The Undergraduate Math Research Seminar (UMRS) is a seminar aimed to create a welcoming undergraduate research community at UMN. We will have opportunities for students to give low-stakes talks on the research and reading they’ve done as well as hold sessions devoted to learning what research is like, the different areas of math research and how to create research presentations. If you’d like to be updated on the seminar, see the mailing list above. To see the 2021-2020 schedule see here, and the 2019-2020 schedule here.
Fall 2021 Talk Schedule:
Abstract: A numerical semigroup is a cofinite subset of the non-negative integers that is closed under addition and contains zero. The Kunz Polyhedra, P_m, are a family of rational polyhedra whose integer points are in bijection with numerical semigroups with fixed smallest nonzero element m. Each face of P_m corresponds to a poset that describes the structure of the numerical semigroups found on that face. Interestingly, some faces of P_m do not have any integer points on them, but are still associated with a poset. In this talk, we will explore why these faces do not have integer points on them by looking at their posets.
Abstract: In the divisor theory of graphs, a finite, connected graph is viewed as a discrete analog of a Riemann surface. A divisor $D$ on a graph is an assignment of integers to each vertex of the graph. An important statistic in this setting is the complete linear system of $D$, which is the collection of effective (non-negative) divisors linearly equivalent to $D$ via the discrete Laplacian operator. Recently, S. Brauner, F. Glebe, and D. Perkinson characterized all complete linear systems on a finite graph $G$ using a system of generating sets, each of which correspond to a cone. We extend their results by using generating sets to compute the subset of effective divisors fixed by a symmetry $\gamma$ of $G$, as well as the subset of effective divisors that are stable under action of $\gamma$. In certain cases, we are able to extend the action of $\gamma$ to an action on the cones arising from these generating sets. When $G$ is the cycle graph, we present a poset isomorphism between a subset of effective divisors under the firing operator and a subposet of Young’s lattice defined by R. Suter, H. Thomas and N. Williams. Using the combinatorics of these divisors, we give an alternate proof of a result by H. Thomas and N. Williams that the scaled simplex exhibits the cyclic sieving phenomenon (CSP), which was introduced by V. Reiner, D. Stanton and D. White.
Abstract: This talk will give a generalization of the exponentiation of monomial ideals. The typical operation only considers raising ideals to a natural power, we extend this to powers of positive real numbers. We do this by generalizing a connection between ideals and convex polytopes. Polytopes are an incredibly simple to understand yet powerful tool in convex geometry. This talk will mainly focus on the polytopal side of real powers and will assume very little background. In particular, algebraic ideas will be expressed using polytopes, so if you’ve seen a square or pyramid before you meet the prerequisites for this talk.
Abstract: Optimal Mass Transportation was first formulated by Gaspard Monge in 1781 and has gained a lot of attention and research effort for the last three decades. Optimal Transport plays an important role in partial differential equations, fluid mechanics, probability theory and functional analysis. Kantorovich later proposed a relaxed version of Monge’s problem and identified a dual formulation. The goal of our project is to produce a duality proof based on arguments accessible to undergraduate level students. Our starting point is a 2018 paper by Haim Brezis who gave an elementary proof to the coincidence in the discrete settings of the three quantities: the original minimal cost introduced by Monge, the relaxed version by Kantorovich and the duality formulation also due to Kantorovich. This research deduced that the whole transport duality theory can be derived from the elementary arguments of Brezis. The project rests on approximating any probability measure by a sequence of averages of Dirac masses, so that the discrete result implies the general coincidence. In addition, the research studied the quotient of the Wasserstein space by the group of rigid transformations, which is used in gradient flow of the entropy.
Abstract: This talk will introduce simplicial complexes and their associated Stanly-Reisner rings. Simplicial complexes are a powerful combinatorial object used to study topological spaces. To better understand the properties of a simplicial complex, we can associate it to an algebraic object called its Stanley-Reisner ring. To motivate our study, we focus our theory on the h-vector of a simplicial complex. The h-vector of a simplicial complex is a wonderful statistic which tells us much about the structure of its associated complex. However, when defined purely combinatorially, it appears to be a seemingly random collection of numbers. By using an algebraic viewpoint, we can remove this confusion and find an enlightening way to encode the h-vector’s data. This talk is intended to be completely introductory, no algebra or topology background required.
Abstract: Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. Permutation triangles are the alternating sign triangles whose entries are either 0 or 1, by analogy with permutation matrices. We will prove a simple characterization of permutation triangles, originally conjectured by Glick. We will also prove the connectivity of alternating sign triangles, which is analogous to the connectivity of alternating sign matrices.