Manifolds have been a central object of study for over a century, and the classification of them has been a core theme for the whole of this time. This talk will give an overview of the successes and failures in this effort, with some illustrative examples.

In this talk we present a new type of Soblev norm defined in the space of functions of continuous paths. Under the Wiener probability measure the corresponding norm is suitable to prove the existence and uniqueness for a large type of system of path dependent quasi-linear parabolic partial differential equations (PPDE). We have establish 1-1 correspondence between this new type of PPDE and the classical backward SDE (BSDE). For fully nonlinear PPDEs, the corresponding Sobolev norm is under a sublinear expectation called G-expectation, in the place of Wiener expectation. The canonical process becomes a new type of nonlinear Brownian motion called G-Brownian motion. A similar 1-1 correspondence has been established. We can then apply the recent results of existence, uniqueness and principle of comparison for BSDE driven by G-Brownian motion to obtain the same result for the PPDE.

1)      The Hardt-Lin's problem and a new approximation of a relaxed energy for harmonic maps.

We introduce a new approximation for  the relaxed energy $F$ of the Dirichlet energy and prove that the minimizers of the approximating functional converge to a minimizer $u$ of the relaxed energy for harmonic maps, and that $u$ is  partially regular without using the concept of Cartesian currents.

2)  Partial regularity in liquid crystals  for  the Oseen-Frank model:  a new proof of the result of Hardt, Kinderlehrer and Lin.

Hardt, Kinderlehrer and Lin (\cite {HKL1}, \cite {HKL2}) proved that a minimizer $u$ is smooth on some open subset
$\Omega_0\subset\Omega$ and moreover $\mathcal H^{\b} (\Omega\backslash \Omega_0)=0$ for some positive $\b <1$, where
$\mathcal H^{\b}$ is the Hausdorff measure.   We will present a new proof of Hardt, Kinderlehrer and Lin.

 3)      Global existence of solutions of the Ericksen-Leslie system for  the Oseen-Frank model.

The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of  the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model,   which generalizes the heat flow for harmonic maps  into the $2$-sphere.   In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank  model in 2D.

1)      The Hardt-Lin's problem and a new approximation of a relaxed energy for harmonic maps.

We introduce a new approximation for  the relaxed energy $F$ of the Dirichlet energy and prove that the minimizers of the approximating functional converge to a minimizer $u$ of the relaxed energy for harmonic maps, and that $u$ is  partially regular without using the concept of Cartesian currents.

2)  Partial regularity in liquid crystals  for  the Oseen-Frank model:  a new proof of the result of Hardt, Kinderlehrer and Lin.

Hardt, Kinderlehrer and Lin (\cite {HKL1}, \cite {HKL2}) proved that a minimizer $u$ is smooth on some open subset
$\Omega_0\subset\Omega$ and moreover $\mathcal H^{\b} (\Omega\backslash \Omega_0)=0$ for some positive $\b <1$, where
$\mathcal H^{\b}$ is the Hausdorff measure.   We will present a new proof of Hardt, Kinderlehrer and Lin.

 3)      Global existence of solutions of the Ericksen-Leslie system for  the Oseen-Frank model.

The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of  the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model,   which generalizes the heat flow for harmonic maps  into the $2$-sphere.   In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank  model in 2D.

Localization and completion of spaces are fundamental tools in homotopy theory. "Zabrodsky mixing" uses localization to "mix homotopy types". It was used to provide a counterexample to the conjecture that any finite H-space which is $A_3$ is also $A_\infty$. The material in this talk will be very classical (and rather basic). I will describe Sullivan's localization functor and demonstrate Zabrodsky's mixing by constructing a non-classical H-space.

A group is called residually finite if every non-trivial element can be homomorphically mapped to a finite group such that the image is again non-trivial. Residually finite groups are interesting because quite a lot of information about them can be reconstructed from their finite quotients. Baumslag showed that if G is a finitely generated residually finite group then Aut(G) is also residually finite. Using a similar method Grossman showed that if G is a finitely generated conjugacy separable group with "nice" automorphisms then Out(G) is residually finite. The graph product is a group theoretic construction naturally generalising free and direct products in the category of groups. We show that if G is a finite graph product of finitely generated residually finite groups then Out(G) is residually finite (modulo some technical conditions)

A Kähler group is a group which is isomorphic to the fundamental group of a compact Kähler manifold. In 2008 Dimca and Suciu proved that the groups which are both Kähler and isomorphic to the fundamental group of a closed 3-manifold are precisely the finite subgroups of $O(4)$ which act freely on $S^3$. In this talk we will explain Kotschick's proof of this result. On the 3-manifold side the main tools that will be used are the first Betti number and Poincare Duality and on the Kähler group side we will make use of the Albanese map and some basic results about Kähler groups. All relevant notions will be explained in the talk.