Let \Gamma be a definable class of forcing posets and \kappa be a cardinal. We define MP(\kappa,\Gamma) to be the statement:

"For any A\subseteq \kappa, any formula \phi(v), for any P \in \Gamma, if there is a name \dot{Q} such that V^P models "\dot{Q}\in \Gamma + dot{Q} forces that \phi[A] is necessary" then V models \phi[A],"

where a poset Q \in \Gamma forces a statement \phi(x) to be necessary if for any \dot{R} such that V^Q \vDash \dot{R} \in \Gamma, then V^{Q\star \dot{R}} models \phi(x). When \Gamma is the class of proper forcings (or semi-proper forcings, or stationary set preserving forcings), we show that MP(\omega_1,\Gamma) is consistent relative to large cardinals. We also discuss the consistency strength of these principles as well as their relationship with forcing axioms. These are variants of maximality principles defined by Hamkins. This is joint work with Daisuke Ikegami.
 

 
In this talk we discuss closed self adjoint extensions of the Laplacian and fractional Laplacian on L2 of Euclidean space minus the origin. In some cases there is a one parameter family of these operators that behave like the original operator plus a potential at the origin. Using these operators, we can construct polymer measures which exhibit interesting phase transitions from an extended state to a bound state where the pinning at the origin due to the potential takes over. The talk is based on joint works with Koralov, Molchanov, Squartini and Vainberg.
 

Shimura varieties are defined over complex numbers and generally have number fields as the field of definition. Motivated by an example constructed by Mumford, we find conditions which guarantee a curve in char. p lifts to a Shimura curve of Hodge type. The conditions are intrinsic in positive characteristics and thus they shed light on a definition of Shimura curves in positive characteristics. 

In this talk, I will start with modular curves, and discuss the moduli interpretation of Shimura curves. Then I will present such a condition in terms of isocrystals. Time permitting, I would show a deformation result on Barsotti-Tate groups, which serves as a key step in the proof. 
 

    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 tree property arises as the generalization of Koenig's infinity lemma to an uncountable cardinal.  The existence of an uncountable cardinal with the tree property has axiomatic strength beyond the axioms of ZFC.  Indeed a theorem of Mitchell shows that the theory ZFC + ``omega_2 has the tree property" is consistent if and only if the theory ZFC + ``There is a weakly compact cardinal" is consistent.  In the context of Mitchell's theorem, we can ask an old question in set theory:  Is it consistent that every regular cardinal greater than aleph_1 has the tree property? In this talk we will survey the best known partial results towards a positive answer to this question.