Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions. I have recently started to explore the statistics of these Robin eigenvalues for planar domains, and the fluctuations of the gaps between the Robin and Neumann spectrum, in part driven by numerical experimentation. Read more about Basic Notions: Zeev Rudnik "The Robin eigenvalue problem: statistics and arithmetic"
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."
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,
