Analysis Seminar: Differential Harnack Bounds, Fractional Heat Flow, and Geometry, part I

Speaker: Mikolaj Sierzega, GMU

Title: Differential Harnack Bounds, Fractional Heat Flow, and Geometry, part I 

Abstract:   Harnack bounds lie at the very heart of regularity theory for partial differential equations. In the case of the classical heat equation, they reach their most elegant and canonical form — the Hadamard–Pini bound. In the 1980s, these ideas were powerfully reimagined in the form of differential Harnack inequalities (also known as Li–Yau estimates), which have since become a cornerstone of modern analysis and played a central role in the resolution of the Poincaré Conjecture.

In recent years, there has been growing interest in fractional heat flows, which arise naturally in the modeling of anomalous diffusions and long-range interactions. It is only natural, then, to ask how the Hadamard–Pini and Li–Yau bounds might extend to this fractional setting. Despite much effort, a satisfactory generalization remains elusive — and it’s not even clear what form such an extension should take. The search for it, however, leads to a deeper understanding and reinterpretation of the classical Li–Yau theory itself.

In this two-part talk, I’ll discuss how the seemingly straightforward problem of estimating solutions to the fractional heat equation leads us into some unexpected areas — including optimal transport, entropy-maximizing flows, the calculus of variations, complex analysis, and Riemann–Finsler geometry. Along the way, we’ll revisit the classical results from a new perspective, outline a broader geometric picture that ties them together, and I’ll also present some concrete answers for the case of the fractional heat flow.

Date/Time: Friday, November 14, 11:30am

Location: Exploratory Hall, Room 4106 or Zoom