Presentations: 1:30-2:30 PM

Breakout Room 1:

Moderator:

1:30-1:45 Immersion Number of Generalized Mycielski Graphs

Megan Heenehan , Eastern Connecticut State University

A graph consists of a set of vertices and a set of edges which are unordered pairs of vertices. Many graph theorists have asked: if a graph has chromatic number $n$ does it contain a clique on $n$ vertices in some way? In 2003, Abu-Khzam and Langston conjectured that the containment is immersion. A graph $G$ has an immersion of a clique on $t$ vertices (a $K_t$-immersion) if we can find $t$ vertices every pair of which are connected by paths that are edge-disjoint. The chromatic number of a graph is the minimum number of colors needed to assign different colors to vertices that form an edge. In 1955, Mycielski provided a construction that when applied to a graph results in a graph with a larger chromatic number. This construction was generalized by Stiebitz and independently by Van Ngoc. In this talk, we explore the immersion number (the largest $t$ such that the graph has a $K_t$-immersion) of generalized Mycielski graphs. We show that the immersion number increases by the same amount as the chromatic number when the construction is applied. This provides an infinite class of graphs that satisfy the Abu-Khzam and Langston conjecture. No prior knowledge of graph theory will be assumed.

1:50 - 2:05 Erdos-Gallai differences and complementary degree sequences of graphs

Michael Barrus, University of Rhode Island

A number of graph families have characterizations in terms of their members' degree sequences. For many well-known examples of these families, a graph $G$ is a member if and only if its complement is a member. Anecdotally, it can be difficult to directly verify this complementarity property of a family just from the degree sequence characterization, though it often turns into a simple observation once other properties of the family are established (such as a forbidden subgraph characterization).

In this talk we prove a complementarity result for a certain type of degree sequence characterization, directly showing that certain Erdos--Gallai differences are always the same for the degree sequences of both a graph and its complement.

2:10-2:25 On Classical and Intuitionistic Logic

Pasquale Doucimo, Worcester State University

We introduce Classical Logic, leading to the Soundness and Completeness Theorems. Then, Intuitionistic Logic is discussed as an alternative logic with special consideration given to the role of proof by contradiction.

Breakout Room 2:

Moderator:

1:30-1:45 Kullback-Leibler Divergence Applied to Data Assimilation

Youssef Qranfal, Wentworth Institute of Technology

Sam Pimentel, Trinity Western University

Data assimilation is typically framed as a problem of minimizing an objective function of weighted $L_2$-norms. This objective function has two terms; the first includes the model-observation differences and the second is a regularizer involving an a priori estimate of the state. In this work we explore the use of an alternative "metric" for both residual minimizer and regularizer, namely the Kullback-Leibler divergence (or cross-entropy "distance"). Within this framework we formulate a Kullback-Leibler divergence based data assimilation approach. We present two iterative algorithms for minimizing functionals involving additive weighted Kullback-Leibler divergence terms. Simple numerical examples are used to demonstrate the application of the methods and comparisons are made to standard data assimilation approaches. We find that the Kullback-Leibler data assimilation method is computationally efficient and can naturally embed a constraint. These algorithms are shown to hold potential for data assimilation applications in geophysical fluid problems where we are interested in time-varying variables of large-scale systems and where physical constraints and computational resources present challenges.

1:50-2:05 Interdisciplinary Mathematical Concepts in the Context of AI for Social Good

Thomas Y. Chen, Academy for Mathematics, Science, and Engineering

The world faces a variety of humanitarian challenges today, from increasingly severe and frequent natural disasters due to climate change, to refugee crises and human trafficking. The application of mathematics in machine learning, deep learning, and computer vision solutions to tackle these issues has exploded in recent years. For example, deep learning-based remote sensing algorithms are trained on multitemporal satellite imagery to detect change, assess damage, and inform the timely allocation of resources in the case of natural disasters. In general, techniques range from simple linear regression, to random forest ensemble models, to convolutional neural networks, to generative adversarial networks. In essence, these various machine learning algorithms, to varying extents, require the use and understanding of linear algebra, analytic geometry, matrix decompositions, vector calculus, optimization, probability and statistics. Furthermore, mathematical methods are integral to creating more explainable/interpretable models, allowing humans to understand the inner decision-making processes of the algorithms. In this talk, we overview the interplay between the mathematical foundations of machine learning and its potential for humanitarian good in a wide range of capacities.

2:10-2:25 Proving theorems in spherical geometry using the quaternions

Marshall Whittlesey, California State University San Marcos

It is well known that the complex numbers can be used to do transformation geometry in the plane. In particular, rotation by angle $\theta$ about the origin is accomplished via multiplication by the complex number $e^{i\theta}=\cos(\theta)+i\sin(\theta)$. It is less well known that the quaternion algebra (consisting of expressions of the form $a+bi+cj+dk$ with $i^2=j^2=k^2=-1$) can be used to do similar transformations in three dimensional space. In this talk we show how to use quaternions to prove a significant classical theorem in spherical geometry. These methods are featured in the speaker's new book with CRC Press {\it Spherical Geometry and its Applications}, which the author hopes will be attractive for use in topics courses in geometry.