August 27: Takumi Murayama (Purdue University)
Location: STON 215
Title: Embeddings of projective varieties and Fujita's conjecture
Abstract: All smooth projective curves (which, over the complex numbers, are the same thing as Riemann surfaces) can be embedded into a projective space. However, it is difficult to find an explicit embedding of a given smooth projective curve into a projective space.
In this talk, I will discuss how to find embeddings of smooth projective curves into projective spaces. This motivates Fujita's very ampleness conjecture, which predicts there is a uniform way to use an ample divisor on a smooth projective variety X to find an embedding of X into a projective space. Fujita's very ampleness conjecture is known in dimensions at most 2 over the complex numbers. It remains a difficult open problem even in dimension 3.
September 3: Zhiyuan Jiang (Purdue University)
Location: STON 215
Title: Complex analytic geometry and positivity
Abstract: I will review some basic concepts in analytic geometry with a focus on the comparison between algebraic geometry and analytic geometry. I will also introduce the concept of positivity in complex analytic geometry and the role they play in birational geometry, bimeromorphic geometry, and the minimal model program.
September 10: Donu Arapura (Purdue University)
Location: STON 215
Title: Perverse Sheaves
Abstract: Perverse sheaves were invented by Beilinson, Bernstein and Deligne a bit over 40 years ago. The first thing to point out (as the inventors do) is that they aren’t perverse and they aren’t sheaves. What they are are really nice objects of the constructible derived category of a variety.
September 17: Fanjun Meng (UC San Diego)
Location: STON 215
Title: Wall crossing for moduli of weighted pointed stable curves
Abstract: I will review Hassett’s classical result on wall crossing for moduli of weighted pointed stable curves.
September 24: Hannah Larson (University of California, Berkeley)
Location: Zoom (https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1)
Title: Linear series on curves
Abstract: I will explain the correspondence between maps to projective space and linear series. Then, we'll discuss some concrete examples in low genus.
October 8: Wanchun (Rosie) Shen (Harvard University)
Location: STON 215
Title: Introduction to singularities in algebraic geometry
Abstract: We give a gentle introduction to the singularities in algebraic geometry. Through examples, we will see how ideas from calculus (differentiation and integration) provide us tools for measuring and classifying singularities.
October 15: Siqing Zhang (Yale University)
Location: STON 215
Title: Differential operators and Azumaya algebras
Abstract: I will introduce the sheaf of differential operators. Then I will give an idea for why, in characteristic p, the sheaf of differential operators becomes something similar to a finite rank matrix algebra, i.e., an Azumaya algebra. Essentially, this is saying that, in characteristic p, calculus becomes very similar to linear algebra.
October 22: Riku Kurama (University of Michigan)
Location: STON 215
Title: Liftability of varieties from positive characteristic to characteristic 0
Abstract: For a variety in positive characteristic, one natural question to ask is whether the variety can be placed in a flat family of varieties, along which the original variety continuously deforms to another variety which lives over a field of characteristic 0. In this talk, I will motivate this question of lifting varieties from positive characteristic to characteristic 0 and then discuss some examples.
October 29: Florin Ambro (Simion Stoilow Institute of Mathematics of the Romanian Academy)
Location: Zoom (https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1)
Title: An invitation to toric varieties
Abstract: Toric varieties provide a bridge between algebraic geometry and combinatorics. From a lattice and a cone or polytope, one constructs a toric singularity or polarized variety, whose invariants are controlled by combinatorics. Such explicit examples can be used to test open problems in algebraic geometry, and gain intuition for the general case. Conversely, combinatorial problems can be solved by using the geometry of the associated toric variety. The aim of this short talk is to present some basic examples of toric varieties, and their corresponding combinatorial data.
November 5: Casimir Kothari (University of Chicago)
Location: STON 215
Title: Introduction to group schemes
Abstract: Group schemes provide a systematic way to study p-torsion phenomena in characteristic p. In this talk, I will give an introduction to the theory of group schemes over a field, and discuss many examples.
November 12: Andreas Höring (Université Côte d’Azur)
Location: Zoom (https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1)
Title: Nef line bundles on K3 surfaces and abelian varieties
Abstract: In the pretalk I will present examples of K-trivial manifolds in dimension and two, that is elliptic curves, two-dimensional tori and K3 surfaces. We will then do some Riemann-Roch computations to solve the nonvanishing problem on K3 surfaces.
November 19: Botong Wang (University of Wisconsin-Madison)
Location: STON 215
Title: Alexandrov-Fenchel inequalities
Abstract: will give a brief introduction to mixed volumes and the Alexandrov–Fenchel inequalities, and present some examples showing how other inequalities can be derived from them.
December 3: Shreya Sharma (University of South Carolina, Columbia)
Location: STON 215
Title: Equivariant tools and Fano threefolds
Abstract: This pre-talk will introduce the key concepts from equivariant algebraic geometry, including G-linearizations of line bundles and cohomological equivariant birational invariants, setting the stage for the main talk. Time permitting, we will also review the classification of smooth Fano threefolds over algebraically closed fields as given by Iskovskikh, Mori, and Mukai.
December 10: Lingyao Xie (University of California, San Diego)
Location: STON 215
Title: Some simple computations of Pic^0:
Abstract: We will do some examples of Pic^0 of reduced curves.