Graduate Student Seminar Tuesdays 4:00-5:00, Chase 319

Abstract: A model for species abundance based on ecological survey data must properly account for the presence of spatial and temporal correlations in order to provide accurate abundance index estimates. Data on lobster abundance collected by trawl survey programs in the Bay of Fundy poses additional modelling complications, including the use of different types of sampling gear with unknown catchability coefficients, varying survey coverage by program and year, and a highly skewed data distribution with a high proportion of zero observations. The negative binomial geostatistical generalized additive mixed model fit to these lobster data includes Gaussian random fields for modelling spatial and spatiotemporal effects. The model is fit with the sdmTMB R package, which performs maximum likelihood estimation using the Laplace approximation to the marginal likelihood. The Gaussian random fields have a Matérn covariance function, and are fit as Gaussian Markov random fields using a triangular mesh laid across the study area. Model selection is informed by the use of residual analysis, Akaike’s Information Criterion, and spatial block cross-validation. The main results of interest include the estimates of the model parameters, as well as maps of the predicted abundance at all locations in the study area for every year, and an index whose value at any year is the total abundance in the study area for that year. Simulation studies are also performed to examine the accuracy of the model under different scenarios of reduced data quality or quantity.

Abstract: This talk will be about one of my favorite facts, that if a set has two compatible multiplication maps then these operations agree, and it is commutative, and it's one (or three) line proof. By moving from sets to categories the (proof of the) Eckmann-Hilton argument becomes the mere structure of a braiding in the category. If the category has three multiplications maps instead, then it does acquire the property of being commutative. This story has a nice pictorial interpretation in terms of the topology of embedded points and strings, and the real goal of the talk is to explore these ideas in terms of this more palatable point of view. -->

Abstract: When studying differential equations, a very natural question to ask is whether a given system of equations has a unique solution. A different question that we can ask is: Given a polynomial F, what are the partial differential equations that have F as a solution? It turns out that specific cases of questions like this can be understood using techniques from Commutative Algebra and have applications to several areas of Mathematics, including Combinatorics, Geometry, Topology and Representation Theory. In this talk, we will focus on some problems in (algebraic and differential) Geometry that can be studied from this perspective. If there is enough time, we will also talk about how graph theory can help us in this setting.

Abstract: The aim of this presentation is to generate a list of all positive integers ordered alphabetically. Using recursive tree technology, we will show that an ordered tree can be generated for any number naming system, but it cannot always be described as an ordered list.

Abstract: Metric-affine connection theory provides various mechanisms for measuring geometric deformations, with the curvature of the Levi-Civita connection being the most well-known, though not the only one. To explore this topic, we utilize the language of tensors and introduce them through an index-based approach without delving into their deeper geometric significance. Instead, we focus on defining these quantities based on their transformational properties. Using basic calculus, we develop the concepts of covariant derivatives, connections, and metrics, ultimately leading to the connection decomposition theorem and the fundamental tensorial quantities—non-metricity, curvature, and torsion—that describe geometric deformations.