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.

Lecturers and abstracts of minicourses

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)

TBA

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

Schedule

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

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

Venue and transportation

The conference/masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen. See detailed instructions on how to reach Copenhagen and the conference venue.

Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation.

A journey planner in English is available.

More information on the "find us" webpage.

Accommodation

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.