Rings of small rank over a Dedekind domain and their ideals

In 2001, M. Bhargava stunned the mathematical world by extending Gauss's 200-year-old group law on integral binary quadratic forms, now familiar as the ideal class group of a quadratic ring, to yield group laws on a vast assortment of analogous objects. His method yields parametrizations of rings of degree up to 5 over the integers, as well as aspects of their ideal structure, and can be employed to yield statistical information about such rings and the associated number fields. In this paper, we extend a selection of Bhargava's most striking parametrizations to cases where the base ring is not Z but an arbitrary Dedekind domain R. We find that, once the ideal classes of R are properly included, we readily get bijections parametrizing quadratic, cubic, and quartic rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss composition for which Bhargava is famous. We expect that our results will shed light on the analytic distribution of extensions of degree up to 4 of a fixed number field and their ideal structure.Comment: 39 pages, 1 figure. Harvard College senior thesis, edite

Measuring Greatness in the NBA

The “greatest player ever” debate stems from the controversy of how to measure a player’s effectiveness and contributions. Some analysts focus their arguments on a player’s statistics and advanced analytics. Another analyst may argue that awards play the largest role in a player’s worth to a team. Even though the tendency is to focus on one category of comparison, a player’s career is too complex to use only one category to rank players. In basketball, there are also so many exceptions in these points of comparison due to the team aspect of the sport. These factors all play a role in determining who is the greatest NBA player of all-time

‘Pushing Through’ in Plato’s Sophist: A New Reading of the Parity Assumption

At a crucial juncture in Plato’s Sophist, when the interlocutors have reached their deepest confusion about being and not-being, the Eleatic Visitor proclaims that there is yet hope. Insofar as they clarify one, he maintains, they will equally clarify the other. But what justifies the Visitor’s seemingly oracular prediction? A new interpretation explains how the Visitor’s hope is in fact warranted by the peculiar aporia they find themselves in. The passage describes a broader pattern of ‘exploring both sides’ that lends insight into Plato’s aporetic method

The Effect of the United Nations Convention Against Torture on the Scope of Habeas Review in the Context of International Extradition

This Note considers the law underlying the question addressed in Trinidad: can habeas courts review an extraditee’s Article Three claims? In turn, this Note considers how courts should interpret the CAT in the extradition context. Part I explores the important conceptual components of the question posed in Trinidad,including US extradition practice, habeas petitions in extradition proceedings, and the CAT’s implementation in the United States. Building on this, Part II examines competing interpretations of Article Three claims in US courts, highlighting how these claims touch on much deeper issues that remain unsettled by several hundred years of habeas corpus jurisprudence. Finally, Part III posits a simple answer to the straightforward question posed in Trinidad. Neither the CAT, its implementing laws or regulations, nor the United States Constitution allows courts to hear an extraditee’s Article Three claims. Therefore, unless Congress changes the current state of the law, Article Three claims are the exclusive purview of the Secretary

Intrastate Crowdfunding in Alaska: Is There Security in Following the Crowd?

This Note analyzes the potential of crowdfunding for the State of Alaska. Crowdfunding can open up new sources of revenue for small businesses while simultaneously providing an avenue for Alaskans to invest in their own communities. The potential, however, must be weighed against the risk of fraud, poorly run businesses, and the lack of protection for investors. It is the responsibility of the Alaska legislature, the State’s securities administrators, and the Securities and Exchange Commission to ensure that investors are adequately protected. This Note discusses Alaska’s crowdfunding legislation, the Alaska Intrastate Crowdfunding Exemption, and recommends changes to the legislation that account for the risks involved in crowdfunding while still capturing its potential

Suicide Rates as They Vary by Region, Sexuality, and Gender

Suicide rates are not consistent worldwide. They vary in a wide variety of ways. Region, sexuality, and gender are all factors that influence suicide. This essay examines the manners in which region, sexuality, and gender influence suicide by themselves and, in some cases, with each other. It offers explanations for why they do so. Finally, this paper aims to give suggestions on future research regarding suicide and future policies to help reduce suicide rates

Aristotle's Platonic Response to the Problem of First Principles

how does one inquire into the truth of first principles? Where does one begin when deciding where to begin? Aristotle recognizes a series of difficulties when it comes to understanding the starting points of a scientific or philosophical system, and contemporary scholars have encountered their own difficulties in understanding his response. I will argue that Aristotle was aware of a Platonic solution that can help us uncover his own attitude toward the problem.Aristotle's central problem with first principles arises from the fact that they cannot be demonstrated in the same way as other propositions. Since demonstrations proceed from prior and better-known principles, if the principles themselves were in need of..

Combinatorial Sutured TQFT as Exterior Algebra

The idea of a sutured topological quantum field theory was introduced by Honda, Kazez and Mati\'c (2008). A sutured TQFT associates a group to each sutured surface and an element of this group to each dividing set on this surface. The notion was originally introduced to talk about contact invariants in Sutured Floer Homology. We provide an elementary example of a sutured TQFT, which comes from taking exterior algebras of certain singular homology groups. We show that this sutured TQFT coincides with that of Honda et al. using $\Z_2$-coefficients. The groups in our theory, being exterior algebras, naturally come with the structure of a ring with unit. We give an application of this ring structure to understanding tight contact structures on solid tori

A Family of Partially Ordered Sets with Small Balance Constant

Given a finite poset $\mathcal P$ and two distinct elements $x$ and $y$, we let $\operatorname{pr}_{\mathcal P}(x \prec y)$ denote the fraction of linear extensions of $\mathcal P$ in which $x$ precedes $y$. The balance constant $\delta(\mathcal P)$ of $\mathcal P$ is then defined by $$\delta(\mathcal P) = \max_{x \neq y \in \mathcal P} \min \left\{ \operatorname{pr}_{\mathcal P}(x \prec y), \operatorname{pr}_{\mathcal P}(y \prec x) \right\}.$$ The $1/3$-$2/3$ conjecture asserts that $\delta(\mathcal P) \ge \frac13$ whenever $\mathcal P$ is not a chain, but except from certain trivial examples it is not known when equality occurs, or even if balance constants can approach $1/3$. In this paper we make some progress on the conjecture by exhibiting a sequence of posets with balance constants approaching $\frac{1}{32}(93-\sqrt{6697}) \approx 0.3488999$, answering a question of Brightwell. These provide smaller balance constants than any other known nontrivial family.Comment: 11 pages, 4 figure