Scientific

The under representation of women, especially women of color, has been persistently well documented (see for example the data dashboard on www.womendomath.org). One reason that this is a problem is that it can be difficult for women to identify role models - this can in turn make it harder for women to envision their own success. During my career, I found it very helpful to learn the stories of women in STEM and to draw on aspects of their success to try to invent my own path. In this talk, I will retell twelve stories of women in STEM that influenced me. I can’t promise that the stories will be historically accurate, but I will try to say what I learned from the stories as I heard them and what lessons I hope others might take from them as well.

Topological insulators are materials that exhibit unique physical properties due to their non-trivial topological order. One of the most notable consequences of this order is the presence of protected edge states as well as closure of bulk spectral gaps, which is known as the bulk-edge correspondence.

In this talk, I will discuss the mathematical description of topological insulators and their related spectral properties. The presentation will begin with an overview of Floquet theory, Bloch bundles, and the Chern number. We will then examine the bulk-edge
correspondence in topological insulators before delving into our research on closure of bulk spectral gaps for topological insulators with general edges. This talk is based on a joint work with Alexis Drouot.

The locally symmetric diffusions, also known as Brownian motions, on generalized Sierpinski carpets were constructed by Barlow and Bass in 1989. On a fixed carpet, by the uniqueness theorem (Barlow-Bass-Kumagai-Teplyaev, 2010), the reflected Brownians motion on level $n$ approximation Euclidean domain, running at speed $\lambda_n\asymp \eta^n$ with $\eta$ being a constant depending on the fractal, converges weakly to the Brownian motion on the Sierpinski carpet as $n$ tends to infinity. In this talk, we show the convergence of $\lambda_n/\eta^n$. We also give a positive answer to a closely related open question of Barlow-Bass (1990) about the convergence of the renormalized effective resistances between two opposite faces of approximation domains. This talk is based on a joint work with Zhen-Qing Chen.

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.