Includes congruences, quadratic reciprocity, Diophantine equations, and number-theoretic functions, among others. Prerequisite: MATH 3354 or instructor permission.
TBA - TBA
Philippe Gille (Lyon-Bucharest) - New homogeneous spaces with a 0-cycle of degree one and without rational points
This is a report on joint work with C. Beli (Bucharest) and T.-Y. Lee (Lausanne). Given a connected affine algebraic group G defined over a field k, B. Totaro raised the question whether a homogeneous space X under G having a 0-cycle of degree one has a rational point. The question is widely open for principal homogeneous spaces and there are counterexamples by Florence and Parimala respectively with finite stabilizers and parabolic stabilizers. The goal of this talk is to present a new class of homogeneous spaces without rational points but with quadratic and cubic points. Those homogeneous spaces are geometrically isomorphic to G/T , that is the quotient of a group G of type G2 by a maximal torus.Add to Google Calendar
TBA - TBA
Julia Bennett (UT Austin) - Large R^4's in Stein surfaces
While many open 4-manifolds are known to admit uncountably many
diffeomorphism classes of smooth structures, it is still unknown if this
behavior can be expected in general. The goal of this talk is to introduce
a family of "large" R^4's with the property that each is contained in some
compact Stein surface. We will discuss why these new R^4's are relevant to
understanding smoothing theory on more general open 4-manifolds and
outline their construction. Along the way, we will define Casson handles
and describe their use in cut-and-paste arguments.