Date 
Spring 2015 Schedule 
April 10 
Speaker: Cindy Tsang Title: Galois Modules and Embedding Problems Abstract: Let K be a number field and G a finite group. Given a Galois extension L/K with Gal(L/K)\simeq G, one asks whether its ring of integers O_L is free over O_K and whether it has an O_Kbasis whose elements are Galois conjugates. This is the study of Galois modules. On the other hand, given a Galois subextension K/k with Gal(K/k)=\Sigma and a \Sigmaaction on G, one asks whether there exists a Galois extension L/k with Gal(L/K)\simeq G such that the \Sigmaaction on G via conjugation in Gal(L/k) agrees with the given action. This is the embedding problem. In this talk, I will explain how these two problems are related. 
April 17 
Speaker: Nadir Hajouji Title: Constellations Abstract: Abstract: A kconstellation of degree n is a sequence of
k permutations g_1, ..., g_k in S_n satisfying two basic properties:

April 24 
Speaker: Yingying Wang Title: Examples of Abelian Extensions Abstract: This talk is going to be an exposition on two specific cases
of Abelian Extensions(Galois extensions with abelian Galois groups): cyclotomic extensions
and Kummer extensions. The following is an introduction to Kummer Extensions.

May 1 
Speaker: Ebrahim Ebrahim Title: From semisimple Lie Algebra to root space decomposition Abstract: I will give an overview of the description of a semisimple Lie algebra in terms of its root space decomposition, and I will show a sample of the proofs of major theorems that are needed to establish the decomposition. 
May 8 
Speaker: Jason Murphy Title: Fully Homomorphic Encryption Abstract: Suppose Alice has some secret data that she wants to send to Bob, but she needs to perform some computations on it first. There's a large amount of data, so she'll need to do the computations remotely with the help of Eveazon Web Services. She uploads her data on the cloud as an encrypted file, to keep the contents hidden from the wicked EWS CEO Eve; but what next? As soon she decrypts the data for the purpose of computation, Eve will know all her secrets. But if there was a way to encrypt the data which preserved addition and multiplication of bits (and every computer program boils down to these two operations), she could just tell Eve to perform the arithmetic operations on the encrypted data and it would decrypt to the same operations being performed on the unencrypted data. With such an encryption scheme, Alice can simultaneously keep her secrets hidden from Eve and really stick it to Eve by having her do all the work. In the talk, we will be exploring how to construct this socalled Fully Homomorphic Encryption scheme using quotients of polynomial rings and the RingLWE problem. 
May 15 
Speaker: Christian Bueno Title: Tropical Varieties of Amoebas and Other Tropical Adventures Abstract: The tropical semifield is the real numbers R adjoined with infinity and then endowed with two binary operations: tropical addition and tropical multiplication. Tropical addition is minimization, and tropical multiplication is the classical addition of numbers. Naturally one can construct polynomials over this semifield and one can study tropical varieties: analogs of varieties in algebraic geometry. The metaphor also works in the other direction. One can define a logarithmic transformation of certain complex algebraic varieties which are called amoebas, and these geometric objects carry with them a tropical variety, which provides insight through it's more combinatorialthananalytic nature. These connections and more have yielded powerful results in enumerative geometry. Lastly, in a whole other direction, the built in minimization properties of the tropical algebra give interesting applications to optimization problems. 
May 22 
Speaker: Glen Frost Title: Luroth's Theorem Abstract: This talk will present a proof of Luroth's Theorem. Also interesting examples of fields. 
May 29 
Speaker: David Wen Title: I scheme, you scheme, we all scheme for... Abstract: The advent of schemes by Grothendieck in his tour de force of EGA and SGA brought about modern algebraic geometry, establishing the foundations of the modern school in the language of schemes and cohomology. Yet redevelopement of foundations has the tendecy to obliterate history which, at times, strip insight and motivation into the objects in question. The definition of scheme is fairly abstract and seems, to varying levels, disconnected to the objects they can represent, for example ring of integers of number fields, algebraic curves and compact Riemann surfaces. For this talk, I will attempt to give the insight and motivation for using the language of schemes and bridge the connections between algebraic number theory, classical algebraic geometry and complex geometry. 
June 5 
Speaker: Cindy Tsang Title: Diophantine Equations and the Hasse Principle Abstract: Let f be a polynomial with rational coefficients. If f=0 has a solution over Q, then it has a solution over R and Q_p for all primes p. A natural question to ask is whether the converse is true, i.e. if f=0 has a solution everywhere locally, does it imply that it has a solution globally? A localtoglobal principle of this type is known as the Hasse principle, which appears everywhere in number theory. In this talk, I will explain the idea behind this principle in more detail and give some examples for which it fails/holds. 