Undergraduate Seminars II - Math UN3952 - Section 002.
Spring 2020.
Columbia University.
Name: Daniele Alessandrini.
Contact: daniele.alessandrini@gmail.com.
Office: 624 Mathematics.
Xuan Wu.
Raymond Cheng.
Henry Liu.
There will be 4 sections of the Undergraduate Seminars. The students need to choose one section to follow, where they also will give talks. The sections are:
We will study algebraic properties of finitely generated groups by constructing group actions on geometric objects. The interplay between the algebra and the geometry gives a new way to understand the structure of the groups.
A permutation of set S={1, 2,3, ..., n} is an arrangement of the numbers in S. There are totally n! of them. If we randomly pick one, what we see depends on the lens we look at. For example, we could either see a uniform measure or a poisson random variable. In this seminar, we will explore the statistical behaviors of random permutations together with a focus on two problems. The first one is the length of the longest increasing subsequence and the second one is the number of steps it takes to suffle a deck of cards.
Elliptic curves—a breakfast item or a cubic curves, depending on who you ask—are beautiful and ubiquitous objects in number theory. This seminar is a tour beginning from the rich arithmetic theory of elliptic curves, and ending with a panoramic view of the proof of Fermat’s Last Theorem, in which elliptic curves make a startling appearance via modular forms. Prior experience in calculus and linear algebra will be sufficient for this journey.
This seminar will use knot theory, specifically knot invariants, as a theme to explore various areas of mathematics and physics. These areas may include: low-dimensional topology, quantum groups, statistical mechanics, categorification, and more (depending on participant interest).
We will study algebraic properties of finitely generated groups by constructing group actions on geometric objects. The interplay between the algebra and the geometry gives a new way to understand the structure of the groups.
Name: Daniele Alessandrini.
Contact: daniele.alessandrini@gmail.com.
Office: 624 Mathematics.
Clara Löh, Geometric Group Theory An Introduction.
The electronic version of the book is available at the Columbia Library. You can download it at the given link from the Columbia network.
Wed 12:30pm-2:30pm.
Classroom: 622 Mathematics.
| No | Date | Title | Speaker | Reference |
|---|---|---|---|---|
| 0 | 1/29 | Prerequisites and organization | Daniele | |
| 1 | 2/12 | Introduction to groups | Hobart | Sec 2.1.1 from Def 2.1.2, Sec 2.1.3 without Ex 2.1.28, Sec 2.2.1. |
| 2 | 2/12 | Free groups | Giovanni | Sec 2.2.2 |
| 4 | 2/19 | Metric Spaces and quasi-isometries | Lalita | Sec 5.1 until Prop 5.1.11 |
| 5 | 2/26 | Graphs as metric spaces | Matthew | Sec 3.1 until Ex 3.1.7, Sec 5.3.1 until Ex 5.3.2, Sec 5.3.2 until Prop 5.3.8 |
| 3 | 3/04 | Generators and relations | Daniele | Sec 2.2.3, Sec 2.2.4 only Def 2.2.25 |
| 6 | 3/04 | Cayley graphs | Kyle | Sec 3.2 until Rem 3.2.3, Sec 5.2 from Def 5.2.3 |
| 7 | 4/01 | Group actions | Clara | Sec 4.1.1 without Ex 4.1.6 and 4.1.7, Sec 4.1.2 without Ex 4.1.13 and 4.1.15 without Prop 4.1.16 and 4.1.17, Sec 4.1.4 without Outlook 4.1.21 |
| 8 | 4/01 | Svarc-Milnor Lemma | Lalita | Sec 5.3.1 from Def 5.3.4, Sec 5.4 until Proof of Prop 5.4.1 |
| 9 | 4/08 | Cayley Graphs of free groups | Giovanni | Sec 3.1 from Def 3.1.8, Sec 3.3 without Cor 3.3.6 and 3.3.7 without Outlook 3.3.8 |
| 10 | 4/08 | Free groups and actions on trees | Kyle | Sec 4.2 until Proof of Thm 4.2.1, part II |
| 11 | 4/15 | Subgroups of free groups and Ping-pong lemma | Clara | Sec 4.2.3 without Outlook 4.2.13 and Cor 4.2.15, Sec 4.3 |
| 12 | 4/15 | Free matrix groups | Matthew | Sec 4.4.1 without Outlook 4.4.3, Sec 4.4.2 |
| 13 | 4/22 | Topological Svarc-Milnor Lemma and applications | Hobart | Sec 5.4 from Corollary 5.4.2, Sec 5.4.1 without Caveat 5.4.9 |
| 14 | 4/22 | |||
| 15 | 4/29 | |||
| 16 | 4/29 |