Topological field theories (TFT's) are physical theories depending only on the topological properties of spacetime as opposed to also depending on the metric of spacetime. This talk will introduce topological field theories, and the work of Freed and Hopkins on how a class of TFT's called "invertible" TFT's describe certain states of matter, and are classified by maps of spectra. Constructions of field theories corresponding to specific maps of spectra will be described.

# Past Junior Geometry and Topology Seminar

Despite its fame there appears to be little literature outlining Lurie's proof sketched in his expository article "On the classification of topological field theories." I shall embark on the quixotic quest to explain how the cobordism hypothesis is formalised and give an overview of Lurie's proof in one hour. I will not be able to go into any of the motivation, but I promise to try to make the talk as accessible as possible.

Moduli spaces attempt to classify all mathematical objects of a particular type, for example algebraic curves or vector bundles, and record how they 'vary in families'. Often they are constructed using Geometric Invariant Theory (GIT) as a quotient of a parameter space by a group action. A common theme is that in order to have a nice (eg Hausdorff) space one must restrict one's attention to a suitable subclass of 'stable' objects, in effect leaving certain badly behaved objects out of the classification. Assuming no prior familiarity, I will elucidate the structure of instability in GIT, and explain how recent progress in non-reductive GIT allows one to construct moduli spaces for these so-called 'unstable' objects. The particular focus will be on the application of this principle to the GIT construction of the moduli space of stable curves, leading to moduli spaces of curves of fixed singularity type.

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.

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.