This seminar, organized by Aniket Shah and Yilong Zhang, intends to provide a platform for graduate students to practice presenting their research or current topics of study in Algebraic Geometry.

Date | Speaker | Topic | Abstract and references |
---|---|---|---|

01/15/2019 | Aniket Shah | An Introduction to the Grothendieck Group | K-theory, as originally developed by Grothendieck, is fairly elementary to define but has curious geometric properties. We give an example-heavy introduction to try to illustrate some of its properties and how it can be used in algebraic geometry. |

01/22/2019 | Yilong Zhang | A Tour of Hodge Theory on Curves | Any two compact Riemann surfaces of fixed genus are diffeomorphic to each other, but as complex structure varies, they can far from being biholomorphic to each other. Hodge Theory develops a way to capture such difference for complex projective varieties, and it behaves well when varieties vary in a family. This talk is an introduction of Hodge Theory in dimension one. |

01/29/2019 | Minyoung Jeon | Schubert Varieties in Flag Varieties for the Classical Groups | The aim of this talk is to describe the Schubert varieties in flag varieties for type A, B, C and D. We will review the definitions of flag varieties and Schubert varieties for type A with basic examples. We will also describe isotropic flags to define the Schubert varieties for the other classical groups. If time permitted, we will introduce degeneracy loci for type A with some properties. |

02/05/2019 | Gabriel Bainbridge | Motivic Spaces | Mathematicians have long wanted to replicate the successes of homotopy theory in algebraic geometry. Although schemes have a geometric flavor, many topological operations and constructions are impossible to replicate in any reasonable category of schemes. Voevodsky's category of motivic spaces provides a solution to this challenge. Not only are we free to take pushouts, quotients, smash products, and internal homs in this category, but we also get a nice homotopy theory. Our work on the homotopy theory of motivic spaces can then solve problems in the original category of schemes. In this talk I will introduce the category of motivic spaces and do some hand-waving about their homotopy theory. |

02/12/2019 | Qingsong Wang | A Characterization of Projective Space | We will introduce Mori's "bend and break" technique and his resolution of Hartshorne Conjecture: A smooth projective variety with ample tangent bundle is isomorphic to projective space. |

02/19/2019 | Charles Koenig | Tropical Geometry | Tropical Geometry is algebraic geometry where our familiar fields are replaced by the tropical semiring: R with addition and multiplication replaced by min and plus. This leads to piece-wise linear varieties with many surprising analogies to classical varieties. In this talk, we'll start with arithmetic and some pictures to get comfortable. Tropical analogs of the Fundamental Theorem of Algebra and Bezout's theorem will be given. They will be connected back to classical algebraic geometry via the Fundamental Theorem of Tropical Varieties. |

02/26/2019 | Ian Cavey | The Hilbert Scheme of points on C^2 | Hilbert Schemes are moduli spaces which parameterize certain kinds of closed subschemes of a fixed scheme. Even in the case of zero-dimensional subschemes on surfaces, these objects are already complicated and carry lots of information about the original space. In the first half of this talk we will see how general Hilbert Schemes arise naturally when studying subschemes. In the second half we will try to see what is happening in the case of points on C^2. |

03/05/2019 | Junjie Chen | Structure of Berkovich Analytic Curves | Let X be a smooth curve over a complete and algebraically closed non-archimedean field, one can construct the Berkovich analytic space X^an. In this talk, I will introduce the skeletal theory of Berkovich curves. We’ll define semistable vertex sets of X and their associated skeleta, then describe the natural one-to-one correspondence between semistable vertex sets and semistable models of X. We'll also see how to define a canonical metric on X^an-X(K) using the skeletal theory. |

03/12/2019 | (Spring break, no seminar) | TBA | TBA |

03/19/2019 | Chen Chen | An Introduction to Toric Varieties | Toric varieties form an important class of algebraic varieties whose development has created a rich dictionary between combinatorial geometry and algebraic geometry. In this talk, we will first construct the affine toric variety associated to a cone, and then create more general toric varieties by gluing together affine toric varieties containing the same torus T. The constructions are motivated by examples from which more general methods can be deduced. |

03/26/2019 | (Zassenhaus lecture, no seminar) | TBA | TBA |

04/02/2019 | Ruize Yang | An Introduction to Toric Varieties, II | This talk will be the continuation of the previous one. We will see how to construct normal toric varieties from fans explicitly and how fans can be generated from polytopes. Several classical examples will be given. Then we will get to our main result “Orbit-cone correspondence” and some corollaries of that. |

04/09/2019 | Yilong Zhang | Line bundles on Singular Curves | On a smooth curve, line bundles are parameterized by its Picard group. Among them there is a unique one, the canonical bundle, which governs the Serre duality. What about the singular curves? In this talk, we will review classical results on smooth curves and focus on curves with at most nodal singularities. We will study Picard group and the "canonical bundle" of such curves. |

04/16/2019 | TBA | TBA | TBA |

04/23/2019 | TBA | TBA | TBA |