# Outreach (Spring 2021)

When: Thursdays 1:30-2:30pm CDT or 12:30-1:30pm CDT (depending on speaker)

The Undergraduate Mathematics Research Seminar (UMRS) is a seminar hosted by the University of Minnesota math department for undergraduates pursuing research, interested in research, or who are just interested in learning more mathematics. For the Spring 2021 semester the seminar will be held online and we welcome those interested from outside universities. This semester, there are two seperate time blocks. The primary time block is 1:30-2:30pm CDT, but there is a secondary time block 12:30-1:30pm CDT available for speakers who are not available for the primary. If you’d like to be on the mailing list or want to give a talk, see the google form above. To see the 2019-2020 schedule see here and for the 2018-2019 schedule see here.

• Henry T. – 2/11 (1:30-2:30 CDT) – Root systems attached to moments

Recent work has shown that certain root systems (special sets of vectors) arising from a family of real-valued matrices play a critical role in deep number-theoretic objects called moments. This tells us that geometry is describing moments in a setting thought to have no attached geometry. Classification of the roots would lead to a deeper understanding of approximations for moments. To this end, progress has been made in classifying roots using their geometry via root strings. We will discuss the roots, their geometry, and methods of classifying them.

• Neelima B. – 2/18 (1:30-2:30 CDT) – The Eichler-Shimura relation

To relate different versions of the Modularity Theorem we look modulo $p$ and express both the Fourier coefficients of the new form associated to an elliptic curve $E$ and the number of solutions of a Weierstrass equation for $E$ in terms of the Frobenius map

• Antonino T. – 3/4 (12:30-1:30 CDT) – Classical integrability and fluid dynamics

Classical integrability in the sense of Liouville is a cornerstone of nonlinear PDE as well as more recent notions of integrable systems. After reviewing Poisson structures, basic Hamiltonian systems, and formal notions of charge and conservation, a version of the Euler equations of fluid dynamics will be shown to admit an Hamiltonian form. For free, we recover an integrability “dictionary” for an old classification problem in representation theory. If time permits, we will explore algebrao geometric aspects of Lax pairs and some recent results regarding an effective theory to describe certain turbulent fluids.

• Delanna D. – 3/11 (1:30-2:30 CDT) – Generating SpotIt Decks

SpotIt decks are a set of cards such that every pair of cards within the deck have exactly one item in common. By exploring the close relationship between mutually orthogonal Latin squares to games of SpotIt, we can algorithmically generate these decks.

• Trent N. – 3/18 (1:30-2:30 CDT) – Toeplitz operators on Hartogs triangles

Toeplitz operators are a class of linear functional operators which are often discussed in the theory of differential equations. For many Toeplitz operators, their properties don’t just depend on the specific operator, but the ambient space upon which functions the Toeplitz operators are defined upon. We discuss the $L^{p}$ regularity of a couple Toeplitz operators over generalized Hartogs triangles.

• Ethan P. – 3/25 (1:30-2:30 CDT) – Real powers of monomial ideals

This talk will give a generalization of the exponentiation of monomial ideals. The typical operation only considers 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.

• Griffin M. – 4/1 (1:30-2:30 CDT) – Finding ratios $p/q$ in a Set: a sequel to disjoint unions, Utah, and weird fractions

We show that for any set $A$ of naturals which isn’t terribly small, we can find distinct primes $p$, $q$ and elements $x, y \in A$ such that $\frac{x}{y} = \frac{p}{q}$. The method is general, we can replace “primes” with any set $B$ of coprime natural numbers such that $\sum\frac{1}{b}$ diverges. Come if you love the probabilistic method of showing an object exists!