Numerical Analysis Seminar
Speaker: Prof. Carsten Carstensen
Host: Neela Nataraj
Title: Computation of Plates
Time, day and date: 3:00:00 PM, Wednesday, March 12
Venue: Ramanujan Hall
Abstract: The general and short title might be better specified and then stands for the mathematical foundation of the adaptive computation of plates or simply the numerical treatment of the biharmonic equation with conforming and nonconforming schemes. In the spirit of John H. Argyris (1913-2004) the complicated conforming finite element scheme marks the beginning of the finite element area with a straightforward mathematics and an involved implementation.
The presentation discusses the simplest lowest-order nonconforming finite element schemes with an easy implementation and a more involved mathematics. In fact, 30 lines of Matlab suffice in a program for basic Morley finite element simulations.
The talk concerns a larger class of popular (piecewise) quadratic schemes for the fourth-order plate bending problems based on triangles are the nonconforming Morley finite element, two discontinuous Galerkin, the C0 interior penalty, and the WOPSIP schemes. The first part of the presentation discusses recent applications to the linear bi-Laplacian and to semi-linear fourth-order problems like the stream function vorticity formulation of incompressible 2D Navier-Stokes problem and the von Karman plate bending problem. The role of a smoother is emphasised and reliable and efficient a posteriori error estimators give rise to adaptive mesh-refining strategies that recover optimal rates in numerical experiments. The last part addresses recent developments on adaptive multilevel Argyris finite element methods. The presentation is based on joint work with B. Grass le (University of Zurich) and N. Nataraj (IITB in Mumbai) partly reflected in the references below.
The eye-catcher is a photo from the Monash campus and illustrates that the plate simulation may fail because of interactions with other loadings and
related to simulations in [8].

Combinatorics Seminar

Speaker: Prof. Anand Srivastav (Christian-Albrechts-Universität zu Kiel, Kiel, Germany)
Host: Sudhir Ghorpade
Title: The k-Hamilton Cycle Maker-Breaker Game
Time, day and date: 4:00:00 PM, Wednesday, March 12
Venue: Ramanujan Hall
Abstract: We study the Maker-Breaker k-Hamilton cycle game, k an integer constant, on the complete graph on n nodes, where the aim of Maker is to build k Hamilton cycles, while Breaker wishes to prevent it. This is a two-person perfect information game on a finite board, namely the edges of the complete graph.
The game is played under the following rules. Maker and Breaker alternately choose edges of the complete graph not taken by any of the players so far. Maker starts, and chooses one edge of the complete graph. Thereafter, Breaker may choose upto b free edges. The game ends without a draw latest after all edges have been choosen by the two players.
In such games, the challenging problem is to find the threshold bias b*, an integer,  so that for b < b* there is a winning strategy for Maker, but for b > b* Breaker has a winning strategy, and to present such strategies.
Krivelevich (J. AMS 2010) determined in a breakthrough paper, extending foundational work of Chvatal and Erdös (1978), the asymptotially exact threshold bias to be (1 - o(1))n/ln(n) for k = 1.  Brüstle, Clusiau, Narayan, Ndiaye, Reed & Seamone (2023) showed that for k = 1 the game can be won by Maker in at most n + Cn/sqrt(ln(n)) many rounds, C a constant, if b < n/ln(n) - cn/ln(n)^{3/2}. This is an asymptotically optimal round complexity.
The game for k > 1 is much more complicated because we must ensure edge-disjointness of the k Hamilton cycles, while these cycles are competing for favorable edges. We show that Maker wins the k-Hamilton cycle game, if b < n/ln(n) -c n/ln(n)^{3/2}, in at most kn + c'n/sqrt(ln(n)) rounds, c, c' being constants depending on k only.
This round complexity is asymptotically optimal as well.

(joint work with Jan Geest, Department of Mathematics, Kiel University)