Spring 2014 Schedule

April 18

Speaker: Cindy Tsang

Title: Primes of the form x^2+ny^2

Abstract: A very old theorem of Fermat states that an odd prime p can be written as x^2+y^2 if and only if p\equiv 1 (mod 4). What about primes of the form x^2+ny^2 for an arbitrary positive integer n? In this talk, I will try to give an overview of the study of this problem. For example, one approach is to use the quadratic reciprocity (I believe that in fact the study of this problem led to the discovery of the quadratic reciprocity), but it only answers the question for certain values of n. On the other hand, using Class field theory, one can obtain an essentially complete answer to this question. This talk should be very accessible.

April 25

Speaker: Jason Murphy

Title: Complex Multiplication and Modular Functions

Abstract: Last time, Cindy spoke about primes of the form x^2 + ny^2, where the punch line was that with finitely many exceptions, p = x^2 + ny^2 if and only if (-n/p)=1 and f_n(x) = 0 mod p has an integer solution (where f_n(x) is the minimal polynomial of a primitive element of the ring class field of Z[sqrt(-n)]). In my talk, we will introduce some of the theory of complex multiplication of elliptic curves and modular functions to get an algorithm for constructing the polynomial f_n(x) that solves p=x^2 + ny^2.

May 2

Speaker: Nathan Saritzky

Title: Moduli Spaces

Abstract: Everyone knows that an algebraist's favorite activity is classifying things. In especially favorable circumstances, we can accomplish this with finitely many discrete invariants (see, for example, the classification of finitely-generated abelian groups). But for more complex problems, we might want to "continuously parameterize" our objects. Fine and course moduli give us a way to do just that, although one must ask a lot even for these to exist. We'll explore the examples regarding hypersurfaces and vector space endomorphisms.

May 9

Speaker: Joe Ricci

Title: Braid Groups

Abstract: Braid groups were formally introduced by Artin in 1925 and since then much work has been done to understand their properties and applications. While interesting as an abstract group, braid groups have many topological interpretations and uses as their name suggests. I will draw some pictures and hopefully convince you of this. More recently, the braid groups paved the way towards the discovery of the Jones polynomial, the celebrated knot and link invariant.

May 16

No seminar

May 23

Speaker: Drew Jaramillo

Title: Subgroups in Quantum SL

Abstract: In this talk I will discuss some quantum subgroups of SL_n. In particular, I will discuss quantum standard Borel and quantum standard parabolic subgroups and their unipotent radicals. We will see how these quantum unipotent radicals arise as the algebra of coinvariants of a natural coaction.

May 30

No seminar

June 6

Speaker: David Wen

Title: High school math to Bezout's Theorem

Abstract: In high school, we have seen polynomials in one variable and their graphs usually described algebraically as "y = f(x)". We may have also seen that given two polynomials that their graphs will intersect in at most the maximum of their degrees. Since this is not high school we look at a more general case. Since the graphs are nothing more than zero sets of polynomials of two variables, we ask: How many intersections are their in the zero set of two arbitrary polynomials in two variables? Is there an upper bound? If there is an upper bound what are the condition necessary to always achieve it? The answers to these questions is provided by Bezout's Theorem which also gives way to some interesting applications.