Archive | problems

cyclicity and meditation

Ben Zimmerman, a neuroscientist working with us on the Slow Cortical Waves project, published a wonderful medium post reflecting on this work and much much more.

…for some reason the “default-mode” of the brain, what the mind automatically does at rest, seems to be to imagine and plan — to linger in the future or the past. This requires processing information from high-level semantic representations backwards to their constituent sensory components…

simulating the opening

Will the campus reopen in Fall?

Whether the schools will reopen in Fall is now a matter of prediction markets bets, but the planning at UofI is already underway.

The focus of the planning is, understandably, on the students, staff and faculty safety. Yet there is an aspect of the process going beyond the campus.

Indeed, UIUC is a primary campus of a large state school with a significant fraction of the students from Illinois, and, more to the point, students returning home on a regular basis.…

Introduction

As Covid-19 takes over the country, many organizations move their teams to work from home. Often, it is necessary to keep an office presence. This can be done in various ways: split your team into smaller units and let them alternate days, or weeks, or do completely random assignments (essentially, toss a coin for who will be in the office three days from now), etc. Or, one can abandon the fixed teams, and shuffle employees, again on a random or periodic basis…

These scenarios are apriori quite different in terms of their impact on infection propagation. …

Biparametric persistence for smooth filtrations

$$\def\Real{\mathbb{R}} \def\phd{\mathtt{PH}} \def\CAT{\mathtt{CAT}}$$
The goal of this note is to define the biparametric persistence diagrams for smooth generic mappings $$h=(f,g):M\to\Real^2$$ for smooth compact manifold $$M$$. Existing approaches to multivariate persistence are mostly centered on the workaround of absence of reasonable algebraic theories for quiver representations for lattices of rank 2 or higher, or similar artificial obstacles.

Singularities of mappings into the plane

We will rely on the standard facts about generic smooth mappings into two-dimensional manifolds: for such mappings, the set of critical points is a smooth curve $$\Sigma$$ in $$M$$, which is immersed outside of a finite number of pleats: near generic point of $$\Sigma$$, there are local coordinates on $$M$$ in which the mapping is locally given by
$y_1=x_1, y_2=q(x_2,\ldots,x_m)$
(folds), and near isolated points of the curve of critical points, in some coordinates the mapping is given by
$y_1=x_1, y_2=x_2^3+x_1x_2+q(x_3,\ldots,x_m),$
(pleats).…

There is no excerpt because this is a protected post.

Protected: cache choice conundrum

There is no excerpt because this is a protected post.

Shuffling the sheep

$$latex \def\xx{\mathbf{x}}\def\Real{\mathbb{R}}$$

We consider the problem of flock control: what are the natural constraints on the steering agents in reconfiguring an ensemble of agents?

Our basic setup is following: the agents are modeled as points $latex \xx=(x_1,\ldots,x_n), x_k\in X, k=1,\ldots, N$, with $latex X$ some terrain (for simplicity, we constrain ourselves here to Euclidean plane, but in general it can be some reasonably tame subset of a manifold). Quasi-static behavior is assumed, under which the system, for each value of the controlling parameters quickly settles into (locally minimizing) the equilibrium whose basin of attraction contains current position.…

Protected: Curvilinear Origami

There is no excerpt because this is a protected post.

Fact of the day: Brownian centroids

$latex \def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\def\xx{\mathbf{x}}\def\ex{\mathbb{E}}$

Consider $latex n$ points in Euclidean space, $latex \xx=\{x_1,\ldots, x_n\}, x_k\in \Real^d, n\leq d+1$.

Generically, there is a unique sphere in the affine space spanned by those points, containing all of them. This centroid (which we will denote as $latex o(x_1,\ldots,x_n)$) lies at the intersection of the bisectors $latex H_{kl}, 1\leq k\lt l\leq n$, hyperplanes of points equidistant from $latex x_k, x_l$.

Assume now that the points $latex x_k,k=1,\ldots,n$ are independent standard $latex d$-dimensional Brownian motions. What is the law of the centroid?…

Fact of the day: Condorset domains, tiling permutations and contractibility

$$\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}$$

Consider a collection of vectors $$e_1,\ldots,e_n$$ in the upper half-plane, such that $$e_k=(x_k,1)$$ and $$x_1>x_2\gt \ldots \gt x_n$$. Minkowski sum of the segments $$s_k:=[0,e_k]$$ is a zonotope $$\Z$$. Rhombus in this context are the Minkowski sums $$\Z(k,l)=s_k\oplus s_l, 1\leq k\lt l\leq n$$ of pairs of the segments. Tilings of $$\Z$$ by $$n\choose 2$$ rhombuses $$\Z(k,l)$$ define also an oriented graph (formed by the edges of the rhombuses, and oriented up in the plane).…