Jekyll2021-06-05T03:44:46+00:00https://scoskey.org/feed.xmlSamuel CoskeyDepartment of Mathematics, Boise State University, 1910 University Dr, Boise, ID 83725-1555.
Zariski geometries and quantum mechanics2021-05-15T00:00:00+00:002021-05-15T00:00:00+00:00https://scoskey.org/masters-thesis/zariski-geometries-and-quantum-mechanics<p>A master’s thesis by Milan Zanussi, Spring 2021<!--more--></p>
<p><em>Abstract</em>: Model theory is the study of mathematical structures in terms of the logical relationships they define between their constituent objects. The logical relationships defined by these structures can be used to define topologies on the underlying sets. These topological structures will serve as a generalization of the notion of the Zariski topology from classical algebraic geometry. We will adapt properties and theorems from classical algebraic geometry to our topological structure setting. We will isolate a specific class of structures, called Zariski geometries, and demonstrate the main classification theorem of such structures. We will construct some Zariski structures where the classification fails by adding some noncommuting structure to a classical one. Finally we survey an application of these nonclassical Zariski structures to computation of formulas in quantum mechanics using a method of structural approx- imation developed by Boris Zilber.</p>A master’s thesis by Milan Zanussi, Spring 2021Four color theorem2021-05-01T00:00:00+00:002021-05-01T00:00:00+00:00https://scoskey.org/senior-thesis/four-color-theorem<p>A senior thesis by Samantha Decker, Spring 2021<!--more--></p>
<p><em>Abstract</em>: Francis Guthrie was born in England in 1831. In 1852, Guthrie conjectured and tried to prove what we now know of today as the Four Color Theorem. He sent his work to his earlier mentor, Augustus de Morgan, who gave Guthrie the credit whenever this conjecture came up. It remained one of the most famous unsolved problems in topology and graph theory, until it was finally proven in 1976 by Kenneth Appel and Wolfgang Hakenwith the help of a computer.</p>A senior thesis by Samantha Decker, Spring 2021Pythageorean theorem2021-05-01T00:00:00+00:002021-05-01T00:00:00+00:00https://scoskey.org/senior-thesis/pythagorean-theorem<p>A senior thesis by Kennedy Burgess, Spring 2021<!--more--></p>
<p><em>Abstract</em>: This document is intended to walk the readers through three proofs of the Pythagorean Theorem. The intended audience is younger levels of education, mainly middle school and high school. The Pythagoreans are remembered for two monumental contributions to mathematics. The first was establishing the importance of, and the necessity for, proofs in mathematics: that mathematical statements, especially geometric statements, must be verified by way of rigorous proof. Prior to Pythagoras, the ideas of geometry were generally rules of thumb that were derived empirically, merely from observation and occasionally measurement. Pythagoras also introduced the idea that a great body of mathematics could be derived from a small number of postulates. Clearly Euclid was influenced by the Pythagoras. The second great contribution was a discovery of and proof of the math that not all numbers are commensurate. More precisely, the Greeks prior to Pythagoras believed would be profound and deeply held passion that everything was built on the whole numbers. Fractions arise in a concrete manner: as ratios of the sides of triangles with integer length. Pythagoras proved the result now called the Pythagorean Theorem.</p>A senior thesis by Kennedy Burgess, Spring 2021Category, measure, and forcing: Set theory lecture notes2021-01-23T00:00:00+00:002021-01-23T00:00:00+00:00https://scoskey.org/forcing<p>With Erik Holmes. <em>Open Math Notes</em>, American mathematical society.<!--more--></p>
<p><em>Abstract</em>: Notes based on a graduate course in set theory. The first part covers measure, category, the continuum hypothesis, and cardinal characteristics of the continuum. The second part introduces the method forcing, and concludes by showing how forcing can prove independence results about the continuum hypothesis as well as the values of the cardinal characteristics.</p>With Erik Holmes. Open Math Notes, American mathematical society.Math 314, Spring 20212021-01-01T00:00:00+00:002021-01-01T00:00:00+00:00https://scoskey.org/course/2021s-314<p>Foundations of analysis<!--more--></p>
<p><em>Catalog description</em>: The real number system, completeness and compactness, sequences, continuity, foundations of the calculus.</p>Foundations of analysisMath 387, Spring 20212021-01-01T00:00:00+00:002021-01-01T00:00:00+00:00https://scoskey.org/course/2021s-387<p>Introduction to combinatorics<!--more--></p>
<p><em>Catalog description</em>: Covers basic enumerative techniques and fundamentals of graph theory. Additional content may include further topics in enumerative techniques or graph theory, extremal combinatorics, Ramsey theory, the probabilistic method, or combinatorial algorithms.</p>Introduction to combinatoricsInternal sorting methods2020-12-17T00:00:00+00:002020-12-17T00:00:00+00:00https://scoskey.org/senior-thesis/internal-sorting-methods<p>A senior thesis by Rebekah Bitikofer, Fall 2020<!--more--></p>
<p><em>Abstract</em>: Internal sorting methods are possible when all of the records to be accessed fit in a computer’s high speed internal memory. There are quite a few (Knuth’s third volume of The Art of Computer Pro- gramming covers 14 in total) but I will go over the four I found to be most versatile and useful.</p>A senior thesis by Rebekah Bitikofer, Fall 2020The history and application of Benford’s law2020-12-17T00:00:00+00:002020-12-17T00:00:00+00:00https://scoskey.org/senior-thesis/the-history-and-application-of-benford-law<p>A senior thesis by Hunter Clark, Fall 2020<!--more--></p>
<p><em>Abstract</em>: Benford’s Law is also known as thefirst digit law. It is a law that states the number 1 in lists or a set of data will show up approximately 30% more often than other numbers.</p>A senior thesis by Hunter Clark, Fall 2020Bernoulli jumps of equivalence relations2020-11-15T00:00:00+00:002020-11-15T00:00:00+00:00https://scoskey.org/presentation/bernoulli-jumps-of-equivalence-relations<p>University of Florida Logic Seminar, Gainesville, November 2020<!--more--></p>
<p><em>Abstract</em>: There are several well-studied “jump operators” on the class of Borel equivalence relations under Borel reducibility, such as the Friedman–Stanley jump and the Louveau jumps. I will define and discuss the new (ish) Bernoulli jumps; for each countable group Gamma there is a Gamma-jump operator. I will discuss which groups give rise to proper jumps (E is strictly below the Gamma-jump of E). Finally I will present applications to the classification of countable scattered orderings, and to finding new benchmarks in the Borel complexity hierarchy. This is joint work with John Clemens.</p>University of Florida Logic Seminar, Gainesville, November 2020Classification of countable models of ZFC2020-09-09T00:00:00+00:002020-09-09T00:00:00+00:00https://scoskey.org/presentation/classificaation-of-countable-models-of-zfc<p>CUNY Models of PA seminar, September 2020<!--more--></p>
<p><em>Abstract</em>: In 2009 Roman Kossak and I showed that the classification of countable models of PA is Borel complete, which means it is as complex as possible. The proof is a straightforward application of Gaifman’s canonical I-models. In 2017 Sam Dworetzky, John Clemens, and I showed that the argument may also be used to show the classification of countable models of ZFC is Borel complete too. In this talk I’ll outline the original argument for models of PA, the adaptation for models of ZFC, and briefly consider several subclasses of countable models of ZFC.</p>CUNY Models of PA seminar, September 2020