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_K-basis 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 \Sigma-action on G, one asks whether there exists a Galois extension L/k with Gal(L/K)\simeq G such that the \Sigma-action 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 k-constellation of degree n is a sequence of k permutations g_1, ..., g_k in S_n satisfying two basic properties:
(1) g_1g_2 ... g_k = 1
(2) The subgroup generated by g_1, ..., g_k acts transitively on n.
I will show various ways one can produce constellations, and discuss how those constructions are related to each other. One of the constructions will allow you to produce explicit generators for certain sporadic groups, "even in the middle of a desert". I will also talk about how the category of dessins d'enfants, mentioned in today's colloquim, is equivalent to the category of 3-constellations.

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.
Let k be a field that contains n-th root of unity and has characteristic prime to n. Field extensions of k such that the Galois group is cyclic and has exponent dividing n(meaning every element of the Galois group has order dividing n are exactly the extensions k(\sqrt[n]{a})/k where a\in k. These extensions are called Kummer Extensions. We will show that finite abelian extensions of k such that the Galois group has exponent dividing n corresponds to the finite subgroup of k^*/(k^*)^n.

May 1

Speaker: Ebrahim Ebrahim

Title: From semi-simple 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 Eve-azon 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 so-called Fully Homomorphic Encryption scheme using quotients of polynomial rings and the Ring-LWE 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 combinatorial-than-analytic 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 local-to-global 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.