Scientific

Borsuk’s number b(n) is the smallest integer such that any set of diameter 1 in the n-dimensional space can be covered by b(n) sets of a smaller diameter. Exponential upper bounds on b(n) were first obtained by Shramm (1988) and later by Bourgain and Lindenstrauss (1989).

To obtain an upper bound on b(n), Bourgain and Lindenstrauss provided exponential bounds (both upper and lower) in Grünbaum's problem – the problem of determining the minimal number of open balls of diameter 1 needed to cover a set of diameter 1. On the other hand, Schramm provided an exponential upper bound on the illumination number of n-dimensional bodies of constant width. In 2015 Kalai asked if there exist n-dimensional convex bodies of constant width with illumination number exponentially large in n.

In this talk I will answer Kalai’s question in the affirmative and provide a new lower bound in the Grünbaum’s problem. This talk is based on a joint work with Andriy Bondarenko and Andriy Prymak.

Biological populations are responding to climate-driven habitat shifts by either adapting in place or moving in space to follow their suitable temperature regime. The shifting speeds of temperature isoclines fluctuate in time and empirical evidence suggests that they may accelerate over time. We present a mathematical tool to study both transient behaviour of population dynamics and persistence within such moving habitats to discern between populations at high and low risk of extinction. We introduce a system of reaction–diffusion equations to study the impact of varying shifting speeds on the persistence and distribution of a single species. Our model includes habitat-dependent movement behaviour and habitat preference of individuals. These assumptions result in a jump in density across habitat types. We build and validate a numerical finite difference scheme to solve the resulting equations. Our numerical scheme uses a coordinate system where the location of the moving, suitable habitat is fixed in space and a modification of a finite difference scheme to capture the jump in density. We apply this numerical scheme to accelerating and periodically fluctuating speeds of climate change and contribute insights into the mechanisms that support population persistence in transient times and long term.

Vertex operator algebras are the symmetry algebras of two dimensional conformal field theory. In a famous series of papers, Kazhdan and Lusztig proved an equivalence between particular semisimple categories of modules over affine Lie algebras and quantum groups, the former of which can also be realized as modules over a corresponding vertex operator algebra. Such equivalences between representation categories of vertex operator algebras and quantum groups are now broadly referred to as the Kazhdan-Lusztig correspondence. There has been substantial research interest over the last two decades in understanding the Kazhdan-Lusztig correspondence for vertex operator algebras with non-semisimple representation theory. In this talk I will present an overview of this research area and discuss recent results and future directions.

Stochastic parameterizations (SMCM) are continuously providing promising simulations of unresolved atmospheric processes for global climate models (GCMs). One of the features of earlier SMCM is to mimic the life cycle of the three most common cloud types (congestus, deep, and stratiform) in tropical convective systems. In this present study, a new cloud type, namely shallow cloud, is included along with the existing three cloud types to make the model more realistic. Further, the cloud population statistics of four cloud types (shallow, congestus, deep, and stratiform) are taken from Indian (Mandhardev) radar observations. A Bayesian inference technique is used here to generate key time scale parameters required for the SMCM as SMCM is most sensitive to these time scale parameters as reported in many earlier studies. An attempt has been made here for better representing organized convection in GCMs, the SMCM parameterization is adopted in one of the state-of-art GCMs namely the Climate Forecast System version 2 (CFSv2) in lieu of the pre-existing simplified Arakawa–Schubert (default) cumulus scheme and has shown important improvements in key large-scale features of tropical convection such as intra-seasonal wave disturbances, cloud statistics, and rainfall variability. This study also shows the need for further calibration the SMCM with rigorous observations for the betterment of the model's performance in short term weather and climate scale predictions.

Idempotent semi-rings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semi-rings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semi-rings and modules over them.

Mini-Course

This lecture is the second part of a mini-course, please see also

• Lecture 1