\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.

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).…

\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.

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). Here \(q\) is a quadratic form of the remaining variables (absent in the pleat case, if \(m=2\).

The image of the critical curve \(\Sigma\) is a curve in the \((f,g)\)-plane called *contour*; generically, \(h\) is an immersion of \(\Sigma\) outside of the pleats; at the pleat points, the contour has cusps.

(The contour on the left is a – rotated and somewhat distorted – rendition of Chicago Millenium Park’s Bean sculpture, with the cusps made visible.)

As the coordinates on the \(f,g\) plane are given, more special points emerge: namely those where either of the functions has a critical point. We will be referring to these special points as V- (vertical) or H- (horizontal) points.

These H- and V-points sit on the critical curve \(\Sigma\). Generically, the functions \(f,g\) are Morse, and the contour has non-vanishing curvature at them. (This follows from the fact that at a critical point of, say \(f\) the restriction of Hessian matrix of \(f\) to the hyperplane \(\{dg=0\}\), is nondegenerate.)

The V- and H-points partition the critical curve into segments which map to the segments of the contour curve where its slope is \(\neq 0,\infty\). We will refer to the segments where the slope is negative as Pareto points (for obvious reasons).

We augment the Pareto segments by attaching at their boundaries the rays (referred to as *extension rays*) pointing up (at V-points) or right (at H-points). We will call the V- or H-points where this attachment results in \(C^1\) curve, the *pseudosmooth* points; those points where the curve loses smoothness, the *pseudocusps*.

We will refer to the resulting union of Pareto segments and extension rays as the *life contour* for the filtration defined by the pair \(f,g\).

At the points of Pareto segments one can coorient the contour, declaring positive the side where both\(f\) and\(g\) can increase. This allows one to define the *index* of a point \(x\) on the smooth part of Pareto segment: choose a function of \(f,g\) vanishing on contour near the point, and increasing in the positive direction; this function has a (degenerate) critical point at \(x\); its index is the index of the point \(p\).

The index at extension rays is defined just as the index of the critical point of \(f\) or \(g\), where the ray was attached to a Pareto segment.

It is easy to see that the index is constant along the smooth parts of the Pareto segments, and jumps at cusps and pseudocusps in a predictable way: the higher with respect to \(f,g\) point has higher index.

We will be referring to the (Pareto, i.e. such that the branches of the contour involved have negative slopes) pseudocusps, cusps and the double points *of the same index*) as *obstacles*.

Consider an *increasing curve* \(\gamma:\Real\to\Real^2\) in the \(f,g\)-plane (this means that both functions strictly increase along the curve). To fix the gauge, we will assume that the curve is parameterized by \(f+g\).

An increasing curve \(\gamma\) defines a usual \(I\)-indexed filtration of \(M\), and correspondingly, persistent homologies and persistent diagrams \(\phd_k(\gamma), k=0,\ldots,m\), which we interpret as a collections of distinguishable points in planes \(\{b\lt d\}\).

(Remark that the approach to multiparametric persistence through restriction to increasing *straight* lines has been considered by several authors. Here we allow arbitrary increasing curves.)

The main (somewhat tautological) result of this note is

Theorem: For the path-connected collection of increasing curves in the plane avoiding obstacles, there is a section of the space of persistent diagrams: that is for any two such curves \(\gamma,\gamma’\), there is an identification

\[

I(\gamma’,\gamma):\phd_*(\gamma)\to\phd_*(\gamma’),

\]

of the bars in the persistent diagrams corresponding to each of the curves, and these identifications are consistent:

\[

I(\gamma”,\gamma’)I(\gamma’,\gamma)=I(\gamma”,\gamma).

\]

The figure below shows six homotopy non-equivalent increasing paths avoiding the obstacles, – five green and the sixth, blue.

The bars shown on the lower left (recall that the curves are parameterized by \(f+g\) correspond to the filtration that the blue curve defines: the births and deaths happen where the curve intersects the life contour.

One can readily see what happens when one moves from one homotopy class to another, across a (pseudo)cusp with the indices of branches \((k,k+1)\): a bar in \(\phd_k\) appears or disappears.

As the curve crosses a self-intersection point of the life contour, where the branches of equal indices cross, one has a less prominent event, where the charges in the persistence diagram \(\phd_k\) bounce of a common vertical or horizontal line (we can call the corresponding bars *interacting*).

