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Biparametric persistence for smooth filtrations

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),

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dying for a cause

During the invasion phase of the Iraq war in 2003, the chances for a US soldier to die were 139/248000\(\approx\) 51 per 100,000.

In 2018, the chances for a woman in the US state of Georgia to die from causes related to her pregnancy are 46 per 100,000.…

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Protected: lake wobegon academic publishing

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Protected: cache choice conundrum

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MATH 199, Monday April 29: Around Arrow

We will be talking about Arrow’s Impossibility theorem, and its topological relatives. The span of the talk will be between American political landscape and elementary algebraic topology – feel free to peruse the links and their vicinity!

Follow-up exercise: try to model your decision process next time you are trying to converge with your friends or relatives, where to go out, or what to watch together. What is the topology of your movie-space?…

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Fact of the day: Functions with given merge tree


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.,…

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Mapping department strength

What is your math department strength? One of those rather polarizing questions, or an opportunity to gossip without guilt.

Still, important to understand where your department is on the map, if one wants to steer it in some particular direction, or to keep it where it is.

I wrote a short python script, using BeautifulSoup (thank you!) and campus-wide subscription to MathSciNet (thank you too!) – and downloaded the raw numbers: how many items (whichever MathSciNet is indexing: articles, books, theses,…) are authored by folks from ‘1-IL’ (our code) vs.…

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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.…

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Protected: Curvilinear Origami

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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?…

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