North meets South Colloquium

9 March 2018
Radu Cimpeanu and Liana Yepremyan

Speaker: Radu Cimpeanu
Title: Crash testing mathematical models in fluid dynamics

Abstract: In the past decades, the broad area of multi-fluid flows (systems in which at least two fluids, be they liquids or mixtures of liquid and gas, co-exist) has benefited from simultaneous innovations in experimental equipment, concentrated efforts on analytical approaches, as well as the rise of high performance computing tools. This provides a wonderful wealth of techniques to approach a given challenge, however it also introduces questions as to which path(s) to take. In this talk I will explore the symbiotic relationship between reduced order modelling and fully nonlinear direct computations, each of their strengths and weaknesses and ultimately how to use a hybrid strategy in order to gain an understanding over larger subsets of often vast solution spaces. The discussion will take us through a number of interesting topics in fluid mechanics on a wide range of scales, from electrohydrodynamic control in microfluidics, to nonlinear waves in channel flows and violent drop impact scenarios.

Speaker: Liana Yepremyan
Title: Turan-type problems for hypergraphs

Abstract: One of the earliest results in extremal graph theory is Mantel's Theorem  from 1907, which says that for given number of vertices, the largest triangle-free graph on these vertices is the complete bipartite graph with (almost) equal sizes. Turan's Theorem from 1941 generalizes this result to all complete graphs. In general, the Tur'\an number of a graph G (or more generally, of  a hypergraph) is the largest number of edges in a graph (hypergraph) on given number of vertices containing no copy of G as a subgraph. For graphs a lot is known about these numbers,  a result by Erd\Hos, Stone and Simonovits determines the correct order of magnitude of Tur\'an numbers  for all non-bipartite graphs. However, these numbers are known only for few  hypergraphs. We don't even know what is the Tur\'an number of the complete 3-uniform hypergraph on 4 vertices. In this talk I will give some  introduction  to these problems and brielfly describe some of the methods used, such as the stability method and the Lagrangian  function, which are interesting on their own.