Date |
## Winter 2017 Schedule |

January 19 |
Abstract: Algebraic Geometry has always related the geometry of geometric spaces with the algebra of algebraic structures. This is first seen with the Nullstellensatz relating ideals and closed subvarieties. The next level would be the relation between divisors and line bundles. This relation allows us to extract geometric information out of our spaces as well as define morphisms on our geometric spaces that can have many different type of behavior. This talk will give a little more insight into these techniques of divisors and line bundles, more specifically of a nice theorem on fixed divisors. |

January 26 |
Abstract: This talk will be an introduction to arithmetic geometry. No background knowledge of required. |

February 2 |
Abstract: We introduce Grassmannian manifolds, which are a natural generalization of projective space. We introduce the Schubert calculus on homological cycles and demonstrate a few geometric conclusions you can draw from it. |

February 9 |
Abstract: Around the turn of the 19th century, the Italian school of algebraic geometry with the likes of Cremona, Segre, Castelnuovo and Enriques began a study of the birational geometry of algebraic surfaces, eventually leading to a classification of algebraic surfaces via minimal models by Enriques and formalized and expanded upon by Kodaira. The surface case then became a guide towards a birational classification of all varieties called the minimal model program; first realized by Mori when he generalized the techniques of the surface case and applied them to threefolds. Much has been done since then but there is still very much to do and currently the minimal model program is an active area of research. This talk will begin a discussion of the birational geometry and the minimal model program of the classical surface case before the works of Mori. |

February 16 |
Abstract: While reading* about the Weil conjectures, I came across the following exercise: Let n be a positive integer, and choose e_1, ..., e_n in the set {1, -1}. Prove that for all p sufficiently large (relative to n), there exists an integer x such that x+k is a quadratic residue mod p if e_k=1, and x+k is a nonresidue if e_k = -1. After giving some background on the Weil conjectures, I will present a solution to the exercise. *cf. www-math.mit.edu/~poonen/papers/Qpoints.pdf, Exercise 7.7 |

February 23 |
Abstract: An algebraic integer is a solution to a polynomial with integer coefficients. For example, 3, 5/12, the cube root of 2, etc., are all algebraic integers, but numbers like pi and e are not. A natural generalization of this concept leads to algebraic functions: they are defined as solutions to polynomial equations with polynomial coefficients. Algebraic functions play an important role in both pure and applied mathematics. In this talk, we will first look at the historical developments of algebraic functions and, then, we will survey the one variable case to get a feel for the subject. |

March 2 |
Abstract: In algebraic geometry, a moduli space is a way of viewing a particular class of schemes as a scheme itself. We introduce the notion of a moduli problem and its framing in category-theoretic language. We then introduce the Hilbert scheme associated to a polynomial p(t), which parametrizes the family of subschemes of projective space that have Hilbert polynomial p(t), and sketch its construction. |

March 9 |
Abstract: We continue the discussion of the surface case of the minimal model program by establishing a cohomological definition of the intersection pairing. This allows for describing the self intersection of curves and the geometric interpretations that arises from negative self intersection. Lastly, we will establish some invariants that sets up Castelnuovo's contraction theorem. |

March 16 |
Abstract: A Leavitt Path Algebra is a type of algebra which can be constructed from an arbitrary directed graph. I will briefly summarize my talk on them from last quarter, and then discuss some results regarding which posets appear when looking at their (graded) prime ideals ordered by set inclusion. |