The project produced in 27 publications, which include 24 published and 2 accepted refereed papers (23 refereed journals and 3 refereed conference proceedings, including 7 papers written by students where the PI is not a co-author), and a PhD thesis. In addition, several preprints were posted in arXiv and submitted for publication.

 During the reported period the PI supervised 4 PhD students (Dan Kelleher defended in 2014 and obtained a postdoctoral position at Purdue University, and 3 other PhD students are expected to defend in 2016-17). The PI also supervised 3 postdoctoral fellows (Joe P. Chen and Leonard (Jay) Wilkins as UConn postdocs, and Michael Hinz as a Feodor Lynen Research Fellow, Alexander von Humboldt Foundation). The PI co-organized a Conference on Analysis and Probability on Fractals at Cornell University in 2014 and an AMS Special Sessions on Analysis, Probability and Mathematical Physics on Fractals; co-organized and supervised research by undergraduate students in the NSF REU programs; was an editor of the Journal of Fractal Geometry, EMS. Also the PI was involved in international cooperation: Universities of Bielefeld and Jena, Germany; Sapienza University of Rome, Italy; Technion, Israel. The PI gave several invited short lecture courses and research talks.

The broader impacts of the project included: integration of research and education, REU and other research with students, and innovative research-oriented mathematics courses; integrating diversity into NSF programs, projects, and activities; international cooperation; integration of various areas of mathematics, such as analysis, probability, group theory, geometry, PDEs with other sciences, such as physics, chemistry, biological sciences, computer science, and with engineering.

The PI published papers in the major mathematics journals: J.Eur.Math.Soc., Stochastic Process.Appl., Trans.Amer.Math.Soc., J.Funct.Anal., J.d'AnalyseMathematique, J.Noncommut.Geom., Math.Res.Lett., and in some of the top (mathematical) physics journals: Phys.Rev.Lett., J.Phys.A:Math. Theor., Phys.Rev.E, Lett.Math.Phys. The main results include the following.

With Michael Hinz the PI made a discovery how one can define (local) magnetic Schrodinger Hamiltonians and proved their (essential) self-adjointness for local Dirichlet forms and corresponding diffusions. This work opens wider possibilities for developing mathematical physics, both classical and quantum, on non-smooth spaces. Essentially we developed vector analysis for Dirichlet forms without approximation by smooth spaces, and without rectifiable curves. Furthermore, we started to develop the stochastic version of the classical differential cohomology on DMMSs. The motivation for this work came from different directions. On one hand, cohomologies appears in many equations of mathematical physics related to fluid dynamics and some areas of quantum physics. On the other hand, in order to claim that one has notions of differential geometry on an non-smooth DMMS, one needs to show that some analogs of the important classical theorems can be proved for such DMMSs. We proved the Hodge decomposition for differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. In this context the analog of the Hodge theorem says that the orthogonal complement to the space of all exact 1-forms coincides with the closed span of all locally harmonic 1-forms. Although in some sens...

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