help wanted

$\def\Real{\mathbb{R}}\def\Proj{\mathbb{P}}$

This page is intended to be a collection of problems from faculty member interested to start a conversation with a graduate student.
These mini-ads might lead to a reading course or advising, or nothing at all, – but at the very least, a yet another window into what our faculty interested in advising have to offer.

richard sowers

• TRACES OF THE UKRAINE CONFLICT IN OPEN STREET MAP.
OpenStreetMap is a crowdsourced database of transportation-related points of interest. How does conflict affect Open Street Map, as the adversaries try to achieve their aims, respond to evolving events, and the international community deals with migration resulting from the conflict? We want to look at the Russian invasion of Ukraine in 2022. Who is making changes and what is being changed?
• TOPOLOGICAL DATA ANALYSIS AND BIODIVERSITY
Topological data analysis is a way of looking at data from the standpoint connectivity and its extensions. It has found applications in a wide variety of fields [1]. Can we use Topological Data Analysis to study biodiversity, one of the pressing questions of this generation? The Global Biodiversity Information Facility [2] contains records of wildlife observations. We would like to extend some of the ideas of topological data analysis to describe biodiversity and compare biodiversity in different places and times.
• BOUNDARY CONDITIONS FOR KOLMOGOROV DIFFUSIONS
In 1934, Kolmogorov formalized the dynamics of a particle subjected to a white noise force. Kolmogorov diffusions are now a familiar part of the landscape of stochastic processes. The partial differential equation corresponding to these dynamics is
$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial y^2}+y\frac{\partial u}{\partial x}$
A surprising lacuna in the theory is the effect of boundary conditions; what happens if we impose boundary conditions in space. A first step, imposing Dirichlet boundary conditions in position was achieved recently. There are a number of natural next questions, e.g., Neumann and Robin conditions and also more complicated boundary geometries.

yuliy baryshnikov

• CONFIGURATION SPACES OF LARGE PARTICLES
Consider the configuration space of $$n$$ distinct, nonoverlapping balls of radius $$r$$ in a domain $$D\subset \Real^d$$. When $$r$$ is too large, the space is empty, and when it is very small, the space is homotopy equivalent to the standard configuration space of $$n$$ particles in $$\Real^d$$. When $$n$$ is somewhere in between, the topology of the resulting configuration space becomes convoluted.
One daunting question is whether the configuration space is connected in the regime of small density (when $$nr^d$$ is small compared to the volume of $$D$$). Not in general (think of $$D$$ being a unit disk in the plane, and balls aligning along its boundary in a ball bearing fashion). But what about rectangular boxes?
The equivalent problem, – whether globally stable configurations of disks of arbitrarily small density in rectangles $$D$$ exist – is wide open even in dimension $$2$$ despite quite a bit of attention to it from statistical mechanics community (after all, Boltzmann-Ehrenfest ergodicity depends on it).
Another interesting question is whether a construction of a disconnected configuration space in the low density regime is possible in strictly convex 3D domains (generalizing the ball bearing planar example).
• REAL RANDOM ENUMERATIVE GEOMETRY AND THE SQUARE ROOT PHENOMENON.
We all know that a generic complex plane curve of degree $$d$$ has $$3d(d-2)$$ inflections. Since Klein it is also known what the (strict) upper bound on the number of real inflections in real plane curves of degree $$d$$ is: $$d(d-2)$$, one third of the number of complex inflections.
But what about the average number?
Turns out, it is exactly $$\sqrt{3d(d-2)}$$: the expected real number is the square root of the (constant) complex one.
Another famous example of this square root phenomenon is the result of Shub-Smale that the expected number of solution of a system of $$n$$ random homogeneous equations in $$\Proj V$$ of degrees $$d_1,\ldots,d_n$$, drawn from the corresponding Kostlan ensembles is equal to
$\sqrt{d_1\cdots d_n}.$
Of course, such a perfect matching does not happen always: say the expected number of lines on the surfaces of degree $$(2n-3)$$ in the $$n$$-dimensional projective space, for the Kostlan ensemble, is just asymptotically growing as the square root of its complex counterpart.
It should be of interest to find when the square root heuristic is precise.
The natural candidate is the number of godrons (generalizations of cusps; the name is thanks to Kergosien and Thom), i.e., the points on a surface in 3D projective space at which its projective dual has a swallowtail singularity.
It is known that the number of such points in a generic surface of degree $$d$$ over $$\mathbb{C}$$ is $$2d(d-1)(11d-24)$$ (see, e.g. here). What is it over the reals?