Rob Jencks (Boston University)
Turing Patterns in a Spatially Discrete Zebrafish

• Zebrafish have been used as a model organism in many areas of biology, including the study of pattern formation. As such, many models have been proposed to describe this process incorporating various effects such as cell movement and context-dependent switching of cell behavior. In 2021, Konow et al proposed a set of simplified models, including a coupled ODE system describing the expected evolution of pigment cells on a discrete ring, similar to Turing’s original work. This talk will discuss the analysis of this model focusing on parameters for the number of cells, length of distant-neighbor interactions, and rates related to birth and death of different cells. Under appropriate, physically realistic conditions, a Turing bifurcation occurs and the model predicts patterns qualitatively similar to those in nature.

KEYNOTE: Kathryn Lindsey (Boston College)
The Thurston Set: Teapots, Dragons, and Rings of Fire

• The Thurston set is a strikingly beautiful, fractal-like subset of the complex plane whose geometry, topology, and arithmetic reveal deep connections across several mathematical domains. It bridges the theories of interval self-maps, iterated function systems, and power series with prescribed coefficient patterns. I will discuss recent progress in understanding its structure and the new insights it provides into these areas—particularly its implications for the Mandelbrot set. I will also highlight some open questions that remain mysterious.

Thejani Gamage (University of Massachusetts Amherst)
The optimal dividend problem under the continuous time diffusion model

• The theory of Reinforcement Learning can essentially be considered as an stochastic control problem, and thus it is natural to consider Reinforcement Learning as an approach to solve stochastic control problems. One of the approaches is to set up an auxiliary problem, known as the “entropy regularized exploratory control problem” that can be solved via two main techniques from Reinforcement Learning literature, Policy Iteration and Policy Evaluation. We consider the specific stochastic control problem, “the optimal dividend problem under the continuous time diffusion model with the dividend rate being restricted in a given interval [0, a]”, one of the well studied problems in actuarial science. Unlike the standard literature, we shall particularly be focused on the case when the parameters (e.g. drift and diffusion coefficients) of the model are not specified so that the optimal control cannot be explicitly determined. To approximate the optimal strategy, we follow the Reinforcement Learning approach. We shall first carry out a theoretical analysis of the entropy-regularized control problem and prove that the value function is the unique bounded classical solution to the corresponding HJB equation. This allows us to use a policy improvement argument, along with policy evaluation devices to construct approximating sequences of the optimal strategy.

Nicholas Arosemena (Brown University)
Computing the Identity Element in Chip-Firing Dynamics

• Chip-firing is a discrete dynamical system with rich combinatorial properties, originally studied in physics as the Abelian Sandpile Model. The critical group of a chip-firing system is well understood as a finite abelian group, but computing its identity element remains challenging. In this talk, we present a new method for determining the identity element, which appears as a pulse of height μ. This parameter μ and its multiples naturally arise in various graph invariants, including the largest invariant factor of the critical group. Our results extend to both traditional graphs and the (L,M) pairs introduced by Guzmán and Klivans.

Malavika Mukundan (Boston University)
Marked cycle curves for quadratic polynomials and rational maps

• Given any holomorphic dynamical system with a marked periodic cycle, we may track the changes in the marked cycle under perturbations of the system in some ambient parameter family. This gives rise to a marked cycle curve, which is a branched covering over the parameter family. In this talk, we discuss marked cycle curves of fixed period over two families: quadratic polynomials, and quadratic rational maps with a critical 2-cycle. The topology of these curves is highly related to the topology of the Mandelbrot set. In particular, we shall give cell-decompositions for these curves based on the combinatorics of special parameters in the Mandelbrot set.