In this workshop we will discuss techniques to facilitate group work, and make discussion sessions more active. We will also introduce tasks aimed at actively involve students in various phases of the learning process (e.g., introduce/explore/review a topic, learn the steps to solve a particular problem or the lay out of a proof).

We describe a number of related questions at the interface of set theory and homology theory, centering on (1) the additivity of strong homology, and (2) the cohomology of the ordinals. In the first, the question is, at heart: To how general a category of topological spaces may classical homology theory be continuously extended? And in the tension between various potential senses of continuity lie a number of delicate set-theoretic questions. These questions led to the consideration of the Cech cohomology of the ordinals; the surprise was that this is a meaningful thing to consider at all. It very much is, describing or suggesting at once (i) distinctive combinatorial principles associated to the nth infinite cardinal, for each n, holding in ZFC, (ii) rich connections between cofinality and dimension, and (iii) higher-dimensional extensions of the method of minimal walks.

 In this talk, we first discuss existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representation. We then talk about sharp two-  sided estimates for fundamental solutions of general time fractional equations in metric measure spaces. This is a joint work with  Zhen-Qing Chen(University of Washington, USA), Takashi Kumagai (RIMS, Kyoto University, Japan) and Jian Wang (Fujian Normal University, China).

In this talk, I will discuss recent results produced with co-authors Ivan Blank (KSU) and Brian Benson (UCR) regarding a formulation of the Mean Value Theorem for the Laplace-Beltrami operator on smooth Riemannian manifolds. We define the sets upon which mean values of (sub)-harmonic functions are computed via a particular obstacle problem in geodesic balls. I will thus begin by discussing the classical obstacle problem and then an intrinsic formulation on manifolds developed in our recent paper. After demonstrating how the theory of obstacle problems is leveraged to produce our Mean Value Theorem, I will discuss local and global theory for our family of mean value sets and potential connections between the properties of these sets and the geometry of the underlying manifold.

A reaction-diffusion initial-boundary problem with a Caputo time derivative of order $\alpha\in (0,1)$ is considered. The solution of such a problem is discussed; it is shown that in general the solution has a weak singularity near the initial time $t=0$, and sharp pointwise bounds on the derivatives of this solution are derived. These bounds are used in a new analysis of the standard L1 finite difference method for the time derivative combined with a standard finite difference approximation for the spatial derivative. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading to solve the problem numerically.