|January 29||Colin Crowley|
|Can it work?|
|February 5||Asvin Gothandaraman|
|Title: Lefschetz hyperplane section theorem via Morse theory|
|Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.|
|Title: An introduction to unirationality|
|Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected.|
|Title: Blowing down, blowing up: surface geometry|
| Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.)
|Title: Intro to Toric Varieties|
|Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.|
|Title: An introduction to Grothendieck-Riemann-Roch Theorem|
|Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.|
|Title: Intro to Stanley-Reisner Theory|
|Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.|
|Title: Embrace the Singularity: An Introduction to Stratified Morse Theory|
| Abstract: Early on in the semester, Colin told us a bit about Morse
Theory, and how it lets us get a handle on the (classical) topology of smooth complex varieties. As we all know, however, not everything in life goes smoothly, and so too in algebraic geometry. Singular varieties, when given the classical topology, are not manifolds, but they can be described in terms of manifolds by means of something called a Whitney stratification. This allows us to develop a version of Morse Theory that applies to singular spaces (and also, with a bit of work, to smooth spaces that fail to be nice in other ways, like non-compact manifolds!), called Stratified Morse Theory. After going through the appropriate definitions and briefly reviewing the results of classical Morse Theory, we'll discuss the so-called Main Theorem of Stratified Morse Theory and survey some of its consequences.
|Title: Birational geometry: existence of rational curves|
|Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight.|