Overview

The 2025 Rocky Mountain Mathematics Consortium Summer School will bring together leading experts in analysis, partial differential equations, and their applications to summarize recent progress in these topics, exchange ideas towards the solution of open questions, and formulate and develop new problems and avenues of research.

Principal Lecturers

Eugenia Malinnikova - Stanford University

Zero sets of solutions to elliptic and parabolic equations

In these lectures we will consider solutions to parabolic and elliptic equations and discuss the doubling property and the frequency function. First, assuming a doubling inequality and analyticity of coefficients, we will obtain a sharp estimate for the Hausdorff measure of the zero set. Then, for general elliptic equations with Lipschitz coefficients,  we will discuss an upper bound of the size of the zero set. Finally, we will deal with parabolic equations with nonanalytic coefficients. The lectures are partly based on the following articles: F. Lin, Nodal sets of solutions to elliptic and parabolic equations, 1991, Comm Pure and Appl. Math; A. Logunov, Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Annals, 2018, Y. Huang and W. Juang, The nodal sets of solutions to parabolic equations, arXiv, 2024

Susanna Terracini - University of Turin

Schauder and boundary Harnack principles via degenerate elliptic equations

In this course, after revisiting the classical Schauder theory for elliptic equations, we will focus on two types of results. We will first address Schauder theory at the boundary for equations that are degenerate or singular over a portion of the boundary. Subsequently we will derive the boundary Harnack inequalities of the first order and higher orders. Finally we will examine the regularity of the ratio of solutions of elliptic equations sharing the same zero set.

Invited Speakers

Stefano Decio – ETH, Zurich, Switzerland

Alessandra De Luca - University of Turin, Italy

William Feldman - University of Utah

Benjamin Foster - Stanford University

Nicola Garofalo - Arizona State University

Connor Mooney - University of California at Irvine

Petronela Radu - University of Nebraska - Lincoln

Giorgio Tortone - University of Pisa, Italy

Zihui Zhao - The Johns Hopkins University

Jiuyi Zhu - Louisiana State University

Registration

The registration deadline for the 2025 RMMC Summer School is April 30, 2025.There is no registration fee, but registration is required.

https://forms.thecollege.asu.edu/RMMCSS2025-registration

Financial Support

We aim to provide financial support to all participants, including graduate students and recent Ph.D. recipients, covering local expenses and up to $500 for travel. However, funding is limited, and we may not be able to accommodate all requests.

If you would like to apply for financial support, please indicate this in the registration form. To receive full consideration, submit your registration and application by April 30, 2025. Priority will be given to graduate students, junior researchers, and participants without access to travel grants.

Note: Request for financial support should be accompanied by a brief CV and a Research Statement. Graduate students requesting support should have their advisor to send a short Letter of Support by email to [email protected].

Organizers

Donatella Danielli
Arizona State University

Sponsors

Funding for this program is supported by the National Science Foundation, School of Mathematical and Statistical Sciences at Arizona State University, and the Clay Mathematics Institute.