Colloquium

Olivia Chu, Bryn Mawr College

Abstract: Evolutionary dynamics shape social and biological systems across scales, from the evolution of multicellularity to the emergence of underground fungal symbioses to the formation and maintenance of animal groups and human societies. In these complex adaptive systems, small-scale interactions and associations can lead to emergent, large-scale phenomena. These interactions are often greatly influenced by various forms of heterogeneity, such as personality differences in human populations and variation in altruistic tendencies in animals. In this talk, I will present several models of complex social and biological systems, motivated by real-world phenomena and observations. These models are driven by evolutionary game theory, opinion dynamics frameworks, and agent-based modeling, and employ tools from stochastic processes, differential equations, and dynamical network analysis. I will discuss applications such as the evolution of cooperation, social group formation, the effects of environmental shocks on political opinions and activism, and altruistic tensions in social insect populations.

Thomas Koberda, University of Virginia

Abstract: It is a difficult and deep problem to understand countable groups that can act by homeomorphisms on compact manifolds, especially in dimension two or more. I will discuss some new ways of investigating groups acting on manifolds through ideas from mathematical logic. This talk will include work that is joint with Sang-hyun Kim and J. De la Nuez Gonzalez.

Milen Yakimov, Northeastern University

One of the major approaches to representation theory is via support theories. They come in different flavors and have different origins. The talk will be a gentle introduction to noncommutative tensor triangular geometry, which is designed as a universal approach to support varieties. No algebraic or categorical background will be assumed. The talk is based on joint works with Dan Nakano (Univ Georgia) and Kent Vashaw (UCLA).

Consider a point mass traveling in a polygon. It travels in a straight line, with constant speed, until it hits a side, at which point it obeys the rules of elastic collision. What can we say about this? When all the angles of the polygon are rational multiples of $\pi$, the travel of any trajectory is trapped in an invariant surface and we know a lot about it. In the case when at least one of the angles is irrational, it is much less understood, though from approximating with the rational case we know a couple of things. Kerckhoff, Masur and Smillie proved that there exists a billiard in an irrational polygon where the billiard flow is 'ergodic' with respect to the natural measure. This means that the amount of time the typical trajectory spends in a given box in the table (or even a cube in the three dimensional unit tangent bundle) is proportional to its area (or volume). This talk will present two results, both concerning a strengthening of ergodicity called ‘weak mixing’:

1) A strengthening of Kerckhoff, Masur and Smillie’s result: There exists a polygon where billiard flow is weakly mixing with respect to the natural volume on the unit tangent bundle.

2) A classification of the rational polygons where the billiard flow is weakly mixing with respect to the natural area on the invariant surfaces in the unit tangent bundle.

This talk will introduce ergodic theory and weak mixing and connect billiards in rational polygons to translation surfaces. Open questions will be presented. No previous knowledge of billiards, ergodic theory nor translation surfaces will be assumed.