The P-partition generating function of a (naturally labeled) poset P is a quasisymmetric function enumerating order-preserving maps from P to Z^+. Using the Hopf algebra of posets, we give necessary conditions for two posets to have the same generating function. In particular, we show that they must have the same number of antichains of each…

In this talk, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, I will provide a framework from which we can provide conditions, which are straightforward to check, for the solvability of a large number of nonlinear scalar boundary value problems. I will begin by giving the general strategy which involves a reformulation of our…

Affine Lie algebras are infinite dimensional analogs of finite dimensional simple Lie algebras. It is known there are finitely many maximal dominant weights for any integrable highest weight representation of an affine Lie algebra. However, determining these maximal dominant weights is a nontrivial task. So far only the descriptions of these weights are known for…

This discussion includes an eclectic mix of convex analysis, subdifferential calculus, and nonlinear analysis. We will introduce modern theoretical tools and algorithms used to solve non-differentiable optimization problems across the sciences. This discussion will focus more on the theory, proofs, and analysis (not applications).

Spectral sequences are an algebraic tool for computing (co)homology of differential graded algebras (DGAs) developed by by Leray in the 1940's and have since found applications in various fields which utilize DGAs to compute useful invariants (e.g. Algebraic Topology, Knot Theory, ect.). This talk will provide a broad overview and attempt to answer some basic…

In this talk I will introduce the ideas of singularities of a polynomial and resolution of those singularities, and then develop the blowup process algorithm. For a polynomial f, a point is singular if f and all its first partials vanish at that point. A resolution of the singularities of f gives a non-singular polynomial…

The study of hyperplane arrangements begins at fundamental questions such as: how many pieces can one cut a cake into with n slices? Quickly, hyperplane arrangements generate many intricately interesting mathematical structures and connections to many seemingly unrelated topics appear. I will attempt to show how matroids offer a convenient framework to study hyperplane arrangements…

In this talk I'll introduce a new random tree model that associates a family of probability distributions on binary trees to each binary tree shape. We'll then use this new model, convex geometry, and the combinatorics of multisets and trees to obtain a finite form of a deFinetti-type theorem for rooted binary trees. This talk…

Rooted binary trees are used in evolutionary biology to represent the evolution of a set of species where the leaves denote the existing species and the internal nodes denote the unknown ancestors. Maximum agreement subtree is used as a measure of discrepancy between two trees. In this talk, I will define the notion of "maximum…

Let  be a free associative algebra over  generated by  for  in some indexing set  and let  be a poset.  For  and   we define an action of   on  (the complex vector space with basis )  in a way such that  either annihilates   or sends it to  where  covers  and we extend multiplicatively and linearly.  Let  be the two-sided ideal which annihilates all elements of .  We characterize  when  is Young's Lattice and we discuss the…

The Italian Renaissance painters began to incorporate perspective into their drawings in the 1400’s, but our eyes naturally understand depth from the 2-dimensional image on the back of our eyeball. It’s this projection on the retina that allows mathematicians to represent field of view with the projective plane, and in this talk we’ll investigate what makes it so difficult to…

Hierarchical models are statistical log-linear models that record the dependency relations among random variables in statistics. Diaconis and Sturmfels in '98 propose using ideals to study these models. I will start by introducing hierarchical models and their ideals. In this talk we'll show that studying an entire family of ideals simultaneously can lead to greater insight than studying them individually. No…

The associahedron is a well-studied polytope. For n dimensions, its vertices are counted by the n-th Catalan number, a sequence starting 1,1,2,5,14,42,... and which counts many, many, many combinatorial objects, such as Dyck paths, planar binary trees, noncrossing set partitions, and polygonal triangulation. There is a well-known generalization of the associahedron, called the graph associahedron,…

An affine variety  is said to be well-poised if  is prime for every . Arrangement varieties are a special class of -varieties built from a hyperplane arrangement decorated by polyhedra. We will show that arrangement varieties always have a well-poised embedding and explore their toric degenerations coming from their tropicalizations. As a class of examples, we realize the Cox…

Factor systems are a tool for working on the extension problem for algebraic structures such as groups, Lie algebras, and associative algebras. We construct the Leibniz algebra analogue to a series of group-theoretic results from W. R. Scott’s Group Theory. Fixing a pair of Leibniz algebras A and B, we develop a correspondence between factor systems…

Let X and Y be random variables. Quasi-independence models are log-linear models that describe a situation in which some states of X and Y cannot occur together, but X and Y are otherwise independent. We characterize which quasi-independence models have rational maximum likelihood estimator, or MLE, based on combinatorial features of the bipartite graph associated…

A linear compartmental model is a linear ODE model which can be visualized by a directed graph.  Identifiability is the study of determining whether a model's parameter values can be inferred from the defining "input-output equation" under perfect conditions.  In this talk, I present a novel combinatorial formula for the computation of the coefficients of these…

Are you a math undergrad or grad student and interested in doing an internship related to math?  This event is for you!  The Graduate Training Module for Friday, October 8th is titled, "What is an Internship?" and consists of a panel of current graduate students who have done several different types of internships,  and are ready and…

In this talk we will introduce a geometric inverse problem arising from geophysical imaging. Take (M,g) to be a smooth, compact, and strictly convex Riemannian manifold with smooth boundary. Let Γ be a nonempty open subset of the boundary. For each p in M we obtain the boundary distance function d(p,z), where z is in…