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Contents
Spring 2020
Date  Speaker  Title (click to see abstract) 
January 29  Colin Crowley  Lefschetz hyperplane section theorem via Morse theory 
February 5  Asvin Gothandaraman  An Introduction to Unirationality 
February 12  Qiao He  Title 
February 19  Dima Arinkin  Blowing down, blowing up: surface geometry 
February 26  Connor Simpson  Intro to toric varieties 
March 4  Peter  An introduction to GrothendieckRiemannRoch Theorem 
March 11  Caitlyn Booms  Intro to StanleyReisner Theory 
March 25  Steven He  Title 
April 1  Vlad Sotirov  Title 
April 8  Maya Banks  Title 
April 15  Alex Hof  Embrace the Singularity: An Introduction to Stratified Morse Theory 
April 22  Ruofan  Birational geometry: existence of rational curves 
April 29  John Cobb  Title 
January 29
Colin Crowley 
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. 
February 5
Asvin Gothandaraman 
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. 
February 12
Qiao He 
Title: 
Abstract: 
February 19
Dima Arinkin 
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.) 
February 26
Connor Simpson 
Title: Intro to Toric Varieties 
Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blowups, and more. 
March 4
Peter Wei 
Title: An introduction to GrothendieckRiemannRoch Theorem 
Abstract: The classical RiemannRoch 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. 
March 11
Caitlyn Booms 
Title: Intro to StanleyReisner Theory 
Abstract: StanleyReisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (StanleyReisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how StanleyReisner theory can help us investigate algebrageometric questions. 
March 25
Steven He 
Title: 
Abstract: 
April 1
Vlad Sotirov 
Title: 
Abstract: 
April 8
Maya Banks 
Title: 
Abstract: 
April 15
Alex Hof 
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 noncompact manifolds!), called Stratified Morse Theory. After going through the appropriate definitions and briefly reviewing the results of classical Morse Theory, we'll discuss the socalled Main Theorem of Stratified Morse Theory and survey some of its consequences. 
April 22
Ruofan 
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. 
April 29
John Cobb 
Title: 
Abstract: 