Dissertation Completion Fellowship Award

The Dissertation Completion Fellowship Award (DCFA) offers financial support to PhD students in the final phase of their degree during the current academic year. DCFA applicants may request funding for up to two quarters. A key component of the application is a clear and detailed timeline that shows you are in the final stages of your program, actively writing your dissertation, and on track to graduate.

Per the terms of the award, recipients must graduate within the quarter for which the fellowship is granted and will not be eligible for further university employment afterward.
Both the Dissertation Advisor and Graduate Advisor must verify the student's graduation timeline and confirm their plan to complete the degree within the award period.

Graduate Funding
 

2024-2025 Award Recipients

The 2024-25 Dissertation Completion Fellowship Award winners are: Aaron Goodwin, Will Hoffer, and Rahul Rajkmar. Thank you so much, we truly appreciate your extensive support of the UCR math community this year.

Aaron Goodwin

Dissertation Title: Compact Moduli Spaces of Marked Plane Curves

Dissertation Advisor: Dr. Patricio Gallardo Candela, Assistant Professor

Will Hoffer

Research Synopsis: Can you hear the shape of a drum? How fast do snowflakes melt? What is a non-integer dimension? The interplay of these questions is central to the objective of this dissertation, namely, how do geometric properties of a shape relate to spectral quantities like heat and sound. Fractals are a type of shape with very rough and irregular boundaries, appearing in all sorts of natural shapes like trees, mountains, lightning, and snowflakes. In this work, we analyze fractals with the property of self-similarity, where each part of the fractal is a smaller copy of the shape at large, using the theory of complex dimensions. We prove that these complex dimensions, which describe the geometric oscillations present in the fractal, are determined by the scaling properties of self-similarity. Then, we analyze two types of quantities: the size of the interior of fractals near their boundary and the amount of heat flow into or out of a region with fractal boundary. Our main results are explicit formulae for these quantities in terms of the possible complex dimensions of the fractal, showing the relationship between geometry and heat content of self-similar fractals in dimension two and above.

Dissertation Title: Shape and Spectrum: On the Heat and Volume of Self-Similar Fractals

Dissertation Advisor: Dr. Michel L. Lapidus, Distinguished Professor

Website: https://willhoffer.com/

Rahul Rajkumar

Research Synopsis: Rahul "Raj" Rajkumar is a probability theorist studying stochastic processes in $p$-adic and local fields, investigating analogues of classical processes in these settings. With his advisor, David Weisbart, he has investigated the properties of Brownian motion analogues, as well as their approximation by scaling limits of random walks. As a Postdoctoral Fellow at Academia Sinica, Rahul will continue to explore generalizations of Brownian motion in the form of $p$-adic Dunkl processes, as well as Brownian motion in related spaces such as the rational adeles.

Dissertation Title: Brownian Motion in Non-Archimedean Spaces

Dissertation Advisor: Dr. David Weisbart, Associate Professor of Teaching

Website: https://rahulrajkumar.github.io/