Seminars from until

We will present some classical facts about the relationship between monoids and monads. We will use ordinal sums of categories and the join product of topological spaces to define the abstract and topological simplices. Along the way we show how the simplicial identities can be obtained. Time permitting we will indicate a 2-categorical generalization of this circle of ideas.

Resurgence is a method used to solve differential equations with a wide range of applications. It is based on the so called alien calculus. The talk will give a brief insight on what resurgence is used for. Then, via example, a short introduction to alien calculus is given.

I am going to present Blake Temple's and my recent breakthrough regarding optimal metric regularity: We recently derived a set of nonlinear elliptic equations, with differential forms as unknowns, (the "Regularity Transformation equations" or "RT-equations"), and proved existence of solutions. The RT-equations determine whether optimal metric regularity can be achieved in General Relativity. Our existence result applies to connections in and Riemann curvature in $W^{m,p}$, $m\geq1$, $p>n$, and thus yields that such connections can always be smoothed to optimal regularity (one derivative above their curvature) by coordinate transformation. Extending this existence theory to the case of GR shock waves, when the connection is in $L^{\infty}$, is subject of our ongoing research. Our current existence result demonstrates that the method of determining optimal metric regularity by the RT-equations works.

We survey our recent and ongoing work [1,2] on finite element methods for contact problems. Our approach is to first write the problem in mixed form, in which the contact pressure act as a Lagrange multiplier. In order to avoid the problems related to a direct mixed finite element discretisation, we use a stabilised formulation, in which appropriately weighted residual terms are added to the discrete variational forms. We prove that the formulation is uniformly stable, which implies an optimal a priori error estimate. Using the stability of the continuous problem, we also prove a posteriori estimates, the optimality of which is ensured by local lower bounds. In the implementation of the methods, the discrete Lagrange multiplier is locally eliminated, leading to a Nitsche-type method [3].

For the problems of a membrane and plate subject to solid obstacles, we present numerical results.

Joint work with Tom Gustafsson (Aalto) and Juha Videman (Lisbon).

References

1. T. Gustafsson, R. Stenberg, J. Videman. Mixed and stabilized finite element methods for the obstacle problem. SIAM Journal of Numerical Analysis 55 (2017) 2718–2744
2. T. Gustafsson, R. Stenberg, J. Videman. Stabilized methods for the plate obstacle problem. BIT– Numerical Mathematics (2018) DOI: 10.1007/s10543-018-0728-7
3. E. Burman, P. Hansbo, M.G. Larson, R. Stenberg. Galerkin least squares finite element method for the obstacle problem. Computer Methods in Applied Mechanics and Engineering 313 (2017) 362–374

We show that the elliptic genus of the higher rank E-strings can be computed based solely on the genus $0$ Gromov-Witten invariants of the corresponding elliptic geometry.

We discuss the application of Siegel paramodular forms to the counting of polar states in symmetric product orbifold CFTs.

We present microstate counting formulae for extremal black holes in the $N=2$ STU-model.