The Northfield Undergraduate Mathematics Symposium (NUMS) is an annual event sponsored jointly by Carleton and St. Olaf. In 2018, eight students spoke at the symposium on research they did around the country over the summer.
Drawing Three Trees at Once: Edge-disjoint Tree Representations of Three Tree Degree Sequences
Ian Seong (Carleton)
Can you draw a tree if given a sequence of degrees of vertices? Sure, as long as the numbers are all positive integers which add to twice the number of edges, which is 2(v-1) (v being the number of vertices). Such a sequence is called “tree degree sequence”.
Now, given three tree degree sequences with the same number of vertices, can you draw a combination of three trees so that no two edges from different trees connect the same two vertices (we call the last condition “edge-disjoint”)?
Do you know if it is even possible for you to draw the combination of trees? In this talk, I will outline a sketch of a proof that as long as the total degree of each vertex is at least 4, it is possible for one to draw the edge-disjoint combination of three trees.
On Solvable Leibniz Algebras with an Abelian Nilradical
Kyla Pohl (St. Olaf)
This talk is devoted to the classification of solvable Leibniz algebras with an abelian nilradical. We consider a k-1 dimensional extension of k-dimensional abelian algebras and classify all 2k-1-dimensional solvable Leibniz algebras with an abelian nilradical of dimension k.
Spectral analysis of a neural field mode
Keenan Ronayne (Carleton)
During various activity states, the brain produces oscillations of set frequencies. Oscillations within certain regions of the brain can be manipulated through techniques such as deep brain and optogenetic stimulation. Modeling the oscillations caused by optogenetic stimulation has proven a difficult task in the field of theoretical neuroscience. A common approach is to use a Wilson-Cowan neural field model which utilizes coupled excitatory and inhibitory neuron populations to produce cortical-like oscillations.
To model this, a pair of nonlinear differential equations is used. Recently, a form of this model was used to explain in vivo gamma oscillations caused by optogenetic stimulation. This was done by creating a neural medium that supported oscillating waves and incorporating stimulation parameters. The parameters of the system required fine tuning to achieve the desired output. This resulted in a narrow model that only described one observed response under a single stimulation regime. Moreover, this model neglected to include spatial delays present in the cortex.
To address these issues, a new model with synaptic and axonal delays was created. In addition to the introduction of the new delay parameters to the system, an in depth spectral analysis was done on the various stimulation regimes seen in literature. These included periodic and single pulse stimulation of varying intensity and duration. Understanding the exact dynamics of the cortex under specific stimulation is essential to the design and implementation of experiments requiring optogenetics.
Linear Factorization of Hypercyclic Functions for Differential Operators
Jakob Hofstad (St. Olaf)
It is known that there exists a differentiable function f such that if one repeatedly differentiates f and creates a list out of the output functions, that any continuous function (within a finite interval) can be approximated by a function from this list to however small of an error that we desire. It is also known that similar results exist if the list is generated by other combinations of differentiation, which includes translation to the left or right by any distance. In our project, we construct such a function for each of the cases above as an infinite product of linear functions, which answers an open question posed by operator theory specialists.
Special Sets of Vertices in Paley Graphs
Clara Buck (Carleton)
Strongly regular graphs have three distinct eigenvalues which give bounds on the average degree of an induced subgraph. Tight sets are sets of vertices that induce a subgraph with the highest or lowest possible average degree relative to the parameters and eigenvalues of the graph. A particular family of examples of strongly regular graphs is formed by the Paley Graphs P(q). The vertex set of a Paley Graph is a finite field of size q $equiv$ 1 mod 4, and two vertices are adjacent when their difference is a nonzero perfect square in the considered field.
We utilize the connection between Paley graphs $P(q^2)$ and finite affine planes to study tight sets of $P(q^2)$ in a geometric context. In this model, each line of the affine plane corresponds to either a clique or an independent set in the graph. Selecting only those lines which correspond to cliques, we construct a partial affine plane in which two vertices are adjacent in the graph if and only if they are on the same line in the affine plane. We work towards classifying the tight sets which are not the disjoint union of smaller tight sets.
Lattice Based Cryptography and Fully Homomorphic Encryption
Ani Nadiga (Carleton)
Although current encryption schemes are secure against attacks from classical computers, they are all insecure against quantum attacks. As such, if quantum computers ever become a reality, we will need to rethink how we do cryptography. Lattice-based cryptography is conjectured to be quantum secure as well as classically secure. In addition, these schemes allow for entirely new cryptographic tools, such as fully homomorphic encryption, which allows for computation on encrypted data.
I will present a high-level overview of the foundations and applications of lattice-based cryptography.
Constructing Generalized Gelfand-Graev Representations
Julie Yuldasheva (St. Olaf)
Generalized Gelfand-Graev representations (GGGRs) have originally been introduced by Kawanaka in 1985. They are important for number theory because special vectors from these representations appear in integral realizations of automorphic L-functions. L-functions have been a central object of study in number theory for over 150 years.
In this project, we focus on the structure of GGGRs of GL(n), the group of all invertible n x n matrices, defined over a finite field. In particular, GGGRs are induced from the semi-direct product of a unipotent subgroup and the stabilizer subgroup of a character on the unipotent subgroup. The result of our project is a simple formula in terms of partition statistics for this elusive stabilizer subgroup.
Exploring Upper Bounds of Graph Proper Diameters
Nathaniel Sauerberg (Carleton)
In networks of communication towers, having a single tower receive and transmit on the same frequency may lead to interference. We can model this situation with an edge-colored graph, with vertices representing the towers and edges present where towers can communicate directly. Properly colored paths, paths in which no consecutive edges are of the same color, generate no interference. We use this notion to define proper distance and proper diameter, analogs of distance and diameter in uncolored graphs.
A natural upper bound for proper diameter is the length of the longest possible path in the graph, the number of vertices in the graph minus 1. Looking specifically at when this upper bound is attainable in 2-connected graphs on 2 colors leads to the construction of the Τau graph family. We give intuitive justification that 2- connected graphs attain the maximum possible proper diameter if and only if they are Tau graphs and some observe other interesting properties of Τau graphs.