Javier Gonzalez Anaya: What is a moduli space?
A natural endeavour in mathematics is to classify objects according to their properties. For example, we can immediately recognize straight lines in the plane, or distinguish different kinds of triangles depending on their symmetries. Less intuitive, however, is that given a class of mathematical objects, it is often possible to construct a geometric space parametrizing those objects. Known as "moduli spaces," the study of these spaces has been a major driving force of modern geometry. In this talk we will explore some of the main ideas behind moduli theory through examples, ranging from the moduli space of lines in the plane to the one of points on the sphere. We will conclude by discussing some of my work in the area, including work with students.
Esther Banaian: Cluster algebras from surfaces and orbifolds: combinatorial expansion formulas and bases
Cluster algebras are rings with a distinguished set of generators which are defined recursively through a process called mutation. Many cluster algebras with desirable properties can be seen as coming from a triangulated surface, and in this setting mutation is realized by a simple combinatorial operation. We will discuss formulas regarding cluster algebras from surfaces which can be used to circumvent the recursive definition. Then, we will discuss how these formulas can be used to exhibit two bases of surface cluster algebras. Time-permitting, we will extend parts of the story to a wider class of algebras arising from orbifolds. No familiarity with cluster algebras will be assumed for the talk. The new results discussed will be from projects with Wonwoo Kang and Elizabeth Kelley.
Hector Banos: Inferring Evolutionary Relationships Among Hybridizing Species: What Can We Really Infer?
Phylogenetics is the branch of evolutionary biology that studies the evolutionary relationships of organisms. In this talk, I will discuss mathematical models used to study evolution in the presence of hybridization. Hybridization occurs when two distinct species merge genetically to create a new species. In such cases, species networks serve as mathematical representations of species relationships. The network multispecies coalescent model (NMSC) is a widely used probabilistic framework for modeling how gene trees form under hybridization and incomplete lineage sorting, making it a key tool for inferring species networks. I will discuss which features of species networks are identifiable and which are not under the NMSC using various representations of genomic data.
Denae Ventura: Counting Monochromatic Solutions of 3-variable Linear Equations
A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations E, there is a threshold value Rk(E) (the Rado number of E) such that for any k-coloring of the integers in the interval [1, n], with n ≥ Rk(E), there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of n? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.