# Seminars & Events for Algebraic Topology Seminar

##### Action-Dimension of Groups

*This is a combined Topology/Algebraic Topology seminar. * For a group G, we define a notion of dimension in terms of dimension change of the of the top homology between a free G space X and it's quotient X/G. We show that this is well defined and calculate this "Action-dimension" for certain groups, including finitely generated solvable groups, free groups, finite groups, and connected Lie groups. As a consequence we give a positive answer to a conjecture of J Kollar: Let f: R^n ---> T^n be the universal covering and M a closed submanifold in T^n = (S^1)^n such that f^{-1}(M) has the homotopy type of a finite complex, then M = T^n.

##### Splittings of the polyhedral product functor

I will talk about joint work with Bahri, Cohen and Gitler. We showed that the polyhedral product functor (a generalization of the Moment Angle Complex) stably splits into pieces which are understood for restricted cases. I will talk about further splitting of the pieces that appear in the splitting and describe the homology group.

##### Poincare and Hodge rings and their applications

We determine the structure of the rings of Poincare and Hodge polynomials, and analyze the tautological comparison map between them. This leads to interesting results about the Hodge numbers of Kaehler manifolds and of algebraic varieties, and in particular to the complete solution of a classical problem of Hirzebruch's.

##### Combinatorial number theory arising from algebraic topology.

We will show how studying v1-periodic homotopy groups of SU(n) led to the following question. Let f(n) denote the sum of the reciprocals of the binomial coefficients (n choose i). For which p-adic integers x does the

sequence f(x_n) approach a p-adic limit? Here x_n are the partial sums for x. The answer when p is odd is quite simple, but when p=2 is complicated and not completely understood.

##### Deformations of periodic frameworks

A d-periodic bar-and-joint framework is an abstraction (and generalization to arbitrary dimension d) of an atom-and-bond crystal structure. We describe a deformation theory for this type of frameworks

and various extensions to volume or symplectic analogues. Questions of generic rigidity highlight the role of sparsity conditions on the underlying quotient graph. This is joint work with Ileana Streinu, Smith College.

##### Embeddings of Rational Homology Balls

We will begin with a description of the rational homology balls appearing in Fintushel and Stern's rational blow-down procedure for smooth 4-manifolds, a generalization of the standard blow-down operation. We will then discuss various smooth and symplectic embedding results of these rational homology balls, as well as a description of a symplectic rational blow-up operation. THIS IS A JOINT TOPOLOGY/ALGEBRAIC TOPOLOGY SEMINAR. PLEASE NOTE DIFFERENT TIME AND LOCATION.

##### Toric Structures on Symplectic Bundles of Projective Spaces

Recently, extending work by Karshon, Kessler and Pinsonnault, Borisov and McDuff showed that a given symplectic manifold $(M,\omega)$ has a finite number of distinct toric structures. Moreover, McDuff also showed a product of two projective spaces $\mathbb{C} P^r\times \mathbb{C} P^s$ with any given symplectic form has a unique toric structure provided that $r,s\geq 2$. In contrast, the product $\mathbb{C} P^r \times \mathbb{C} P^1$ can be given infinitely many distinct toric structures, though only a finite number of these are compatible with each given symplectic form $\omega$. In this talk, we will discuss how to extend these results by considering the possible toric structures on a toric symplectic manifold $(M,\omega)$ with $\dim H^2(M)=2$.

##### On the conservation of (equivariant) homeomorphism classes of M(J)-manifolds

A /toric manifold / is a closed manifold of dimension 2n which admits a locally standard half dimensional torus action whose orbit space can be identified with an n-dimensional simple polytope. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Wedge operations and torus symmetries

A fundamental result of toric geometry is that there is a bijection between toric varieties and fans. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Power operations and the Kunneth spectral sequence

Power operations have been constructed and successfully utilized in the Adams and Homological Homotopy Fixed Point Spectral Sequences by Bruner and Bruner-Rognes. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Hypergraphs and the Regularity of Square-free Monomial Ideals

This talk represents joint work with Kuei-Nuan Lin. Fix a polynomial ring R over a field and consider a homogeneous ideal I. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Stasheff-Steenrod ideas applied to nonlinearity

We are partially successful in applying such ideas to construct effective theories for evolution processes defined by certain nonlinear PDEs. The effective theories at different scales are meant to be related by mappings

which respect the effective models. In particular we apply these ideas to incompressible fluid flow in 3D.

##### Cohomology for GL(4) and Galois Representations

In a series of papers, Ash, Gunnells and McConnell have studied cohomology groups for congruence subgroups Gamma of SL(4,Z), and have verified experimentally that Hecke eigenclasses for these cohomology groups seem to have attached Galois representations. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### The Geometry of Fox's Free Calculus with Applications to Higher Dimension Knots

This talk with discuss the geometry hidden within Fox's free calculus, and its use in higher dimensional knot theory.

##### Normal Smoothings for Smooth Cube Manifolds

A smooth cube manifold M is a smooth n-manifold M together with a smooth cubification on M. The cube structure provides rays that are normal to the open k-subcubes. Using these rays we can construct "normal charts" in an obvious and natural way. A complete set of normal charts gives a (topological) "normal atlas" on M. If this atlas is smooth it is called a "normal smooth atlas" on M and induces a "normal smooth structure" on M (normal with respect to the cube structure). We prove that every smooth cube manifold has a normal smooth structure, which is diffeomorphic to the original one. This result also holds for smooth all-right-spherical manifolds.

##### Some remarks on the Postnikov tower

We will discuss a geometric approach to the Postnikov tower which admits a natural extension to the homotopy theory of schemes.