January 15: Christian Schnell (Stony Brook University)
Location: SCHM 112
Title: Hyperkahler Manifolds
Abstract: We'll look at some basic facts about hyperkähler manifolds (also known as holomorphic symplectic manifolds) and Lagrangian fibrations, with examples. You can prepare by trying to remember all the examples of hyperkähler manifolds you have seen before.
January 22: Yilong Zhang (Purdue University)
Location: SCHM 112
Title: 27 lines and Schubert Calculus
Abstract: A smooth cubic surface contains 27 lines. These lines can also be described as the zero locus of a section on certain vector bundle on the Grassmannian of lines in P^3, so computing the number of lines is equivalent to determining the top Chern class of this bundle. In this talk, we will explore how the number 27 appears through the intersection theory of Grassmannians.
February 5: Ben McReynolds (Purdue University)
Location: SCHM 112
Title: Nilpotent representations and a theorem of Stallings
Abstract: In this talk, I will review from the start some basic concepts in group theory and universal objects for classes of representations. Next I will discuss a lovely theorem of Stallings that connects the nilpotent representation theory of a group with the first and second integral homology groups of a group. Finally, I will end with an analog of Stallings theorem which is better suited for constructions and is heavily inspired by Stallings.
February 12: Siyang Liu (University of South California)
Location: SCHM 112
Title: Wonderful Compactifications
Abstract: I’ll talk about a wonderful model for resolutions of certain good singularities.
March 5: Emanuel Reinecke (Institut des Hautes Études Scientifiques)
Location: SCHM 112
Title: The rigid unit disk
Abstract: I will give a gentle introduction to rigid-analytic varieties, using Huber's framework of adic spaces. We will look in some detail at the example of the closed unit disk, where everything becomes very explicit.
March 12: Donu Arapura (Purdue University)
Location: SCHM 112 (2 pm-2:20 pm)
Title: Introduction to Hodge-Weil classes.
Abstract: Weil was known to be skeptical about the Hodge conjecture, and as a challenge, he proposed proving the algebraicity of certain Hodge classes now known as Hodge-Weil classes. I will explain the construction, and perhaps a bit of background concerning the conjecture itself.
March 26: Bogdan Zavyalov (Princeton University/Institute for Advanced Study)
Location: SCHM 112
Title: Poincare Duality for compact complex manifolds
Abstract: In this talk, I will discuss a somewhat unorthodox way to prove Poincare Duality for compact complex manifolds (most likely I will stick to the case of complex curves) using the notion of dualizable objects in monoidal categories. This approach generalizes well to many other cohomology theories.
April 2: Ben Tighe (University of Oregon)
Location: SCHM 112
Title: Hodge-theoretic singularities
Abstract: I'll discuss rational and Du Bois singularities, two classes which are closely related to the minimal model program, and why one might want to study them in the context of birational geometry and deformation theory.
April 9: François Greer (Michigan State University)
Location: SCHM 112
Title: Kodaira dimension 0
Abstract: The first birational invariant of varieties is Kodaira dimension. In large Kodaira dimension, classification is hard. In small Kodaira dimension, classification is more tractable.
April 16: Masayuki Kawakita (Research Institute for Mathematical Sciences, Kyoto University)
Location: SCHM 112
Title: Singularities in birational geometry
Abstract: Singularities appear naturally in birational geometry from the point of view of the minimal model theory. I will introduce an invariant of singularities called the minimal log discrepancy and propose conjectures on this invariant.
April 23: Amadou Bah (Columbia University)
Location: SCHM 112
Title: The Ramification theory of local fields: from perfect to imperfect residue fields.
Abstract: The ramification theory of discrete valuation fields plays a crucial role in various subfields of Arithmetic Geometry. In this biref expository talk, I will recall elments of the classical picture (perfect residue field) to introduce the key geometric idea in A. Abbes and T. Saito's generalisation of the theory to arbitrary residue fields. If I have time, I will then explain how such a generalisation enters into Takeshi Saito's extension to arbitrary dimensions of the classical Grothendieck-Ogg-Shafarevich formula for the Euler characteristic of an l-adic sheaf on a curve over a perfect field.
April 30: Mircea Mustaţă (University of Michigan)
Location: SCHM 112
Title: A quick introduction to D-modules
Abstract: I will give a brief introduction to the theory of D-modules on the affine space a.k.a. modules over the Weyl algebra.