A family of problems in Diophantine geometry has the following form: We fix a collection of "special" algebraic varieties among which the 0-dimensional are called "special points". Mostly, if V is a special variety then the special points are Zariski dense in V, and the problem is to prove the converse: If V is an irreducible algebraic variety and the special points are Zariski dense in V then V itself is special.
A family of problems in Diophantine geometry has the following form: We fix a collection of "special" algebraic varieties among which the 0-dimensional are called "special points". Mostly, if V is a special variety then the special points are Zariski dense in V, and the problem is to prove the converse: If V is an irreducible algebraic variety and the special points are Zariski dense in V then V itself is special.
Abstract: In this talk we will present joint work with Nir Avni and Alex Lubotzky concerning the model theory of higher rank arithmetic groups. We will show that many of these groups are determined by their first order theory as individual groups and also as a collection of groups.
For everynumber field K, there is a finite abelian group C called the class group, which serves as an obstruction to unique factorization. Since Gauss, number theorists have tried to understand questions such as how often is C trivial, or how often C contains an element of fixed order (as K varies). In the 1970's, Cohen and Lenstra observed empirically that when the degree and signature of K is fixed, the isomorphism class of C adheres to a natural probability distribution. I'll discuss these Cohen-Lenstra heuristics and survey what is known, Read more about Basic Notions: Ari Shnidman "Randomness in arithmetic: class groups."
Abstract: Inseparable extensions and morphisms are an important feature in positive characteristic. The study of these uses (smooth) foliations in the tangent bundle of derivations, as was first seen in a theorem of Jacobson (1944) on purely inseparable field extensions of exponent 1. In this talk we will state Jacobson's theorem and some of its generalizations: to normal domains, to regular local and non-local rings, and to morphisms of smooth varieties.
Abstract: We will concentrate on two papers by Marina Ratner: "Horocycle flows are loosely Bernoulli". Israel J. Math. 31 (1978) no. 2, 122-132. and "The Cartesian square of the horocycle flow is not loosely Bernoulli". Israel J. Math. 34 (1979). , no. 1-2, 72-96 (1980). We will start from the definition of loosely Bernoulli, then a detailed discussion about the proof that horocycle flow is loosely Bernoulli and finally some hints about the non-loosely Bernoulli proof in the product case.
In this talk I present conjectures and results of joint work with Pavel Etingof and Edward Frenkel on an analytic version of the Langlands correspondence for curves over local fields.
It was shown by C. L. Siegel (1929) that the eigenvalues of the vibrating membrane problem has no non-trivial multiplicities. In this talk we consider the eigenvalues of the vibrating clamped plate problem. This is a fourth order problem. We show that its eigenvalues have multiplicity at most six. The proof is based on a new recursion formula for a Bessel-like function and on Siegel-Shidlovskii Theory. If time permits we also consider the problem of determining the density of the nodal sets of a clamped plate.
Speaker: Eilidh McKemmie Title: The probability of generating invariably a finite simple group Abstract: We say a group is invariably generated by a subset if every tuple in the product of conjugacy classes of elements in that subset is a generating tuple. We discuss the history of computational Galois theory and probabilistic generation problems to motivate some results about the probability of generating invariably a finite simple group. We also highlight some methods for studying probabilistic invariable generation.
Abstract: Unipotent groups form one of the fundamental building blocks in the theory of linear algebraic groups. Over perfect fields, their behavior is very simple. But over imperfect fields, the situation is much more complicated. We will discuss various aspects of these groups, from the fundamental theory to a study of their Picard groups, which appear to play a central role in understanding their behavior.
