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SUMMARY:From phase transitions to minimal submanifolds in low codimensions
  - Davide Parise (Imperial College London)
DTSTART:20260316T140000Z
DTEND:20260316T150000Z
UID:TALK245065@talks.cam.ac.uk
CONTACT:Zoe Wyatt
DESCRIPTION:Minimal submanifolds—critical points of the area functional
 —play a central role in geometric analysis\, with deep connections to di
 fferential geometry\, topology\, and the calculus of variations. A major b
 reakthrough in their existence theory came with Almgren’s min–max fram
 ework in the 1960s\, which provides a powerful but technically intricate a
 pproach to constructing minimal submanifolds in great generality.\n\nIn re
 cent years\, a new PDE-based perspective has emerged\, inspired by models 
 from phase transitions and superconductivity. The guiding principle is to 
 realize minimal submanifolds as limits of nodal sets of critical points of
  elliptic functionals. In this talk\, I will first give an overview of thi
 s variational–PDE framework. Using ongoing series of works with collabor
 ators as examples\, I will show how it can offer new conceptual insights a
 nd\, in some settings\, greater flexibility than classical min–max techn
 iques. In codimension one\, I will focus on the Allen–Cahn functional an
 d explain how it can be used to construct minimal hypersurfaces with free 
 boundary\, i.e. meeting the ambient boundary orthogonally.  We will then m
 ove to higher codimensions (specifically codimensions 2 and 3)\, where the
  theory is much less developed. Here\, I will discuss the abelian and non-
 abelian Yang–Mills–Higgs functionals as natural higher codimension ana
 logues and present recent progress in this setting. I will highlight other
  successes of this theory\, and point to some open problems along the way.
  The content of this talk is based on joint works with Martin Li\, Lorenzo
  Sarnataro\, Alessandro Pigati\, and Daniel Stern.\n\n
LOCATION:MR13
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