Jekyll2019-12-04T21:32:52+00:00http://scoskey.org/feed.xmlSamuel CoskeyDepartment of Mathematics, Boise State University, 1910 University Dr, Boise, ID 83725-1555.
On conjugacy problems for graphs and trees2019-10-11T00:00:00+00:002019-10-11T00:00:00+00:00http://scoskey.org/presentation/on-conjugacy-problems-for-graphs-and-trees<p>Boise Set Theory and Logic Seminar, October 2019<!--more--></p>
<p>Abstract: Given a graph or tree G, the conjugacy problem for G is the conjugacy equivalence relation on the group Aut(G) of automorphisms of G. In this talk we will introduce the basic framework for studying the complexity of the conjugacy problem for G, and then use it to examine a series of examples of graphs and trees G. In particular we will demonstrate that a variety of complexities can occur, from trivial (smooth), up to maximally complex (Borel complete), and in between.</p>Boise Set Theory and Logic Seminar, October 2019Jumps of equivalence relations and scattered linear orders2019-09-09T00:00:00+00:002019-09-09T00:00:00+00:00http://scoskey.org/presentation/jumps-of-equivalence-relations-and-scattered-linear-orders<p>CUNY Set Theory Seminar, September 2019<!--more--></p>
<p>Abstract: This is joint work with John Clemens. We will begin this talk by discussing the problem of classifying the countable scattered linear orders. Here a linear order is called scattered if the rational order doesn’t embed into it. The class of such orders admits a ranking function valued in the ordinals; we will study the corresponding classification problem for each fixed rank. We will show that each increase in rank results in a “jump” in the complexity of the classification problem. In the second part of the talk we will define a family of jump operators on equivalence relations, each associated with a fixed countable group. The jump in the case of scattered linear orders is that associated with the group Z of integers. We will discuss the basic theory of these jump operators. Finally, we will discuss the question of when such a jump operator is proper, in the sense that the jump of E is strictly above E in the Borel reducibility order.</p>CUNY Set Theory Seminar, September 2019Computable reducibility of equivalence relations2019-05-15T00:00:00+00:002019-05-15T00:00:00+00:00http://scoskey.org/masters-thesis/computable-reducibility-of-equivalence-relations<p>A master’s thesis by Gianni Krakoff, Spring 2019<!--more--></p>
<p><em>Abstract</em>: Computable reducibility of equivalence relations is a tool to compare the complexity of equivalence relations on natural numbers. Its use is important to those doing Borel equivalence relation theory, computability theory, and computable structure theory. In this thesis, we compare many naturally occurring equivalence relations with respect to computable reducibility. We will then define a jump operator on equivalence relations and study proprieties of this operation and its iteration. We will then apply this new jump operation by studying its effect on the isomorphism relations of well-founded computable trees.</p>A master’s thesis by Gianni Krakoff, Spring 2019Fractals, what are they?2019-05-01T00:00:00+00:002019-05-01T00:00:00+00:00http://scoskey.org/senior-thesis/fractals-what-are-they<p>A senior thesis by Nicole Reese, Spring 2019<!--more--></p>
<p><em>Abstract</em>: This presentation explores the concept of a fractal, the Hausdorff dimension of a fractal, and fractals generated by iterated function systems.</p>A senior thesis by Nicole Reese, Spring 2019The hyperreals; do you prefer non-standard analysis over standard analysis?2019-05-01T00:00:00+00:002019-05-01T00:00:00+00:00http://scoskey.org/senior-thesis/hyperreal-numbers<p>A senior thesis by Chloe Munroe, Spring 2019<!--more--></p>
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<td><em>Abstract</em>: The hyperreal number system $\ast\mathbb R$ forms an ordered field that contains $\mathbb R$ as a subfield as well as infinitely large and small numbers. A number is defined to be infinitely large if $</td>
<td>x</td>
<td>>n$ for all $n = 1, 2, 3, \ldots$ and infinitely small if $</td>
<td>x</td>
