The Soap Bubble Theorem and a pp-Laplacian overdetermined problem

We consider the $p$-Laplacian equation $-\Delta_p u=1$ for $1<p<2$, on a regular bounded domain $\Omega\subset\mathbb R^N$, with $N\ge2$, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature $H$ of $\partial\Omega$ is constant, then $\Omega$ is a ball and the unique solution of the Dirichlet $p$-Laplacian problem is radial. The main tools used are integral identities, the $P$-function, and the maximum principle.Comment: 18 pages, 0 figure

The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation

In this paper we construct the fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation approach. In addition, we characterize the powers of the fractional sub-Laplacian in the Heisenberg group, and as a byproduct we compute the $k$-th order momenta with respect to the heat kernel

From Empathy to Social Empathy: A Journey in a Diversity, Equity, and Inclusion Workplace Graduate Certificate

This study aimed to explore and understand the impact of the Graduate Certificate in Workplace Diversity, Equity, and Inclusion (DEI Graduate Certificate) on master’s and doctoral students’ social empathy (SE) development. The participants in this certificate engaged in several online activities that increased their knowledge of microaggressions, privilege, and unconscious bias while building their social empathy. This certificate provided various tools and skills to its participants so that they might value diversity, look at the world from another’s perspective, and start a dialogue to promote social change. This case study involved 19 participants. To answer our research questions, we used different data sources: a pre-course survey given at the beginning of the certificate, a social empathy index administered at the beginning of Unit 2, discussion board postings that were part of the course assignments, and one-on-one Zoom interviews. We identified six overarching themes: (a) slow down, (b) be uncomfortable, (c) self-reflection, (d) perspective-taking (PT), (e) contextual understanding of systemic barriers (CU), and (f) behavioral intentions. This research strongly suggested that the participants’ social empathy increased, which was indicated by the exceptional number of behavioral intentions set by them through the online transformative learning experiences of the DEI Graduate Certificate. As Segal (2017) and Mirra (2018) found in their studies, we also found that social empathy development can be developed in a DEI professional development certificate. We recommend university leaders, that is department chairs, professors, and professional development directors, include social empathy development in their curriculums. We agree with the data outcomes of this study in recommending that university leaders include social empathy development in their curriculums and within their professional development programs to promote positive social change

3d Modularity

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs.Comment: 119 pages, 10 figures and 20 table