Wednesdays, 7:30pm; Room 507, Mathematics
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ums [at] math.columbia.edu
| Date | Speaker | Title | Abstract |
| Sept 9 | Anand Deopurkar | Undergraduate seminars |
Introduction to the fall seminars: Rob Castellano (Enumerative geometry and string theory), Anand Deopurkar (Young tableaux in algebra and geometry), Ashwin Deopurkar (Tropical curves), Dan Gulotta (Elliptic curves) Introduction to the spring seminars: Dave Bayer |
| Sept 16 | Alex Cowan | An introduction to elliptic curves | An elliptic curve is a certain kind of polynomial equation which is very important in modern number theory. In the first part of the talk I'll explain why they're important and state some basic facts about the structure of solutions to the equation. In the second part of the talk, I'll explain some of what current research is figuring out about these objects. |
| Sept 23 | Sam Mundy | Number fields and unique factorization | In number theory, one often works with various rings that extend the ring of integers, and these rings tend to behave in many ways like the integers themselves. But there is one important property of the integers which these rings very often fail to have: Unique factorization. Number theorists have many different methods of dealing with this problem, almost all of which involve ideals, or the ideal class group, which quantifies the failure of unique factorization. I will discuss these ideas at an elementary level, and give connections with classical number theory. If time permits, I will also discuss connections with Fermat's Last Theorem. |
| Sept 30 | Melissa Liu | Mirror symmetry for the projective line | I will describe some versions of mirror symmetry relating the symplectic (resp. complex) geometry of the projective line to the complex (resp. symplectic) geometry of its mirror. The talk will begin with definitions of symplectic and complex structures. |
| Oct 7 | Sebastian Mueller | Fractal geometry | In this talk, we will attempt to introduce the colourful and repetitious field of fractal geometry. We will introduce a metric on the set of pictures in the plane, the notion of a iterated function system, and see how these produce a rich class of fractals. We will also look at some other well known fractals that arise in other situations. With any luck, technology will be on our side and we will be able to see plenty of computer generated examples. |
| Oct 14 | Linus Hamann |
Ramification in number theory and geometry |
The talk will begin by a quick introduction to algebraic number theory. In particular, we will discuss the natural arithmetic question of how primes split and ramify in different number fields. Afterwards, we will begin with a discussion of complex geometry, introducing and then proceeding to describe maps between Riemann Surfaces. The geometric definition of ramification will naturally arise from this. With this in mind, I will attempt to describe how these two notions of ramification actually coincide by introducing the concept of an affine scheme. |
| Oct 21 | No talk | ||
| Oct 28 | BoGwang Jeon | Introduction to hyperbolic geometry | In this talk, I'll define the hyperbolic metric and then explain how the topology of 2- and 3-dimensional manifolds are related to this one. |
| Nov 4 | Sebastien Picard | Introduction to a priori estimates | In this talk, we will state the linearized Navier-Stokes equations, prove the existence and uniqueness of weak solutions, and use Lebesgue elliptic estimates to derive regularity results; we will introduce tools from classical functional analysis (i.e. segment property, function spaces, weak convergence) along the way. |
| Nov 11 | Mathematics Open House | ||
| Nov 18 | No talk | ||
| Nov 25 | Thanksgiving break | ||
| Dec 2 | Mikhail Khovanov |
Two-dimensional algebra and link invariants |
We will discuss linearization, categories and monoidal categories. Monoidal categories are examples of two-dimensional algebraic structures. We will zero in on a particular example, a monoidal category known as the Kuperberg sl(3) spider, and explain how it gives rise to invariants of knots and links in the 3-space. |