<td><1/n$ for all $n = 1, 2, 3\ldots$ This number system is built out of the real number system analogous to Cantor’s construcion of $\mathbb R$ out of $\mathbb Q$. The new entities in $\ast\mathbb R$ and the relationship between the reals and hyperreals provides an appealing alternate approach to real (standard) analysis referred to as nonstandard analysis. This approach is based around that principle that if a property holds for all real numbers then it holds for all hypereal numbers, known as the transfer principle. By only using the fact that $\ast\mathbb R$ is an ordered field that has $\mathbb R$ as a subfield, includes unlimited numbers and satisfies the transfer principle the topics of analysis can be explored.</td>
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</table>A senior thesis by Chloe Munroe, Spring 2019The nametag problem2019-05-01T00:00:00+00:002019-05-01T00:00:00+00:00http://scoskey.org/senior-thesis/the-nametag-problem<p>A senior thesis by Christian Carley, Spring 2019<!--more--></p>
<p><em>Abstract</em>: This paper explores what has been termed, ``The Name Tag Problem’’ (NTP). The problem is framed thusly. A group of $n$ people sit around a table and to each person a name tag has been assigned. How can you assign the nametags so that just one person has the correct name tag assigned, and no matter how many times the table is rotated, still just one person has the correct name tag assigned?</p>A senior thesis by Christian Carley, Spring 2019Borel morphisms between cardinal characteristics, parts 1 and 22019-04-26T00:00:00+00:002019-04-26T00:00:00+00:00http://scoskey.org/presentation/borel-morphisms-between-cardinal-characteristics<p>Boise Set Theory Seminar, April 2019<!--more--></p>
<p>Abstract: A cardinal characteristic is the cardinality of some special subset of the continuum. Vojtas introduced a categorical framework in which to study cardinal characteristics and their relationships. In this talk we introduce the objects and morphisms of this framework. We then consider a variation of this framework where the objects are morphisms are Borel definable.</p>Boise Set Theory Seminar, April 2019On splitting and splittable families2019-03-07T00:00:00+00:002019-03-07T00:00:00+00:00http://scoskey.org/splitsplit<p>With Bryce Frederickson, Samuel Mathers, and Hao-Tong Yan.<!--more--></p>
<p><em>Abstract</em>: A set $A$ is said to <em>split</em> a finite set $B$ if exactly half the elements of $B$ (up to rounding) are contained in $A$. We study the dual notions: (1) a <em>splitting family</em>, a collection of sets such that any subset of ${1,\ldots,k}$ is split by a set in the family, and (2) a <em>splittable family</em>, a collection of sets such that there is a single set $A$ that splits each set in the family.</p>
<p>We study the minimum size of a splitting family on ${1,\ldots,k}$, as well as the structure of splitting families of minimum size. We use a mixture of computational and theoretical techniques. We additionally study the related notions of $\mathord{\leq}4$-splitting families and $4$-splitting families, and we provide lower bounds on the minimum size of such families.</p>
<p>Next we investigate splittable families that are just on the edge of unsplittability in several senses. First, we study splittable families that have the fewest number of splitters. We give a complete characterization in the case of two sets, and computational results in the case of three sets. Second, we define the <em>splitting game</em>, and study splittable families for which a splitter cannot be found under adversarial conditions.</p>With Bryce Frederickson, Samuel Mathers, and Hao-Tong Yan.Introduction to continuous logic and model theory for metric structures2019-03-01T00:00:00+00:002019-03-01T00:00:00+00:00http://scoskey.org/presentation/introduction-to-continuous-logic-and-model-theory-for-metric-structures<p>Boise Set Theory Seminar, March 2019<!--more--></p>Boise Set Theory Seminar, March 2019Foundations of analysis2019-01-01T00:00:00+00:002019-01-01T00:00:00+00:00http://scoskey.org/course/1819s-314<p>Math 314, Spring 2019<!--more--></p>
<p><em>Catalog description</em>: The real number system, completeness and compactness, sequences, continuity, foundations of the calculus.</p>Math 314, Spring 2019