On the shape of a high-dimensional random lattice
Abstract/Contents
- Abstract
- The statistics concerning lattices in high dimensions (call "the geometry of numbers" in the literature) used to be studied intensively in the mid-20th century, especially by C.A. Rogers and Wolfgang Schmidt. By perhaps an accident of history, it has become forgotten by mathematicians for almost 50 years. Yet there is still plenty of wonderful mathematics left there to develop, and furthermore, recently people in various sectors — computer science, cryptography, number theory, and even dynamics — are starting to realize the need and benefits of understanding high-dimensional lattices and related topics. This thesis is largely in three parts. In Chapter 1, we give an exposition on the Rogers integration formula and its variants, the main technical device in the study of high-dimensional lattices. In Chapter 2, inspired by the ideas of Schmidt, we prove that the lengths of the first O(n^(1/2)) shortest vectors of an n-dimensional random lattice exhibit a Poisson distribution as n → ∞, improving a recent result of Sodergren. In Chapters 3 and 4, we prove a few previously unknown facts on the statistics of the LLL bases that help answer a few questions that have been raised regarding the peculiar behavior of the LLL algorithm, and then present an experimental result that supports our theoretical conclusions and suggests where future research could be headed to.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2015 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Kim, Seungki |
|---|---|
| Associated with | Stanford University, Department of Mathematics. |
| Primary advisor | Venkatesh, Akshay, 1981- |
| Thesis advisor | Venkatesh, Akshay, 1981- |
| Thesis advisor | Boneh, Dan |
| Thesis advisor | Soundararajan, Kannan, 1973- |
| Advisor | Boneh, Dan |
| Advisor | Soundararajan, Kannan, 1973- |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Seungki Kim. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2015. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2015 by Seung Ki Kim
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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