Mathematics Theses and Dissertations
https://scholarsbank.uoregon.edu/xmlui/handle/1794/2044
Fri, 15 Jan 2021 17:21:05 GMT2021-01-15T17:21:05ZAn Odd Analog of Plamenevskaya's Invariant of Transverse Knots
https://scholarsbank.uoregon.edu/xmlui/handle/1794/25886
An Odd Analog of Plamenevskaya's Invariant of Transverse Knots
Montes de Oca, Gabriel
Plamenevskaya defined an invariant of transverse links as a distinguished class in the even Khovanov homology of a link. We define an analog of Plamenevskaya’s invariant in the odd Khovanov homology of Ozsváth, Rasmussen, and Szabó.
We show that the analog is also an invariant of transverse links and has similar properties to Plamenevskaya’s invariant. We also show that the analog invariant can be identified with an equivalent invariant in the reduced odd Khovanov homology. We demonstrate computations of the invariant on various transverse knot pairs with the same topological knot type and self-linking number.
Tue, 08 Dec 2020 00:00:00 GMThttps://scholarsbank.uoregon.edu/xmlui/handle/1794/258862020-12-08T00:00:00ZA Special Family of Binary Forms, Their Invariant Theory, and Related Computations
https://scholarsbank.uoregon.edu/xmlui/handle/1794/25875
A Special Family of Binary Forms, Their Invariant Theory, and Related Computations
Dethier, Christophe
In this manuscript we study the family of diagonalizable forms, a special family of integral binary forms. We begin with a summary of definitions and known results relevant to binary forms, diagonalizable forms, Thue equations, and reduction theory.
The Thue--Siegel method is applied to derive an upper bound on the number of solutions to Thue's equation $F(x,y) = 1$, where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in the argument to handle forms whose discriminant is small in absolute value. These results are applied to bound the number of integral points on a certain family of elliptic curves.
A proof is given for an alternative classification of diagonalizable forms using the Hessian determinant. Algebraic restrictions are given on the coefficients of a diagonalizable form and divisibility conditions are given on its discriminant. A reduction theory for the family of diagonalizable forms is given. This theory is used to computationally verify that $F(x,y) = 1$, where $F$ is a quintic diagonalizable form with small discriminant, has few solutions.
Tue, 08 Dec 2020 00:00:00 GMThttps://scholarsbank.uoregon.edu/xmlui/handle/1794/258752020-12-08T00:00:00ZAT-algebras from zero-dimensional dynamical systems
https://scholarsbank.uoregon.edu/xmlui/handle/1794/25685
AT-algebras from zero-dimensional dynamical systems
Herstedt, Paul
We outline a particular type of zero-dimensional system (which we call "fiberwise essentially minimal"), which, together with the condition of all points being aperiodic, guarantee that the associated crossed product C*-algebra is an AT-algebra. Since AT-algebras of real rank zero are classifiable by K-theory, this is a large step towards a classification theorem for fiberwise essentially minimal zero-dimensional systems.
Thu, 24 Sep 2020 00:00:00 GMThttps://scholarsbank.uoregon.edu/xmlui/handle/1794/256852020-09-24T00:00:00ZSome Extension Algebras of Standard Modules over Khovanov-Lauda-Rouquier Algebras of Type A, Including A-Infinity Structure
https://scholarsbank.uoregon.edu/xmlui/handle/1794/25676
Some Extension Algebras of Standard Modules over Khovanov-Lauda-Rouquier Algebras of Type A, Including A-Infinity Structure
Buursma, Doeke
We give an explicit description of the category of Yoneda extensions of standard modules over KLR algebras for two special cases in Lie type A. In these two special cases, the A-infinity category structure of the Yoneda category is formal. We give an example to show that, in general, the A-infinity category structure of the Yoneda category is non-formal.
Thu, 24 Sep 2020 00:00:00 GMThttps://scholarsbank.uoregon.edu/xmlui/handle/1794/256762020-09-24T00:00:00Z