Sets, groups, and fields definable in vector spaces with a bilinear form.
Abstract: There is a long history of study of algebraic objects definable in classical mathematical structures. As a prominent example, by results of Weil, Hrushovski, and van den Dries, it is known that the groups definable in an algebraically closed field K are precisely the algebraic groups over K, and the only infinite field definable in K is the field K itself.
One might say that there exist zeta functions of three kinds: the very well-known ones (whose stock example is Riemann's zeta function); some less familiar ones; and at least one type which has been totally forgotten for decades. We intend to mention instances of all three types. The first type is important, if not predominant, in algebraic number theory, as we will try to illustrate by (very few) examples. As examples of the second type we will discuss zeta functions of finite groups.
Abstract: Perturbation theoryworks well for the the discrete spectrum below the essential spectrum. Whathappens if a parameter of a quantum system is tuned in such a way that abound state energy (e.g. the ground state energy) hits the bottom of theessential spectrum? Does the eigenvalue survive, i.e., the correspondingeigenfunction stays $L^2$, or does it dissolve into the continuumenergies?
For every number fieldK, 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,
Abstract: Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute.
