The Graduate Colloquium

Next semester's colloquium!

The graduate colloquium will usually run Tuesdays or Thursdays at 4:30pm.

If you would like to speak in the seminar, please email Alex Duncan.

Abstracts

Introductory Panel - Open discussion of graduate program led by upper-year graduate students and new faculty.

Funding and Travel Panel - Learn about opportunities for funding and travel and how to apply for them.

Invitation Panel - Come learn about graduate courses being offered next year:
G. McNulty: 768 Equational Logic
R. Howard: 732 Algebraic Topology
P. McFaddin: 747 Algebraic Geometry
L. Szekely: 778 Linear Algebra Method in Combinatorics

Matthew Ballard - We'll try to answer the titular question. In the process, we will encounter algebraic varieties, cohomology, and some string theory.

Joshua Cooper - Spectral graph theory has been an invaluable tool in combinatorics and computer science since the mid-1980's. In general, the idea is to connect the eigenvalues/vectors of matrices associated to graphs with combinatorial properties of those graphs: expansion and mixing, diameter, degree sequence, etc. Over the past decade, this program has been carried over to hypergraphs, a simple and natural generalization of graphs. Since hypergraphs are so much more complicated than graphs, the topic is still in its infancy, with many unanswered basic questions. We focus here on the particularly charming topic of adjacency hypermatrices (aka tensors or multidimensional arrays) and their eigenpairs, where it is possible to generalize many classical results from graph theory about the coefficients of characteristic polynomials, subgraphs, spectral symmetry, Cartesian products, and more.

Jesse Kass - How to construct and classify finite-dimensional complex algebras? This is a question that mathematicians have studied since Hamilton’s 1843 construction of the quaternions. Despite this long study, it was only in 1972 that Iarrobino used ideas in algebraic geometry to prove that known constructions fail to produce “most” algebras, in a precise sense. I will explain Iarrobino’s work and discuss some related open questions.

Ralph Howard - Some basic geometric inequalities will be proven, including Bonnesen's strengthened  version of the planar isoperimetric inequality and the Fray-Milnor theorem on the total curvature of knots.  Some other inequalities will be surveyed, including the relation between mixed volumes and algebraic geometry, and some open questions mentioned.  Most of the talk will not require any knowledge beyond undergraduate mathematics.

Tyler Lewis - Singularity theory is a broad field that has fascinating relations to algebraic geometry, commutative algebra, complex analysis, representation theory, Lie groups, topology, and many other fields. In this talk, we will define what a singularity is, look at some interesting examples, and define the main invariants and ideas that go into studying singularities.

Marcus Neal and Patrick Rybarczyk - Marcus Neal and Patrick Rybarczyk will give an overview of the Columbia Math Circle. This will include the philosophy of math circles, types of topics covered at math circles, and how the community can be involved with them. The Columbia Math Circle can provide an enriching experience for graduate students. Mr. Neal and Mr. Rybarczyk will discuss specific ways in which graduate students can participate in and even lead a Columbia Math Circle session.

Robert Vandermolen - Bilinear forms are of interest in many different fields of mathematics such as Algebra, Analysis, and Geometry. In Analysis, bilinear forms show up in the study of Hilbert Spaces. In Algebra, the Killing form, a bilinear form, appears in the study of Lie Groups. We will discuss topologies defined on dual spaces using their bilinear forms. Further, we will discuss the relationship between continuous functions on these new topological spaces and whether or not they are adjointable.

Yi Sun - We present a 3D lattice model to study self-assembly of multicellular aggregates by using kinetic Monte Carlo (KMC) simulations. This model is developed to describe and predict the time evolution of postprinting structure formation during tissue or organ maturation in a novel biofabrication technology--bioprinting. Here we simulate the self-assembly and the cell sorting processes within the aggregates of different geometries, which can involve a large number of cells of multiple types.

Xueping Zhao - We present a computational model to study the pattern formations in tissues with stem cells and differentiated cells. Our model is derived based on the Generalized Onsager Principle, combining one energy dissipative system and several active factors, such as spontaneous polarity states, birth and death of cells, self-propelled motion of differentiated cells and ATP hydrolysis. Linear stability analysis is conducted to reveal the long-wave instability inherent in the neighborhood of the constant steady states. To solve this complex model, we develop an efficient energy stable numerical scheme and implement it on GPU clusters for high-performance computing. Our model introduces a novel way to investigate the interplay of stem cell division, differentiated cell migration and other active factors from environments.

For more information about this seminar and previous semesters' line-ups, see Matthew Ballard's seminar page.