A subset $A$ of a Riemannian symmetric space is called an antipodal set
if the geodesic symmetry $s_x$ fixes all points of $A$ for each $x \in A$.
This notion was first introduced by Chen and Nagano.  In this talk, using
the $k$-symmetric structure, first we describe an antipodal set of a complex
flag manifold.  Tanaka and Tasaki proved that the intersection of two real
forms $L_1$ and $L_2$ in a Hermitian symmetric space of compact type is an
antipodal set of $L_1$ and $L_2$.  We can observe the same phenomenon for
the intersection of certain real forms in a complex flag manifold.
  As an application, we calculate the Lagrangian Floer homology of a pair
of real forms in a monotone Hermitian symmetric space.  Then we obtain
a generalization of the Arnold-Givental inequality.
  This talk is based on a joint work with Hiroshi Iriyeh and Hiroyuki Tasaki.

The study of isoperimetric inequalities has a long history,
it's humble beginnings in Ancient Greek mathematics belying a deep and
rich theory. A major tool in the study of the isoperimetric profile is
the Calculus of Variations. Variational arguments lead to weak
differential inequalities for the isoperimetric profile, which allows
analytical tools such as the maximum principle to be employed. Of
central importance here is the connection with curvature, which is
intimately connected with the topology of isoperimetric regions. I
will survey some of the results in this direction, paying particular
attention to the interplay of the isoperimetric profile and curvature
flows which is the focus of my current research.

An isometric action of a connected Lie group on a Riemannian manifold is called polar if there exists a connected closed submanifold that meets each orbit of the action and intersects it orthogonally. Dadok established in 1985 a remarkable, and mysterious, relation between polar actions on Euclidean spaces and Riemannian symmetric spaces. Soon afterwards an attempt was made to classify polar actions on symmetric spaces. For irreducible symmetric spaces of compact type the final step of the classification has just been achieved by Kollross and Lytchak. In the talk I want to focus on symmetric spaces of noncompact type. For actions of reductive groups one can use the concept of duality between symmetric spaces of compact type and of noncompact type. However, new examples and phenomena arise from the geometry induced by actions of parabolic subgroups, for which there is no analogon in the compact case. I plan to discuss the main difficulties one encounters here and some partial solutions.

Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré
and Geometrization Conjectures, some of its basic properties are still unknown. For
example it has been an open question whether the surgeries eventually stop to occur
(i.e. whether there are finitely many surgeries) and whether the full geometric
decomposition of the underlying manifold is exhibited by the flow as times goes to infinity.

In this talk I will show that the number of surgeries is indeed finite and that the
curvature is globally bounded by C t^{-1} for large t. Using this curvature
bound it is possible to give a more precise picture of the long-time behavior of the
flow.