January 12 
Ozlem Ejder (USC) Let $E$ be an elliptic curve defined over $\Q$. The torsion subgroup of $E$ over the compositum of all quadratic extensions of $\Q$ was studied by Michael Laska, Martin Lorenz, and Yasutsugu Fujita. Laska and Lorenz described a list of $31$ possible groups and Fujita proved that the list of $20$ different groups is complete. In this talk, we will generalize the results of Laska, Lorenz and Fujita to the elliptic curves defined over a quadratic cyclotomic field i.e. $\Q(i)$ and $\Q(\sqrt{3})$. 
January 19 
Brian Hwang (Cornell) A number of questions in Galois theory can be phrased in the following way: how large (in various senses) can the Galois group G of an extension of the rational numbers be, if the extension is only allowed to ramify at a small set of primes? If we assume that G is abelian, class field theory provides a complete answer, but the question is open is almost every nonabelian case, since there is no known way to systematically and explicitly construct such extensions in full generality. However, there have been some programs that are gaining ground on this front. While the problem above is natural and the objects are classical, we will see that to answer certain questions about of this Galois group, it seems necessary to use techniques involving automorphic forms and their representationtheoretic avatars. In particular, it will turn out that some recent results on "harmonic" families of automorphic forms translate to the fact that such number fields, despite not being explicitly constructible by known methods, turn out to "exist in abundance" and allow us to find bounds on the sizes of such Galois groups. 
January 26 
Peter Stevenhagen (Leiden) Artin's conjecture on primitive roots, which was originally formulated for multiplicative groups, has a natural analogue for elliptic curves. In this survey talk, I will discuss the analogy and focus on "new" phenomena such as the existence of "neverprimitive" points. 
February 2 
Michiel Kosters (UC Irvine) Let P: ... > C_2 > C_1 > P^1 be a Z_pcover of the projective line over a finite field of characteristic p which ramifies at exactly one rational point. In this talk, we study the padic Newton slopes of Lfunctions associated to characters of the Galois group of P. It turns out that for covers P such that the genus of C_n is a quadratic polynomial in p^n for n large, the Newton slopes are uniformly distributed in the interval [0,1]. Furthermore, for a large class of such covers P, these slopes behave in an even more regular way. This is joint work with Hui June Zhu.
Niccolò Ronchetti (Stanford) Recently, Venkatesh introduced the derived Hecke algebra to explain extra endomorphisms on the cohomology of arithmetic manifolds: the crucial local construction is a derived version of the spherical Hecke algebra of a reductive padic group. Working with ptorsion coefficients, we will describe a Satake homomorphism for the derived spherical Hecke algebra of a padic group. This will allow us to understand its structure well enough to attack some global questions, which are work in progress. 
February 9 
Jennifer Balakrishnan (Boston University) Elliptic curves defined over the rational numbers are of great interest in modern number theory. The rank of an elliptic curve is a crucial invariant, with many open questions about its behavior. In particular, there is great interest in the "average" rank of an elliptic curve. The minimalist conjecture is that the average rank should be 1/2. In 2007, Bektemirov, Mazur, Stein, and Watkins [BMSW], using wellknown databases of elliptic curves, set out to numerically compute the average rank of elliptic curves, ordered by conductor. They found that "there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either." In joint work with Ho, Kaplan, Spicer, Stein, and Weigandt, we have assembled a new database of elliptic curves ordered by height. I will describe the database and examine some of the questions posed by [BMSW]. I will also discuss ongoing work by a team of undergraduates at Oxford on similar questions about families of elliptic curves. 
February 16 
James Maynard (Oxford) It is a famous conjecture that any one variable polynomial satisfying some simple conditions should take infinitely many prime values. Unfortunately, this isn't known in any case except for linear polynomials  the sparsity of values of higher degree polynomials causes substantial difficulties. If we look at polynomials in multiple variables, then there are a few polynomials known to represent infinitely many primes whilst still taking on `few' values; FriedlanderIwaniec showed $X^2+Y^4$ is prime infinitely often, and HeathBrown showed the same for $X^3+2Y^3$. We will demonstrate a family of multivariate sparse polynomials all of which take infinitely many prime values. 
February 23 
Serin Hong (Caltech) The ladic cohomology of RapoportZink spaces is expected to realize local Langlands correspondences in many cases. Along this line is a conjecture by Harris, which roughly says that when the underlying RapoportZink space is not basic, the ladic cohomology of the space is parabolically induced. In this talk, we will discuss a result on this conjecture when the RapoportZink space is of Hodge type and "HodgeNewton reducible". The main strategy is to embed our RapoportZink space to an appropriate space of EL type, for which the conjecture is already known to hold. If time permits, we will also discuss other applications of this strategy. 
March 2 
Nathan Kaplan (UC Irvine)

March 9 
Amir Mohammadi (UCSD)

March 16 
Lucia Mocz (Princeton)
