CMU Algebra and Combinatorics Seminar

Fall 2016

Organizers

Meeting Times

Wednesdays, 3–3:50pm, in Pearce 332.

Webpages from previous semesters:

Schedule

Date Speaker Affiliation Title
9/7 Ben Salisbury Central Michigan University Rigged Configurations and the Infinity Crystal
9/21 Motohico Mulase University of California, Davis Counting irreducible characters, topology of moduli spaces, and Galois representations
9/28 John Machacek Michigan State University The Boundary Measurement Matrix
10/05 Olivia Dumitrescu Central Michigan University Positivity of divisors on blown-up projective spaces
10/12 David C. Murphy Hillsdale College Generalizing Toric Varieties
10/19 Eric Bucher Michigan State University Quiver Mutations: An Intersection of Algebra, Combinatorics, and Topology
11/2 Charlotte Ure Michigan State University Graded Simple Quotients of the Generalized Clifford Algebra
11/30 Rohini Ramadas University of Michigan Dynamics on moduli spaces
12/7 Sema Güntürkün University of Michigan Conjectures on bounding Hilbert Functions and Betti numbers

Abstracts

Speaker: Ben Salisbury (Central Michigan University)
Title: Rigged Configurations and the Infinity Crystal
Abstract: The infinity crystal models the structure of (one half of) the quantum group (or, quantized universal enveloping algebra) and is built using a complicated algebraic procedure. However, crystals highlight the important combinatorial characteristics of the algebra, and therefore can be modeled using various combinatorial objects. In this talk, the rigged configuration model will be described and some of its important properties will be discussed.

Speaker: Motohico Mulase (University of California, Davis)
Title: Counting irreducible characters, topology of moduli spaces, and Galois representations
Abstract: The talk aims at introducing the audience to the current research frontier on spaces of characters of the fundamental group of a surface. The set of algebraic curves defined over the field of algebraic numbers, on which the absolute Galois group acts, is an example of such a space. We start with explaining a celebrated theorem due to Belyi that connects Galois theory and complex geometry as a motivation. We then illustrate the power of mathematically rigorous "topological quantum field theory" in counting problems on moduli spaces. The talk focuses on the elementary nature of mathematics, sticking to the philosophy that something beautiful must be simple. The talk is based on my joint work with Dr. Dumitrescu, and also our earlier paper with Dr. Safnuk.

Speaker: John Machacek (Michigan State University)
Title: The Boundary Measurement Matrix
Abstract: Given an edge weighted directed graph embedded on a surface, the boundary measurement matrix has entries given by signed sums of paths between vertices. Postnikov (2006) first studied the boundary measurement matrix for graphs embedded on a disk, and a formula for the maximal minors (i.e. Plücker coordinates) was given by Talaska (2008). We will consider graphs embedded on any closed orientable surface with boundary and give a formula of the maximal minors of the boundary measurement matrix.

Speaker: Olivia Dumitrescu (Central Michigan University)
Title: Positivity of divisors on blown-up projective spaces
Abstract: The minimal model program aims at a birational classification of complex algebraic varieties. The classification of surfaces was completed in the beginning of the 20th century by the Italian school of Algebraic Geometry. In the 1980s, minimal model to higher dimension was extended by admitting the presence of suitable singularities. The abundance conjecture and the existence of good models are among the main open problems in the minimal model program. In this talk we study log canonical pairs given by divisors on the blow-up of projective spaces in collections of points in general position. We give a cohomological description of the strict transforms of these divisors in the iterated blow-up along the cycles of the singular locus. Vanishing theorems for the higher cohomologies are used to give a systematic study of semi-ample divisors on these further blown-up spaces. As a consequence, we prove the abundance conjecture for an infinite number of such log canonical pairs, and an explicit construction of good minimal models. This is a joint work with Elisa Postinghel (2015).

Speaker: David C. Murphy (Hillsdale College)
Title: Generalizing Toric Varieties
Abstract: Toric varieties are often introduced and constructed from combinatorial data such as polytopes or cones and fans. In this way, they offer an accessible entry point to algebraic geometry. However, it is important to remember that toric varieties were first viewed as torus embeddings, and it was from this perspective that their classification was derived. It is also from this perspective that a number of attempts to generalize toric varieties and their classification have arisen. In this talk we will discuss toric varieties and one such generalization.

Speaker: Eric Bucher (Michigan State University)
Title: Quiver Mutations: An Intersection of Algebra, Combinatorics, and Topology
Abstract: Quiver mutation is a combinatorial process that takes a directed graph and makes a “local change” to create a new direct graph. Fomin and Zelevinsky in 2003, utilized these combinatorics to understand a class of algebras known as cluster algebras. Mutation has since been utilized to understand objects in an assortment of mathematical areas: including algebraic geometry, number theory, representation theory, topology, and even string theory. In this talk we will introduce quiver mutation and show it's connection to the topology of surfaces. We will then look at a special sequence of mutations known as maximal green sequences.The existence of these sequences has many consequences on the underlying structure. This topic is an interesting intersection of algebra, topology, and combinatorics with many exciting and challenging problems still open; which will be accessible to faculty, graduate students, and undergraduates alike.

Speaker: Charlotte Ure (Michigan State University)
Title: Graded Simple Quotients of the Generalized Clifford Algebra
Abstract: For any homogeneous form $f$ of degree $d$ in $n$ variables, we can define the generalized Clifford algebra $C_f$, which has a natural $\mathbb{Z}/d\mathbb{Z}$-grading. If $f$ is a quadratic form, its Clifford algebra is classical. My talk will be concerned with the case $n=d=3$, i.e. $f$ is a ternary cubic form. We will construct graded simple quotients of $C_f$ that are graded central simple algebras of arbitrarily high dimensions. This talk is based on joint work with Adam Chapman, Casey Machen and Rajesh Kulkarni.

Speaker: Rohini Ramadas (University of Michigan)
Title: Dynamics on moduli spaces
Abstract: The moduli space $M_{0,n}$ is an algebraic variety parametrizing all ways of labeling $n$ distinct points on the Riemann sphere, up to Möbius transformations. In my talk I will introduce Hurwitz correspondences — a certain natural family of multi-valued (like square root) self-maps of $M_{0,n}$. Hurwitz correspondences arise in topology via Thurston's answer to the question: which branched coverings of a topological sphere are homotopic to rational functions on the Riemann sphere? I will discuss their dynamical complexity via numerical invariants called dynamical degrees.

Speaker: Sema Güntürkün (University of Michigan)
Title: Conjectures on bounding Hilbert Functions and Betti numbers
Abstract: Hilbert functions of a homogeneous ideal in a polynomial ring is an important invariant with applications not only in Commutative Algebra but also in other areas such as Algebraic Geometry and Combinatorics. Macaulay showed in 1927 that any Hilbert function of a homogeneous ideal is attained by a lex ideal. Along with Hilbert functions, (graded) Betti numbers are known as important combinatorial invariants for (homogenous) ideals as well. Bigatti-Hulett-Pardue proved the extremal property of lex ideals in terms of Betti numbers among the ideals with same Hilbert functions. Later these significant results in Hilbert functions and Betti numbers motived some conjectures such as the ones by Eisenbud, Green, Harris and Evans. Today in this survey talk we will discuss about well known conjectures on bounding Hilbert functions and Betti numbers.

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