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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 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.

In Bass-Serre theory, one derives structural properties of groups from their actions on simplicial trees. In this talk, we introduce the theory of groups acting on $\mathbb{R}$-trees. In particular, we explain how the Rips machine is used to classify finitely generated groups which act freely on $\mathbb{R}$-trees.

I will look at some decidability questions for subgroups of Aut($F_n$) for general $n$. I will then discuss semisimple actions of Aut($F_n$) on complete CAT(0) spaces proving that the Nielsen moves will act elliptically. I will also look at proving Aut($F_3$) is large and if time permits discuss the fact that Aut($F_n$) is not Kähler

This talk will give an almost complete proof of the h-cobordism theorem, paying special attention to the sources of the dimensional restrictions in the theorem. If time allows, the alterations needed to prove its cousin, the s-cobordism theorem, will also be sketched.

The Nottingham Group of a finite field is an object of great interest in profinite group theory, owing to its extreme structural properties and the relative ease with which explicit computations can be made within it. In this talk I shall explore both of these themes, before describing some new work on efficient short-word approximation in the Nottingham Group, based on the profinite Solovay-Kitaev procedure. Time permitting, I shall give an application to the dynamics of compositions of random power series.