| Tuesday 30 April 2024 | |
| Events for day: Wednesday 24 April 2024 |
| 12:45 - 13:45 Geometry and Differential Equations Seminar Existence of Radial Positive Solutions for Homogeneous and Non-homogeneous Neumann Problems School MATHEMATICS Let $B_{1}$ be the unit ball in $mathbb{R}^{n}$. We analyze the positive solutions to the problem egin{equation}label{aaa} left{egin{array} {ll} - Delta u + u = u vert uvert ^{p - 2}, & mbox{ in } B_{1}, %u>0, & mbox{ in } Omega, dfrac{partial u}{partial u} = 0, & mbox{ on } partial B_{1} , end{array} ight. end{equation} where $p > 2$ and $frac{partial}{partial u}$ is the outward normal derivative. This problem, sometimes referred to as the Lane-Emden equation with Neumann boundary conditions, arises for instance in mathematical models which aim to study pattern formation, and more specifically in those governed by diffusion ... 14:00 - 15:00 Combinatorics and Computing Weekly Seminar Maximum Entropy of Convex Corners and Matroids School MATHEMATICS Entropy of a random variable is a measure of how much a random variable can be losslessly compressed. When certain symbols are allowed to be confused, the notion of graph entropy is used. A central problem in this subject is the question of maximum entropy of a graph and the distribution achieving it. As observed by Changiz-Rezaei and Godsil, this problem is closely related to the notion of fractional chromatic number of graphs and uniform cover of the vertices by maximal independent sets. Simony et al. generalized the notion of graph entropy to other geometric objects known as convex corners. In this talk, we consider the problem of maximum ... 16:00 - 17:00 Mathematics Colloquium Vector Field on Plane in Characteristic Zero and p>0 School MATHEMATICS A polynomial vector field on a complex plane C^2 defines a foliation of the plane and one would like to know when the leaves of this foliation are algebraic, i.e. when the analytic integral curves are algebraic. This is a hard unsolved problem. I will suggest a conjectural approach to this problem which uses the reduction of the vector field modulo primes p. It turns out that the corresponding problem in characteristic p is easy to solve. We then conjecture that the original vector field is algebraic if and only if its reduction modulo a prime p is algebraic for almost all p. We have some partial results towards proving this conjecture. This ... |