- Title: Division algebras and representations of Lie groups.
Speaker: Ben Antieau, University of Washington.
Time and Place: Friday, Jan. 10 at 3:30 in LC 412.
Abstract: I will give a basic introduction to principal bundles over different kinds of spaces, and I will discuss the various spaces that classify these bundles. A key insight is that even topologically these bundles can be classified by maps to complex algebraic varieties. This led Atiyah and Hirzebruch to the first counterexamples to the original, integral Hodge conjecture. I will discuss how a different approach to these algebraic classifying spaces, due to Totaro, led Ben Williams and myself to a method for answering concrete questions about division algebras over fields, and I will explain how these ideas resulted in our answer to a question of Auslander and Goldman in algebra from 1960.
- Title: Classification of algebraic varieties: classical results and recent advances
Speaker: Zsolt Patakfalvi, Princeton University.
Time and Place: Tuesday, Jan. 21 at 4:30 in LC 412.
Abstract: Two algebraic varieties are called birationally equivalent if they admit isomorphic Zariski open subset. The classification of algebraic varieties up to birational equivalence has been in the focus of algebraic geometry since the initial work of the Italian school in the 19th century. The primary invariant used during this classification is Kodaira dimension. It assigns a number between 0 and the dimension or negative infinity to every birational equivalence class of varieties. The bigger this number is the more the variety is thought of as being "hyperbolic". I will present classical results and recent advances in positive characteristic on the behavior of Kodaira dimension in fibrations.
- Title: Syzygies of line bundles on Segre-Veronese varieties
Speaker: Claudiu Raicu, Princeton University.
Time and Place: Thursday, Jan. 23 at 4:30 in LC 412.
Abstract: Syzygies are important classical invariants attached to algebraic varieties. They can be defined iteratively starting from the defining equations of a variety and continuing with the relations between these equations, the relations between the relations etc. In the special case of the Segre-Veronese embeddings of products of projective spaces, the structure of syzygies is particularly rich, due to the presence of a large group of symmetries. Their study can be approached through a variety of techniques, placing them at the confluence of algebraic geometry, commutative algebra, representation theory and combinatorics. I will discuss some recent results on the structure of the syzygies of line bundles on Segre-Veronese varieties, and explain how their vanishing controls the asymptotic vanishing behavior (in the sense of Ein and Lazarsfeld) of syzygies of arbitrary algebraic varieties.
- Title:Linear Spaces of Singular Matrices
Speaker: Nathan Ilten, University of California-Berkeley
Time and Place: Wednesday, Feb. 12 at 3:30 in LC 412.
Abstract: Simple questions in linear algebra often give rise to interesting objects in algebraic geometry. For example, the space of singular $n \times n$ matrices is the classical hypersurface given by the vanishing of the $n\times n$ determinant. This hypersurface has interesting geometric properties. In particular, it contains many linear subspaces, whose study is related to (among other things) geometric complexity theory.
In this talk, I will introduce a geometric object, called a Fano scheme, parameterizing these linear subspaces. We'll look at some basic examples of Fano schemes, and then see a characterization of exactly when our Fano schemes are connected. This is joint work with Melody Chan..
- Title: Singularities of polynomials in characteristic zero and characteristic p
Speaker: Karl Schwede, Penn. State University.
Time and Place: Thursday, March 20 at 4:30 in LC 412.
Abstract: We will consider solution sets to polynomial equations. For example, consider y^2 = x^3, the solution set has a singularity at the origin (a cusp). I will talk about different ways to measure singularities such as this one. First I will discuss the multiplier ideal, a way to measure singularities via integration. Second I will discuss the test ideal, a way to measure singularities if one considers them over a finite field. It has been known for nearly two decades that these two independently introduced notions are closely related. I will discuss recent joint work with Manuel Blickle and Kevin Tucker which shows that these two objects are in fact aspects of the same phenomenon.
- Title:The structure of free complexes.
Speaker: Daniel Erman, University of Michigan.
Time and Place: Monday, April 16, 2012 at 2 pm in LC 312.
Abstract: Over the polynomial ring, a graded free complex consists of a sequence of matrices whose entries are polynomials and where the consecutive products of the matrices equals 0. Graded free complexes arise throughout algebra, geometry, and topology, where they are often used to compute homological invariants. I will discuss a new perspective for understanding the structure of graded free complexes. This is joint work with David Eisenbud.
- Title: Riemann-Roch for Graphs and Applications
Speaker: Matt Baker, Georgia Tech University.
Time and Place: Monday, April 21 at 4:30 in LC 412.
Abstract: We will begin by formulating the Riemann-Roch theorem for graphs due to the speaker and Norine. We will then describe some refinements and applications. Refinements include a Riemann-Roch theorem for tropical curves, proved by Gathmann-Kerber and Mikhalkin-Zharkov, and a Riemann-Roch theorem for metrized complexes of curves, proved by Amini and the speaker. Applications include a "volume proof" of the Matrix-Tree theorem by An-B.-Kuperberg-Shokrieh, a new proof by Jensen and Payne of the Gieseker-Petri theorem in algebraic geometry, and a new Chabauty-Coleman style bound for the number of rational points on an algebraic curve over the rationals by Katz and Zureick-Brown.
- Title: Phylogenetic Algebraic Geometry
Speaker: Seth Sullivant, North Carolina State University.
Time and Place: Thursday April 24 at 4:30 in LC 412.
Abstract: The main problem in phylogenetics is to reconstruct evolutionary relationships between collections of species, typically represented by a phylogenetic tree. In the statistical approach to phylogenetics, a probabilistic model of mutation is used to reconstruct the tree that best explains the data (the data consisting of DNA sequences from genes appearing in all species being analyzed). In algebraic statistics, we interpret these statistical models of evolution as geometric objects in a high-dimensional probability simplex. This connection arises because the functions that parametrize these models are polynomials, and hence we can consider statistical models as algebraic varieties. The goal of the talk is to introduce this connection and explain how the algebraic perspective leads to new theoretical advances in phylogenetics, and also provides new research directions in algebraic geometry. The talk material will be kept at an introductory level, with background on both phylogenetics and algebraic geometry.
- Title: Primes, Zeros, and Random Matrix Theory
Speaker: Hugh Montgomery, University of Michigan.
Time and Place: Friday September 5 at 3:30 in LC 412.
Abstract: From the work of Riemann we understand that the distribution of prime numbers isclosely associated with the locations of the zeros of the Riemann zeta function. We review this,and then discuss more recent work, from which it appears that the spacings between the zeta
zeros resembles the spacings between the eigenvalues of a random unitary matrix.
- Title: Some Constructions of "Mirror" Manifolds and Why They Don't Always Agree
Speaker: David Favero, University of Alberta.
Time and Place: Thursday, November 6, 2014 at 4:30pm in LC 412.
Abstract: As it turns out, some manifolds come in pairs. Well... maybe not. While given a Calabi-Yau manifold, mirror symmetry predicts the existence of a "mirror" Calabi-Yau manifold, the various constructions of the mirror don't always agree. I will give an introduction to mirror symmetry and discuss multiple mirror phenomena in the Berglund-Hubsch-Krawitz construction of mirror pairs. If time permits, I will discuss a similar phenomenon in the Batyrev-Borisov construction. At the end, I will provide some theorems which unify these mirrors through birational geometry and derived categories. This is joint work with Tyler Kelly.