This is a reading seminar for two graduate students.) This talk studies the birational geometry of fibered surfaces, which are integral, projective, flat schemes of dimension 2 over a Dedekind scheme. In contrast to smooth projective curves, birational equivalence for surfaces does not imply isomorphism, which leads to the problem of understanding and selecting canonical representatives within a birational class. We first introduce basic tools for birational surface theory, including blowing-ups, contraction, and desingularization. We then explain how intersection theory on regular surfaces is used to analyze these operations and to identify exceptional curves. This perspective naturally leads to minimal surfaces and to applications of contraction criteria in the construction of canonical models.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

(This is a reading seminar for two graduate students.) This talk studies the birational geometry of fibered surfaces, which are integral, projective, flat schemes of dimension 2 over a Dedekind scheme. In contrast to smooth projective curves, birational equivalence for surfaces does not imply isomorphism, which leads to the problem of understanding and selecting canonical representatives within a birational class. We first introduce basic tools for birational surface theory, including blowing-ups, contraction, and desingularization. We then explain how intersection theory on regular surfaces is used to analyze these operations and to identify exceptional curves. This perspective naturally leads to minimal surfaces and to applications of contraction criteria in the construction of canonical models.

The subfield of low-dimensional topology colloquially called "3.5-dimensional topology" studies closed 3-manifolds through the eyes of the 4-manifolds that they bound. This talk focusses on Casson's question of which rational homology 3-spheres bound rational homology 4-balls. Since rational homology 3-spheres bounding rational homology 4-balls are a rare phenomenon, we will discuss how to construct examples.