Abstract: We shall discuss a class of mean field equations on compact surfaces, which arise from the study of gauge field vortices. We present some existence and non-existence results as well as concentration properties of the solution, according to the topological and geometrical properties of the surface. Uniqueness will be also discussed in certain cases.
This presentation will focus on the ways in which several middle school mathematics teachers revised their interactions with their students while participating in a long-term professional development project, based at Rutgers University. Many teachers want their students to go beyond solving mathematical problems on paper to explaining and justifying their solutions to their peers. I will discuss our research on the ways in which teachers can modify their practice to achieve this goal. I will show video excerpts that highlight critical moments in the classroom, to demonstrate the teacher-student dynamic, and also illustrate how an eighth-grade mathematics class evolved as their teacher changed his approach.
Questions about the distribution of prime numbers, and about the existence of prime numbers of special forms, have been stymieing mathematicians for over two thousand years. It's almost necessary to study two different subjects: the theorems about prime numbers that we have been able to prove, and the (vastly more numerous) conjectures about prime numbers that we haven't yet succeeded at proving. In this talk I'll describe many of the open problems (and a few closed ones) concerning the distribution of primes, mentioning when I can some techniques that have been used to attack them.