We will discuss whether it is possible to reconstruct a phylogenetic tree from sequence data, given a specific model of evolution. To answer this question for a coalescent-based model, we study ideals associated to the model. By observing some facts about the geometry of the model, we can demonstrate that these ideals contain polynomials which may…
Given a determinantal representation by means of a cyclic weighted shift matrix, one can show the resulting polynomial is hyperbolic and invariant under the action of the dihedral group. Chien and Nakazato (2015) asked the converse question. By properly modifying a determinantal representation construction of Dixon (1902), we show for every hyperbolic polynomial with dihedral…
Many important Lie (super)algebras can be constructed using bosonic and fermionic oscillators. We introduce inhomogeneous supersymmetric bilinear forms on a complex superspace and show that they lead to oscillator-like superalgebras. We classify such forms for superspaces up to dimension 7 and mention a few examples of subalgebras obtained from the corresponding superalgebras. This talk is…
I will present an algorithm that, for a given vector of n relatively prime polynomials in one variable over an arbitrary field, outputs an invertible matrix with polynomial entries such that it forms a degree-optimal moving frame for the rational curve defined by the input vector. From an algebraic point of view, the first column…
Let G be a Lie group acting smoothly on the plane. Then two smooth curves C and C' are G-equivalent if there exists some g in G such that gCC'. Can we answer the question, when are two curves G-equivalent? What can we additionally say if we restrict our attention to algebraic curves? In this…
Lie Algebras, manifolds, varieties. We have all heard terms like these, but may not know quite what they mean. This week we will have introductory "What Is?" talks. These talks are intended to build your vocabulary/intuition for future talks. This week we will hear about: Lie Algebras, Leibniz Algebras, Representations, Algebraic Varieties.
Leibniz algebras are a generalization of Lie algebras. In this talk I will discuss some results on characteristic ideals, which are analogs of known results from Lie algebra. I will use these results to prove that the radical of a Leibniz algebra is a characteristic ideal. We will also explore the derivation algebra of cyclic…
Algebraic geometry and combinatorics play an important role in the analysis of phylogenetic models. I will give a brief overview of toric geometry. Then I will introduce a particular phylogenetic model, called the Cavander-Farris-Neyman model with a molecular clock, and I will discuss how we can study this model from the point of view of…
The numerical range has been studied extensively in linear algebra and analysis. We will define and discuss properties of the numerical range, then show every numerical range invariant under rotation is associated with a matrix with nice structure.
The size of a maximum agreement subtree of two phylogenetic trees is a statistic that is often used to test the null hypothesis that no cospeciation occurred between two families of species of interest. The size of the maximum agreement subtree can be computed in polynomial time but the distribution of this statistic is not…
We will meet briefly to sign up for talks.
The Zariski closure of a discrete log-linear statistical model is a toric variety. In the last twenty years, this fact has lead not only to the development of useful algorithms for working with data, but developments in algebraic geometry and combinatorics as well. I will discuss a particular subset of the discrete log-linear models known…
We prove results about the polytope associated to the toric ideal of invariants of the Cavender-Farris-Neyman model with a molecular clock on a rooted phylogenetic tree. For instance, the number of vertices of this polytope is a Fibonacci number and the facets of the polytope can be described using the combinatorial structure of the underlying…
We will talk about bijections between various families of combinatorial objects counted by OEIS A071356. These include certain underdiagonal lattice paths, pattern avoiding permutations, pattern avoiding inversion sequences, and posets. Connections will be made to combinatorial Hopf algebras using some lattice theory. OEIS A071356 is closely related to the Catalan numbers.
Speaker 1: Caprice Stanley Title: Markov Chain Mixing Time Abstract: Informally a Markov chain is a memoryless stochastic process and its mixing time is the time required for the chain to be near its stationary distribution. Depending on the context, analysis of mixing time can be of great importance. For example, random walk-based algorithms for…
Speaker 1: Faye Pasley Title: Determinantal Representations, the Numerical Range, and Invariance We study hyperbolic polynomials with symmetry and express them as the determinant of a Hermitian matrix with special structure. By properly modifying a construction of Dixon (1902), we show for every hyperbolic polynomial of degree $n$ invariant under the cyclic group of order $n$ there…