Abstract: Current research continues to debate the influence of fluctuating selection on genetic diversity within populations. In most previous models of fluctuating selection for studying genetic diversity, the distribution of selection coefficients is assumed to be symmetrical, meaning that the chances of having positive and negative selection coefficients are identical over time. These models predict that selective fluctuations reduce genetic diversity similar to the stochastic influence of genetic drift. Using stochastic simulations and analytical approaches based on diffusion approximations, we analyze the impact of fluctuating selection on genetic diversity when the distribution of selection coefficients over time is not symmetric, but is instead shifted to negative values. This captures the fact that new mutations are more likely to be deleterious. We show that, unlike the symmetric case, selective fluctuations can greatly increase genetic variation when new mutations are deleterious on average. We show that this phenomenon occurs because deleterious mutations that would be kept at low frequency in constant environment are able to transiently attain high frequencies in a changing environment. Our findings suggest that fluctuating selection could be an important force for generating genetic diversity even if it does not lead to long-term coexistence of alternate alleles.

Abstract: Discrete homotopy theory is a homotopy theory designed for studying graphs and for detecting combinatorial, rather than topological, holes.'' Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, subspace arrangements, and topological data analysis. In this talk, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us to extend the existing theory of universal covers to all graphs, and to prove a classification theorem for coverings. We also prove a discrete version of the Seifert--van Kampen theorem, generalizing a previous result of H. Barcelo et al. We then use it to solve the realization problem for the discrete fundamental group through a purely combinatorial construction.

One of the biggest open problems in the subject currently is determining whether the cubical nerve functor provides an equivalence between the discrete homotopy theory of graphs and the classical homotopy theory of spaces. We propose a new line of attack towards this open problem, by breaking it into more tractable problems comparing the homotopy theories of the respective $n$-types, for each nonnegative integer n. We also solve this problem for the first nontrivial case, $n=1$.