Burger's equation is one of the basic model equations in fluid mechanics. In this talk I'll give an introduction to Burger's equation from the perspective of infinite dimensional geometry. We'll discuss how diffeomorphism groups can be endowed with a manifold structure and how with certain Riemannian metrics these groups have Burger's equation as their geodesic equations. We'll also see how different versions of Burger's equation that are simply related from the analytic perspective can have wildly different geometric properties.
Studying Burger's Equation Through the Geometry of Diffeomorphism Groups
Sep. 27, 2016 12:10pm (MATH …
Kempner
Pedro Berrizbeitia (Universidad Simon Bolivar)
X
Let be an integer. Except for finitely many primes , the equation has integer solutions and for all integer . The problem is a classical one in Number Theory. In fact, if is the number of solution of the equation, then .
The analog problem for infinite or more general fields is the following: Let be a multiplicative group of finite index in , the multiplicative group of units of the field , what does look like, in particular, when is ?.
The first publication on the subject appeard in 1989, when Leep and Shapiro, gave a positive answer when the index of in is 3. That is, they proved that except for a small and short list of exceptional fields. They conjectured that the result was true in the case of index 5, for any infinite field . They mentioned that they had not been succesfull even for , the rational numbers.
In the talk we will prove that if is of finite index in , then , that is, every rational number is writen as the difference of two elements of . The proof is a very nice application of the Van der Waerden Theorem on artithmetic progressions, that we will state without proof.
More general results in Ramsey Theory allow to prove the same result of any infinite field, and even for some other more general rings. We will brieffly describe these more general results and consequences.
We will also discuss the behaviour of , of , etc. In fact, we will scketch the solution of a conjecture posed in 1992 by Bergelson and Shapiro on an additive question that was published in 2011, in a joint publication with Florian Luca.
Additive properties of multiplicative groups of finite index in fields