Manifolds with ordinary boundary/corners have found their presence in differential geometry and PDEs: they form Man^b or Man^c category; and for boundary value problems, they are nice objects to work on. Manifolds with analytical corners -- a-corners for short -- form a larger category Man^{ac} which contains Man^c, and they can in some sense be viewed as manifolds with boundary at infinity.

In this talk I'll walk you through the definition of manifolds with corners and a-corners, and give some examples to illustrate how the new definition will help.

# Past Junior Geometry and Topology Seminar

Floer (co)homology, invariant which recovers periodic orbits of a Hamiltonian system, is the central topic of symplecic topology at present. Its analogue for open symplecic manifolds is called symplectic (co)homology. Our goal is to compute this invariant for big family of spaces called Nakajima's Quiver Varieties, spaces obtained as hyperkahler quotients of representation spaces of quivers.

Abstract: The Kontsevich graph weights are period integrals whose

values make Kontsevich's star-product associative for any Poisson

structure. We illustrate, by using software, to what extent these

weights are determined by their properties: the associativity

constraint for the star-product (for all Poisson structures), the

multiplicativity (decomposition into prime graphs), the cyclic

relations, and some relations due to skew-symmetry. Up to the order 4

in ℏ we express all the weights in terms of 10 parameters (6

parameters modulo gauge-equivalence), and we verify pictorially that

the star-product expansion is associative modulo ō(ℏ⁴) for every value

of the 10 parameters. This is joint work with Arthemy Kiselev.

Manifolds with corners are similar to manifolds, yet are locally modelled on subsets $[0,\infty)^k \times R^{n-k}$. I will discuss some of the theory of these objects, as well as introducing $C^\infty$-rings. This will explain the background to my current research in $C^\infty$-Algebraic Geometry. Time permitting, I will briefly discuss my current research on $C^\infty$-schemes with corners and motivation of this research.

Hyperbolic groups were introduced by Gromov and generalize the fundamental groups of closed hyperbolic manifolds. Since a closed hyperbolic manifold is aspherical, it is a classifying space for its fundamental group, and a hyperbolic group will also admit a compact classifying space in the torsion-free case. After an introduction to this and other topological finiteness properties of hyperbolic groups and their subgroups, we will meet a construction of R. Kropholler, building on work of Brady and Lodha. The construction gives an infinite family of hyperbolic groups with finitely-presented subgroups which are non-hyperbolic by virtue of their finiteness properties. We conclude with progress towards determining minimal examples of the "sizeable" graphs which are needed as input to the construction.

The study of 3-manifolds is founded on the strong connection between algebra and topology in dimension three. In particular, the sine qua non of much of the theory is the Loop Theorem, stating that for any embedding of a surface into a 3-manifold, a failure to be injective on the fundamental group is realised by some genuine embedding of a disc. I will discuss this theorem and give a proof of it.