(BK) and R over H^*(BG) are isomorphic as R-modules. I am going to explain that this isomorphism is multiplicative and natural in the pair (G,K) provided that 2 is invertible in R. The essential new ingredient of the proof is the notion of a homotopy Gerstenhaber algebra, whose definition will be recalled.

There are a number of constraints on topological, locally linear actions of elementary abelian p-groups on 4-manifolds, including: (i) conditions for the collapse of the Borel spectral sequence for actions of cyclic groups of prime order, and(ii) upper bounds on the rank of finite groups acting homologically trivially with discrete singular set. [Joint work in progress with Semra Pamuk]

Hessenberg varieties are subvarieties of a full flag variety. Their topology makes connections with other research areas such as hyperplane arrangements and graph theory. In this talk, first I explain an explicit presentation of the cohomology rings of regular nilpotent Hessenberg varieties in type A which is joint work with Hiraku Abe, Megumi Harada, and Mikiya Masuda. Then, I would like to explain that their cohomology rings can be described in terms of hyperplane arrangements which is joint work with Takuro Abe, Mikiya Masuda, Satoshi Murai, and Takashi Sato. Finally, if I have time, then I would like to explain an explicit presentation of the cohomology rings of regular nilpotent Hessenberg varieties in all Lie types from the point of view of hyperplane arrangements which is joint work with Makoto Enokizono, Takahiro Nagaoka, and Akiyoshi Tsuchiya.

A Gelfand--Zetlin system is a completely integrable system on partial flag varieties. In this talk, I will explain how to describe the topology of Gelfand--Zeitlin fibers in terms of ladder diagram combinatorics. Also, I will provide an effective algorithm locating Lagrangian fibers. If time permits, applications of the description on Floer theory will be discussed.

The fat-wedge filtration of moment-angle complexes provide us a topological approach to the Golodness of simplicial complexes. I will talk about this approach specialized to surface triangulations. This talk is based on a joint work with K. Iriye.

Action-angle coordinates are a type of coordinate chart on symplectic manifolds originating from the theory of commutative completely integrable systems and the Liouville-Arnold theorem. For example, every toric manifold has global action-angle coordinates: the pre-image under the moment map of the interior of the Delzant polytope, $\mathring{\Delta}$, is symplectically isomorphic to the product $\mathring{\Delta}\times T$, equipped with the canonical symplectic structure.

Multiplicity-free spaces are the natural non-abelian generalization of toric manifolds. A multiplicity-free space is a symplectic manifold equipped with a Hamiltonian action of a compact connected Lie group that is ``completely integrable'' in the appropriate non-commutative sense. For example, every coadjoint orbit is a multiplicity-free space. Unlike toric manifolds, multiplicity-free spaces do not come readily equipped with nice global action-angle coordinates.

In this talk I will present recent work in collaboration with Anton Alekseev, Benjamin Hoffman, and Yanpeng Li that gives a new construction of action-angle coordinates on a large family of multiplicity-free spaces.

In general, homology groups rarely determine the homotopy type of spaces. It is therefore of interest to find special cases when this happens. Let M be a smooth, orientable, closed, connected 4-manifold such that its first homology group has no 2-torsion. We give a homotopy decomposition of the suspension of M in terms of spheres, Moore spaces and the suspended CP^2. This is used to calculate the K-theory of M as a group and to determine the homotopy type of certain current groups and gauge groups.

This is joint work with Stephen Theriault.

For a space X, the n-th topological complexity (TC) of X, was defined by Rudiak and is related to robotic motion planning. Farber and Oprea put these into a generating function, and conjectured that it has a single pole of degree 2 at x=1. After introducing the basic notions of TC, we show that this conjecture holds for some class of spaces. We also related this to counterexamples to Ganea's conjecture first constructed by Iwase. We describe joint work with Michael Farber and Daisuke Kishimoto and also with Jeff Strom. All of this work was done during the Fields program.

We introduce flag Bott manifolds of general Lie type as the total space of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds and nice toric actions. We calculate the toris equivariant cohomology rings of flag Bott manifolds of general Lie type.

I will report on joint work in progress with Johannes Ebert and Jan-Bernhard Kordass. In this work be strengthen a result of Chernysh on the homotopy type of spaces of metrics of positive scalar curvature on high dimensional manifolds.