School of Mathematics

Troubling Trends in ML Scholarship

Zachary Lipton
Carnegie Mellon University
February 22, 2019
Deep learning has led to rapid progress in open problems of artificial intelligence—recognizing images, playing Go, driving cars, automating translation between languages—and has triggered a new gold rush in the tech sector. But some scientists raise worries about slippage in scientific practices and rigor, likening the process to “alchemy.” How accurate is this perception? And what should the field do to combine rapid innovation with solid science and engineering?

Brief introduction to deep learning and the "Alchemy" controversy

Sanjeev Arora
February 22, 2019
Deep learning has led to rapid progress in open problems of artificial intelligence—recognizing images, playing Go, driving cars, automating translation between languages—and has triggered a new gold rush in the tech sector. But some scientists raise worries about slippage in scientific practices and rigor, likening the process to “alchemy.” How accurate is this perception? And what should the field do to combine rapid innovation with solid science and engineering?

Plateau’s problem as a capillarity problem

Francesco Maggi
The University of Texas at Austin; Member, School of Mathematics
February 21, 2019
We introduce a length scale in Plateau’s problem by modeling soap films as liquid with small volume rather than as surfaces, and study the relaxed problem and its relation to minimal surfaces. This is based on joint works with Antonello Scardicchio (at ICTP Trieste), Darren King and Salvatore Stuvard (at UT Austin).

Lorentzian polynomials

June Huh
Visiting Professor, School of Mathematics
February 19, 2019

Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The class of Lorentzian polynomials contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. I will give several combinatorial applications. No specific background will be needed to enjoy the talk. Joint work with Petter Brändén (https://arxiv.org/abs/1902.03719).

On minimizers and critical points for anisotropic isoperimetric problems

Robin Neumayer
Member, School of Mathematics
February 19, 2019

Anisotropic surface energies are a natural generalization of the perimeter functional that arise in models in crystallography and in scaling limits for certain probabilistic models on lattices. This talk focuses on two results concerning isoperimetric problems with anisotropic surface energies. In the first part of the talk, we will discuss a weak characterization of critical points in the anisotropic isoperimetric problem (joint work with Delgadino, Maggi, and Mihaila). 

An Application of a Conjecture of Mazur-Tate to Supersingular Elliptic Curves

Emmanuel Lecouturier
Tsinghua University
February 14, 2019

In 1987, Barry Mazur and John Tate formulated refined conjectures of the "Birch and Swinnerton-Dyer type", and one of these conjectures was essentially proved in the prime conductor case by
Ehud de Shalit in 1995. One of the main objects in de Shalit's work is the so-called refined $\mathscr{L}$
invariant, which happens to be a Hecke operator. We apply some results of the theory of Mazur's
Eisenstein ideal to study in which power of the Eisenstein ideal $\mathscr{L}$ belongs. One corollary of our

Isoperimetry and boundaries with almost constant mean curvature

Francesco Maggi
The University of Texas at Austin; Member, School of Mathematics
February 12, 2019
We review various recent results aimed at understanding bubbling into spheres for boundaries with almost constant mean curvature. These are based on joint works with Giulio Ciraolo (U Palermo), Matias Delgadino (Imperial College London), Brian Krummel (Purdue), Cornelia Mihaila (U Chicago), and Robin Neumayer (Nothwestern and IAS).

Min-max solutions of the Ginzburg-Landau equations on closed manifolds

Daniel Stern
Princeton University
February 12, 2019
We will describe recent progress on the existence theory and asymptotic analysis for solutions of the complex Ginzburg-Landau equations on closed manifolds, emphasizing connections to the existence of weak minimal submanifolds of codimension two. On manifolds with nontrivial first cohomology group, our results rely on new estimates for the Ginzburg-Landau energies along paths of maps connecting distinct homotopy classes of circle-valued maps, which may be of independent interest.