Studying pathwise quadratic variation---and its extensions---is the focus of this poster presentation. This allows us to obtain a model-free formulation of mathematical finance and extend the current state-of-the-art in volatility modelling established by Gatheral et al. Gatheral's empirical insight is that, across a wide array of assets, volatility processes are "rougher" than Brownian motion and thus not amenable to the theory of stochastic calculus of continuous semimartingales.

This poster will address Gatheral's findings and non-normality of asset processes by presenting the following: (i) A generalisation of the quadratic roughness property recently discovered by Cont and Das to arbitrary $p$-roughness; (ii) A discussion on the properties of the Takagi-van der Waerden functions, a class of fractal functions that serve as rough integrators, in the context of $p$-roughness.

Ultimately, the hope is that the pathwise approach helps us present the current state of volatility research in an elementary fashion.

Also, for every finite group $G$ and for every $\eta \in \left(0, \frac{1}{\operatorname{card} (G)}\right)$, we construct a simple separable unital AH algebra $A$ with stable rank one and a strictly approximately inner action $\alpha \colon G \to \operatorname{Aut} (A)$ such that:

(1) $\alpha$ is pointwise outer and doesn't have the weak tracial Rokhlin property.

(2) $\operatorname{rc} (A) =\operatorname{rc} \left(A \rtimes_{\alpha} G\right)= \eta$.

We present a novel model for the spread of ecosystem engineers as a free boundary problem. We derive the semilinear parabolic equation from an individual random walk model. The Stefan condition for the moving boundary is replaced by a biologically derived two-sided condition that models the movement behavior of individuals at the boundary as well as the process by which the population moves the boundary to expand their territory.

We prove the local existence of solutions for the model. We assign a convex functional to this problem so that the evolution system governed by this convex potential is exactly the system of evolution equations describing the above model. We shall then apply variational and fixed-point methods to deal with this free boundary problem.

We show that, for $q\in[0,1]$, the height of $T_{n,q}$ is asymptotically ${(1+o(1))(c^* \log n + n(1-q))}$ in probability. This yields three regimes of behaviour for the height of $T_{n,q}$, depending on whether $n(1-q)/\log n$ tends to zero, tends to infinity, or remains bounded away from zero and infinity. In particular, when $n(1-q)/\log n$ tends to zero, the height of $T_{n,q}$ is asymptotically of order $c^*\log n$, like it is for random binary search trees. Finally, when $n(1-q)/\log n$ tends to infinity, we prove stronger tail bounds and distributional limits for the height of $T_{n,q}$.

We give tight upper and lower bounds for the $2$-limited broadcast domination of Cartesian products of paths. The upper bounds are established by explicit constructions. The methods to obtain the lower bounds utilize the dual of $2$-limited broadcast domination, $2$-limited multipacking.

We consider a model for population dynamics in a habitat consisting of two homogeneous one-dimensional patches in a coupled ecological reaction diffusion equation. The movement between patches is incorporated into the interface conditions. We establish the existence, uniqueness, and global asymptotic stability of the steady state. Then we explore how the qualitative properties of the steady state depend on movement behavior.

We apply our analysis to a previous result where it was shown that a randomly diffusing population in a continuously varying habitat can exceed the carrying capacity at steady state. We clarify the role of nonrandom movement in this context. In particular, we determine conditions on movement rates and patch preference, so that the steady-state density exceeds the carrying capacity.