The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.
The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.
Institut de Mathématiques de Jussieu
Institute for Advanced Study
Richard Schwartz, Brown University
University of British Columbia and IAS
When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. In this talk we will give an overview of progress that has been made for surfaces with boundary, and contrast this with some recent results in higher dimensions. This is joint work with R. Schoen.
Nick Higham, University of Manchester, UK
There is a growing availability of multiprecision arithmetic: floating
point arithmetic in multiple, possibly arbitrary, precisions.
Demand in applications includes for both low precision (deep learning and
climate modelling) and high precision (long-term simulations and solving
very ill conditioned problems). We discuss
- Half-precision arithmetic: its characteristics, availability, attractions,
pitfalls, and rounding error analysis implications.
- Quadruple precision arithmetic: the need for it in applications, its
cost, and how to exploit it.
As an example of the use of multiple precisions we discuss iterative
refinement for solving linear systems. We explain the benefits of
combining three different precisions of arithmetic (say, half, single, and
double) and show how a new form of preconditioned iterative refinement can
be used to solve very ill conditioned sparse linear systems to high
Donatella Danielli, Purdue University
Obstacle problems play an ubiquitous role in the applied sciences, with applications ranging from linear elasticity to fluid dynamics, from temperature control to financial mathematics. In this talk we will show how seemingly different phenomena can be expressed in terms of the same mathematical model of obstacle type. We will also discuss some recent results concerning the regularity of the solution and of its free boundary. In particular, we will highlight the pervasive role played by some families of monotonicity formulas.
Francis Bonahon, USC
The convenient formula (X+Y)^n = X^n + Y^n is (unfortunately) frequently used by our calculus students. Our more advanced students know that this relation does hold in some special cases, for instance in prime characteristic n or when YX=qXY with q a primitive n-root of unity. I will discuss similar ``miraculous cancellations’’ for 2-by-2 matrices, in the context of the quantum group U_q(sl_2).
Eduardo Teixeira, University of Central Florida
The development of modern free boundary theory has promoted major knowledge leverage across pure and applied disciplines and in this talk I will provide a panoramic overview of such endeavor. The goal of lecture, however, will be to explicate how geometric insights and powerful analytic tools pertaining to free boundary theory can be imported to investigate regularity issues in nonlinear diffusive partial differential equations. This new systematic approach has been termed non-physical free boundaries, and in the past few years has led us to a plethora of unanticipated results.
Phil Gressman, University of Pennsylvania
In the 1970s, E. Stein and other mathematicians studying fundamental questions related to pointwise convergence of Fourier series discovered surprising new links between this very old problem and the geometry of submanifolds of Euclidean space. These discoveries paved the way for many of the questions at the forefront of modern harmonic analysis. A common element in many of these areas is the role of a strange sort of curvature condition which arises naturally from Fourier-theoretic roots but is poorly understood outside the extreme cases of curves and hypersurfaces. In this talk, I will discuss recent work which combines elements of Geometric Invariant Theory, Convex Geometry, Signal Processing, and other areas to shed light on this problem in intermediate dimensions.
John Voight, Dartmouth College
A Belyi map is a finite, branched cover of the complex projective line that is unramified away from 0, 1, and infinity. Belyi maps arise in many areas of mathematics, and their applications are just as numerous. They gained prominence in Grothendieck's program of dessins d'enfants, a topological/combinatorial way to study the absolute Galois group of the rational numbers.
In this talk, we survey computational methods for Belyi maps, and we exhibit a uniform, numerical method that works explicitly with power series expansions of modular forms on finite index subgroups of Fuchsian triangle groups. This is joint work with Jeroen Sijsling and with Michael Klug, Michael Musty, and Sam Schiavone.
Jose Maria Diego Rodriguez, Instituto de Fisica de Cantabria
Dark matter is arguably one of the main mysteries in modern physics. We know how much is there, we know where it is but we don't know what it is. Despite the numerous (and expensive) efforts on Earth to directly detect the alleged and elusive dark matter particle, experimental evidence remains as elusive as the dark matter particle itself. As of today, the strongest (and only) experimental evidence for dark matter still comes from astrophysical probes. One of such probes is gravitational lensing that can be used to map the distribution of dark matter on cosmological scales. I will briefly review the most popular candidates for dark matter and focus on our research that uses gravitational lensing to rule out some of these candidates.