We study how to construct a stochastic process on a finite interval with given `roughness'. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization, we provide a better (path wise) estimator of Hölder exponent. Furthermore, we study the concept of (generalized) p-th variation of a real-valued continuous function along a sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ℓp-norm of the coefficients along a Schauder basis. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.

The onset of turbulence in pipe flows has captivated scientists and mathematicians for over a century. In this talk, we adopt a dynamical systems approach on an experimentally validated predator-prey model for the onset of turbulence. We apply a travelling wave ansatz to a PDE system and conduct a stability analysis of the resulting ODE system. Using fast-slow techniques introduced by Geometric Singular Perturbation Theory, we seek heteroclinic connections between the physically relevant saddles in the system. Finally, we discuss challenges and open research questions set by this study.

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The talk begins with a short introduction to reverse mathematics, a now classical framework that allows to compare the logical strength of results from various mathematical fields. It then discusses the emerging programme of strict reverse mathematics as promoted by Harvey Friedman. This is followed by a case study (work in progress with Nicholas Pischke and Patrick Uftring) that is inspired by though not entirely faithful to this new programme: By re-interpreting the real numbers, we obtain a new picture of analysis in reverse mathematics, which suggests that some theorems may not need the strong logical axioms that are classically associated with them.

This thesis studies coevolutionary games on adaptive networks in which agents are equipped with bounded memory and the ability to rewire. We examine the emergence of cooperation, the evolution of network topology, and the reward disparities that arise across the different experimental scenarios. We depart from the classical well-mixed and memoryless populations used in Classical Game Theory by considering finite populations of heterogeneous agents whose behavior depends on their randomly assigned strategy type. The model is implemented as an agent-based simulation on three representative network topologies: Watts–Strogatz, Barabási–Albert, and Erdős–Rényi. Across all scenarios, the results consistently show that network adaptivity and rewiring promote cooperation and increase the rewards of cooperative agents relative to those who tend to defect, primarily by facilitating the formation of cooperative clusters. The findings highlight the joint importance of memory and adaptive network structure in sustaining cooperation and suggest extensions for future research, including heterogeneous memory capacities and endogenous strategy change.

TBA ______________________________ Invited by Prof. Phan Thành Nam

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