This talk is a survey on results concerning the Teichmuller space of negatively curved Riemannian metrics on M. It is defined as the quotient space of the space of all negatively curved Riemannian metrics on M modulo the space of all isotopies of M that are homotopic to the identity. This space was shown to have highly non-trivial homotopy when M is real hyperbolic by Tom Farrell and Pedro Ontaneda in 2009.

Title: Chang's Conjecture (joint with Monroe Eskew) Abstract: I will review some consistency results related to Chang's Conjecture (CC). First I will discuss some classical results of deriving instances of CC from huge cardinals and the new results for getting instances of CC from supercompact cardinals, and present some open problems. Then, I will review the consistency proof of some versions of the Global Chang's Conjecture - which is the consistency of the occurrence many instances of CC simultaneously. We will aim to show the consistency of the statement: (\mu^+,\mu) -->> ( u^+,

We address the semistable reduction conjecture of Abramovich and Karu: we prove that every surjective morphism of complex projective varieties can be modified to a semistable one. The key ingredient is a combinatorial result on triangulating lattice Cayley polytopes. Joint work with Karim Adiprasito and Michael Temkin. The lecture consists of two parts: first 30 minutes an algebra-geometric introduction by Michael Temkin, and then a one hour talk by Gaku Liu about the key combinatorial result.

Global Chang's Conjecture

Yair Hayut - (joint with Monroe Eskew)

For $\kappa < \lambda$ infinite cardinals let us consider the following generalization of the Lowenheim-Skolem theorem:
"For every algebra with countably many operations over $\lambda^+$ there is a sub-algebra with order type exactly $\kappa^+$".

We will discuss the consistency and inconsistency of some global versions of this statement and present some open questions. 

Let F be a non-Archimedean local field. In the representation theory of GL_n(F), one of the basic problems is to characterize its irreducible representations up to isomorphism. There are many invariants (e.g., epsilon factors, L-functions, gamma factors, depth, etc) that we can attach to a representation of GL_n(F). Roughly, the local converse problem is to find the smallest subcollection of twisted local \gamma-factors which classifies the irreducible admissible representations of GL_n(F) up to isomorphism.