宾夕法尼亚大学物理科研项目

University of Pennsylvania Physical Research

科研主题

Physics【物理学】

Condensed Matter Theory【凝聚态理论】

Mathematics【数学】

Geometry and Physics【几何与物理】

low dimensional materials【低维材料】

科研内容：

OVERVIEW & PURPOSE

The goal of the course is to teach group theory and information theory relating the two through linear algebra and geometry. This combination is somewhat novel/unconventional in its construction, reflecting the instructor’s research interests. However, even when taken separately, these subjects are useful for both pure math, communication, cryptography, and data science purposes, and are good general foundations for students who are interested in these fields.

OBJECTIVES

1.      A basic grasp of group theory and matrix representation

2.      A basic grasp of information theory up to the noisy channel theorem

3.      Implement robust symbol and stream codes

4.      Analysis of the relationship between the noisy channel theorem and stacking density problem( in particular from the perspective of isomorphism)

Week 1-2(Intensive group theory and information theory crash courses, restricted mainly to linear algebra for practical implementation)

1.      Basic definitions of a group, matrices, and vector space

2.      NxN matrices, orthonormal basis, and diagonalization

3.      Justification for entropy as a measure for information

4.      Simple encoding methods in information theory

5.      Relating encoding with linear algebra

Assignments: Construct a program to encode and decode simple data through matrices and vectors

Week 3-4(Open question exploration)

1.      Introduction and implementation of Huffman coding(Symbol coding) and arithmetic coding(Arithmetic coding)

2.      Student analysis and comparison of the two methods

(Student research and open problem exploration, will only explore partially depending on time constraints)

3.      An exploratory discussion on the relation between noisy channel coding and group homomorphism

4.      Discussion on sphere stacking in low dimensions and shannon’s noisy channel coding problem

5.      Discussion on group homomorphism and sphere stacking

6.      Review recent progress made in 8 dimensions and 24 dimension on sphere stacking in 2016 and interpret using information-theoretic and group theoretic tools.