Talk #1: "The Lefschetz Theorem on (1,1) Classes" (Zhiyou Wu)
We will discuss and give a proof of a special case of the Hodge conjecture: that for a smooth projective manifold, every (1,1) cohomology class is analytic. We will require some results from Kähler geometry, such as the Hodge-Dolbeault decomposition, which will be stated and their proofs sketched. If time permits we will make further remarks about the general Hodge conjecture.
Talk #2: "Homotopy of 2-complexes (a Topology - Algebra dictionary)" (David Pazmino Pullas)
Abstract: In speech, people sometimes use words from another language to describe situations that may seem difficult to express or understand in the original one. A similar situation may arise when we approach a mathematical problem. In our case, we will focus on finding some algebraic characterizations for 2-dimensional CW-complexes. The talk will start with a very short reminder of some of the properties of homotopy groups. Next, we will realize that a 2-complex can be regarded as a group presentation, or more precisely as a crossed module (which will be defined in the talk). This will permit us to translate some problems in low dimensional topology to an algebraic language. An application for that translation will be briefly discussed, as well as translations to similar "languages".