# Archive | research

## 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…

## more on matrix-tree formula

Fix a weighted graph $$\Gamma=(V,E), w:E\to \mathbb{R}$$.

The Laplacian $$L$$ of $$\Gamma$$ is the symmetric matrix

$$L_{u,v}=\left\{ \begin{array}{ll} -w(uv)&\mbox{ if }u\neq v,\\ \sum_{u’\neq u} w(uu’)& \mbox{ if }u=v\\ \end{array}\right..$$

(Here we view the weights $$w$$ as formal variables.)

As we all know, any principal minor of $$L$$ equals the sum of the weights of spanning trees of $$\Gamma$$.

Another way to define the principal minor is as the determinant of the restriction the quadratic form given by $$L$$ to any of the coordinate hyperplanes.…

## alons enfants de la patrie

In collaboration with Cynthia Sung, Daniel Feshbach and bloodthirsty French people……

## 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).…

## Protected: lake wobegon academic publishing

There is no excerpt because this is a protected post.

## Protected: cache choice conundrum

There is no excerpt because this is a protected post.

## Fact of the day: Functions with given merge tree

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

Consider a Morse function on $latex f:\Real^n\to \Real$ with controlled behavior at infinity, – say, $latex f=|x|^2$ near infinity. Assume further that all critical values $latex a_1<a_2<\ldots<a_k$ are distinct and that all indices of critical points are $latex 0$ or $latex 1$. (Condition obviously holds in one variable.)

Clearly, there are many function that satisfy these conditions. In the (again, most transparent) univariate case, the enumeration of topological types of functions with given critical values is the subject of a nice thread of papers by Arnold on “snakes” (see, e.g.,…

## Protected: Curvilinear Origami

There is no excerpt because this is a protected post.