Quanta Magazine

Quanta Magazine, from the Simons Foundation, has been publishing some excellent articles about mathematics.  It is not a research journal, so Mathematical Reviews doesn’t cover it.  Nevertheless, if you want to dig deeper into some of the mathematical issues discussed in their articles, MathSciNet is a great tool for doing so.

Photograph of Sylvia Serfaty. Photo by Stefan Falke, used with permission of Quanta Magazine

Sylvia Serfaty
Photo by Stefan Falke.
Used by permission of Quanta Magazine

Quanta has an interview by Siobhan Roberts with Sylvia Serfaty, a mathematician at the Courant Institute who work in analysis, PDEs, and mathematical physics.  Serfaty is by any measure a successful mathematician.  She publishes three or four articles per year in good journals.  She has written a successful book with Étienne Sandier.   She has won some great prizes.  Yet Serfaty does not consider herself a genius.  Moreover, she disputes the idea that you need to be a genius or a prodigy to do mathematics.  While there are geniuses in mathematics, they are rare.  She correctly points out that you don’t have to be in that thin slice to do interesting mathematics.  Rather, to be successful you have to be curious and persistent. Serfaty says, “You enjoy solving a problem if you have difficulty solving it. The fun is in the struggle with a problem that resists.”1

Serfaty is not alone in this point of view.  Terry Tao wrote on his blog:

Does one have to be a genius to do mathematics?

The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities.

Tao cites an article in New Scientist magazine that makes the same point: most people who do incredible work are not necessarily mutants.  Rather, for the most part, they find something that interests them deeply, then work very hard to make progress.

In the spirit of using MathSciNet to dig more deeply into an article in Quanta, below is a copy of the review of the book by Serfaty and Sandier mentioned in the interview.  In the meantime, I recommend Quanta for its articles on mathematics — and other things.

– – – – –

1 When my students used to complain that something was hard, I would tell them, “It wouldn’t be fun if it wasn’t hard.”  This went down better with the graduate students than the undergrads.

Sandier, Etienne(F-PARIS12); Serfaty, Sylvia(1-NY-X)
Vortices in the magnetic Ginzburg-Landau model.
Progress in Nonlinear Differential Equations and their Applications, 70. Birkhäuser Boston, Inc., Boston, MA, 2007. xii+322 pp. ISBN: 978-0-8176-4316-4; 0-8176-4316-8

This book presents a detailed and comprehensive account of the rigorous mathematical analysis of the Ginzburg-Landau model of superconductivity in two dimensions. In the planar Ginzburg-Landau model, the state of a superconductor with cross-section $\Omega\subset\Bbb R^2$ is described by a complex order parameter, $u\in H^1(\Omega;\Bbb C)$ and magnetic vector potential $A\in H^1(\Omega; \Bbb R^2)$, so that the magnetic field $h=\nabla \times A$ is oriented orthogonally to the plane. Assuming the superconductor is exposed to an external magnetic field of constant intensity $h_{\rm ex}$, the physically observable configurations $(u,A)$ should minimize, $$ G_\epsilon (u,A) = \int_\Omega \left\{ \frac12 |(\nabla – iA)u|^2 + {1\over 4\epsilon^2}(|u|^2-1)^2 + \frac12 (h-h_{\rm ex})^2\right\} dx. $$ Here $\epsilon>0$ is the reciprocal of the Ginzburg-Landau parameter. Most results in this book concern the singular limit $\epsilon\to 0$.

The largest part of the book concerns the structure of energy minimizers for external fields $h_{\rm ex}$ nearby the “lower critical field” $H_{c_1}\sim |\ln\epsilon|$, the smallest value of the external field at which vortices are observed. In fact, the book illustrates how complex and interesting the $\epsilon\to 0$ limit actually is, with different types of minimizers appearing depending on the order of $h_{\rm ex}-H_{c_1}$.

For $h_{\rm ex}$ close to $H_{c_1}$, $h_{\rm ex}- H_{c_1}=O(\ln|\ln\epsilon|)$, they prove that minimizers have finitely many vortices, the number being bounded in $\epsilon$. As $\epsilon\to 0$ these vortices will accumulate at specific points in the domain, determined by the minimum of $h$ for vortexless configurations. This result was first proven by the authors in [Calc. Var. Partial Differential Equations 17 (2003), no. 1, 17–28; MR1979114], but here they refine their result by blowing up around the limiting points. They obtain an asymptotic expansion of the critical fields at which additional vortices are produced, as well as a renormalized energy to determine the configuration of the vortices around the concentration point.

If $h_{\rm ex}-H_{c_1}=O(|\ln\epsilon|)$, they show that minimizers will have an unbounded number of vortices as $\epsilon\to 0$, and these vortices will spread out over the sample $\Omega$. They present the result of [E. Sandier and S. Serfaty, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 4, 561–592; MR1832824], proving that the suitably normalized vorticity measure and magnetic field $h$ converge to a solution to an obstacle problem for $h$. The result is presented here in the framework of gamma convergence.

The book presents some original, previously unpublished results for the regime where $\ln|\ln\epsilon| \ll h_{\rm ex}- H_{c_1} \ll |\ln\epsilon|$. In this limit, minimizers have an unbounded number of vortices which nevertheless accumulate at points, as in the first case above. After blowing up around these points, the authors prove that the normalized vortex interaction energy $\Gamma$-converges to a classical Gauss variation problem from potential theory.

Many other results are also presented, including a study of bifurcation branches of local minimizers with fixed numbers of vortices and some new results on vorticity measures for nonminimizing solutions of the Ginzburg–Landau equations.

Several new techniques (and improvements on recent methods) are introduced. The methods used in proof are based on sharp matching upper and lower bounds on the energy. Most results depend on improving lower bounds on the energy, and the authors derive new sharp versions of the “vortex ball” constructions (see [E. Sandier, J. Funct. Anal. 152 (1998), no. 2, 379–403; MR1607928] or [R. L. Jerrard, SIAM J. Math. Anal. 30 (1999), no. 4, 721–746 (electronic); MR1684723].) Another new technique introduced in the book combines the vortex ball construction with the Pohozaev identity to obtain a finer analysis of vortices. The Pohozaev balls method is instrumental in the analysis of configurations with finitely many vortices, and leads to a sharpening and simplification of classical results of F. Bethuel, H. R. Brezis and F. Hélein [Ginzburg-Landau vortices, Birkhäuser Boston, Boston, MA, 1994; MR1269538].

Given the prevalence of Ginzburg-Landau-type models in condensed matter physics, these techniques are likely to find many applications and extensions in other singularly perturbed problems with quantized singularities, such as Bose-Einstein condensates, gauge field theories (such as Chern-Simons-Higgs), ferromagnetism, and liquid crystals.

Reviewed by Stanley A. Alama

About Edward Dunne

I am the Executive Editor of Mathematical Reviews. Previously, I was an editor for the AMS Book Program for 17 years. Before working for the AMS, I had an academic career working at Rice University, Oxford University, and Oklahoma State University. In 1990-91, I worked for Springer-Verlag in Heidelberg. My Ph.D. is from Harvard. I received a world-class liberal arts education as an undergraduate at Santa Clara University.
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