Masterclass: Geometry of Phase Transitions

Copenhagen Centre for Geometry & Topology (GeoTop) at the University of Copenhagen


April 28 to May 2, 2025

This workshop/masterclass will focus on some important recent results on the analysis of the Allen-Cahn and Ginzburg-Landau equations as well as nonlocal minimal surfaces, with an emphasis on applications to minimal surfaces. The masterclass, comprising of three minicourses by experts in their field, is primarily aimed at graduate students and postdoctoral researchers in geometric analysis who already have a strong background in PDE and geometry, with some familiarity with minimal surfaces and geometric measure theory preferred. 

Limited travel funding will be available for participants, with more substantial support available for UK based participants via the INI. 

 

 

  • Joaquim Serra (ETH)
Course Title: Nonlocal minimal surfaces
Abstract: We will give an overview of nonlocal minimal surfaces in Euclidean space and on Riemannian manifolds.
Tentative outline: 
1) The Caffarelli-Roquejoffre-Savin regularity theory for minimizers.
2) The a priori curvature estimates for nonlocal minimal surfaces of finite index.
3) The robust regularity of stable s-minimal surfaces as s approaches 1.
4) Applications to minmax.
5) What are higher codimension nonlocal minimal surfaces?


  • Marco Guaraco (Imperial)
Course Title: The Allen-Cahn Equation and the Classical Plateau Problem
Abstract: We will look at how to approximate minimal surfaces with boundary using Allen-Cahn sections over real line bundles. This setup opens several directions, including how to construct surfaces from sections, questions of regularity near the boundary, the classification of model solutions, and the reverse problem of constructing sections from a given surface. We will spend some time on each of these topics.
Tentative outline:
1) A brief discussion of the Allen-Cahn equation and how it relates to minimal surfaces.
2) A brief discussion of the classical Plateau problem.
3) Double covers and homology classes.
4) Construction and stability of Allen-Cahn solutions that model minimal surfaces (1D model solution) and their boundaries (2D model solution).
5) Boundary regularity for minimisers in dimension 3 and in convex domains.
6) Open problems and future directions.



  • Alessandro Pigati (U of Bocconi)
Course Title: Approximating the area functional in codimension two: Yang-Mills-Higgs energy on U(1) bundles
Abstract: Constructing minimal submanifolds in arbitrary Riemannian ambients has always been one of the most popular and important problems in the calculus of variations and geometric analysis, giving birth in the 60s to the vast field of geometric measure theory. For low dimensional submanifolds (curves and surfaces), a fruitful approach is to parametrize the submanifold, viewing it as the image of a map, typically relaxing area with the Dirichlet energy. In a dual way, in low codimension (one or two), it is useful to view the submanifold as the zero set of a map, and to seek an energy which resembles the area of the level set from a variational standpoint. In codimension one, this was successfully achieved using the Allen-Cahn energy, which models phase transitions between two pure phases. In codimension two, together with Daniel Stern, we showed that a successful candidate is the Yang-Mills-Higgs energy on U(1) bundles, coming from the Ginzburg-Landau model of superconductivity, in a special self-dual regime; additional evidence for this was obtained in later works with Davide Parise and Daniel Stern. The aim of this minicourse is to survey this “level set approach” and some of the functionals which have been proposed so far, with particular emphasis on Yang-Mills-Higgs and the relevant techniques.
Tentative outline:
1) Brief survey on Allen-Cahn, Ginzburg-Landau, abelian Yang-Mills-Higgs
2) Tools from geometric measure theory: rectifiable sets, currents, varifolds
3) Generalized varifolds and Ambrosio-Soner rectifiability criterion
4) Positive and negative results for Ginzburg-Landau (with no magnetic field)
5) Brief review of connections on vector bundles and their curvature. Abelian Yang-Mills-Higgs energy. Regularity of critical points in the Coulomb gauge
6) Analogue of Modica’s inequality. Monotonicity and clearing out
7) Exponential decay of energy density away from the zero set. Quantization and the vortex equations. Integrality of the limit varifold
8) Related results: Gamma-convergence of this energy to area, convergence of its gradient flow to mean curvature flow, analogue of Savin's theorem

 

 

 

 

 

 

More detailed information will be available closer to the dates of the conference.

A tentative sketch:
- Monday, Tuesday, and Thursday will be full days, starting approximately at 10 a.m. and ending around 5 p.m., with 2 x 45 minutes lectures by each of the speakers, including lunch at noon and an afternoon tea.
- Wednesday and Friday will be short days with 1 x 45 minutes lectures by each of the speakers, starting at 9 a.m. and ending at 12 noon. The dinner will be on Thursday, and a reception will be held on Monday.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are Steel House Copenhagen (close to city centre), and CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandinavia (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and  Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.

 

 

 

 

 

 

 

 

 

 

 

 

Registration has now ended.
(Deadline for registration was April 7, 2025.)

If you're local and forgot to register, but still plan to be around,
please email Jan Tapdrup [jt@math.ku.dk] so that he can add you to the counts for e.g. coffee.

 

 

 

 

 

 

 

 

 

 

INI grant holder/UK contact: Huy The Nguyễn, h.nguyen@qmul.ac.uk 

Site administrator: Jan Tapdrup, jt@math.ku.dk 

Niels Martin Møller, nmoller@math.ku.dk 

Alex Mramor, almr@math.ku.dk