This result prompts question about the structure of the space of increasing paths in the plane avoiding a finite collection of obstacles.

Theorem: the connected components of space of increasing curves in the plane avoiding a finite set of obstacles \(o_1,\ldots,o_k\in\Real^2\) are contractible, and their number is equal to the number of chains of obstacles, i.e. subsets \(o_{i_1}\prec\ldots o_{i_l}\), where \(o_m\prec o_n\) is the vector ordering of the points.

Adjacency of these cells is in itself an interesting invariant of the overall structure. Namely, for any collection of obstacles, consider the set of increasing paths passing through those obstacles. Each of these closed strata is manifestly contractible, as is (which is easy to prove the same way the Theorem above is proven) each connected component of the set of the increasing paths passing only through given set of obstacles.

Then the resulting stratification of the space of increasing paths is dual to a \(\CAT(0)\) cube complex: the vertices of the complex correspond to the connected components of teh set of increasing paths avoiding obstacles, and the sets \(h_k\) of increasing paths passing through obstacle \(o_k\) form the hyperplanes (for the nomenclature, see Sageev).

We will be referring to this cube complex as the *characteristic complex* of biparametric persistence structure defined by \(f,g\).

**I would stipulate that the characteristic complex is the key descriptor of biparametric persistence in smooth category.**

This note is but an introduction to the overall problem of understanding the nature of the characteristic complex for biparametric persistences. Here are a few research directions we hope to purse:

- How the characteristic complex changes under the small perturbations of the pair \(f,g\)? It is clear that cubes with all edges large survive (or cannot emerge out of nothing):
- Interacting bars: provide a combinatorial characterization in terms of the life contour.
- It is quite immediate that the characteristic complex survives smooth perturbations of \((f,g)\). As one deforms the pair, the structure of the characteristic complex changes in a predictable way (some vertices can appear/disappear; pairs of hyperplanes can appear/disappear etc). A study of these transformation would provide a good handle on the relation of the patterns of the characteristic complex.
- Relation of the characteristic complex and the invariants of biparametric persistence (as defined here or here) would be, obviously, of interest.

This note is an outline of my talk at the conference on Geometric Data Analysis at U Chicago in May 2019; written up while at IMI at Kyushu University.

My understanding is that Peter Bubenik and Mike Catanzaro were also considering biparametric persistence for smooth maps.

]]>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.…

]]>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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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?…

]]>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?

]]>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., here), relating them to enumeration of ramified coverings of Riemannian spheres, up-down sequences and other cute objects.…

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., here), relating them to enumeration of ramified coverings of Riemannian spheres, up-down sequences and other cute objects.

We are interested however in a somewhat finer enumeration. Notice that under our index condition, the sublevel sets $latex U(c)=\{f\leq c\}$ for non-critical $latex c$ are collections of topological $latex n$-disks. When increasing $latex c$ crosses a value corresponding to a critical point of index $latex 0$, a new disk is born; when the index is $latex 1$, two disks merge (an alternative would be to generate an element in 1-dimensional homology in the sublevel set for which we don’t have any ammunition – critical points of index 2 – to kill later on).

Combining these births of components and their merges results in a graph which is referred to as “merge tree” and coincides, in dimension $latex n\gt 1$ with the so-called Reeb (a.k.a Reeb-Kronrod) tree.

(Remark: This merge tree completely determines the 0-dimensional persistent diagram corresponding to the filtration by the sublevel sets. The algorithm of decomposing a merge tree into a collection of bars is simple: find the lowest critical point and connect it to the root. The span of the resulting path defines the longest bar. Remove the path and iterate on the resulting subtrees. We note that the bars in those subtrees will be fully contained in the bar corresponding to the removed stem.)

Can one reconstruct the function back from its merge tree? Yes, in the *univariate* case. Indeed, if the tree is a plane tree, that is if for any merging branches, we know their order left-to-right. In this case, the original function can be obtain using the contour or height walk, called sometimes the Dyck path, see Le Gall’s survey. The height walk will be equivalent (up to reparametrization of the domain) to the original function.

So, how many functions will produce the same merge tree in univariate case: well, the merge tree is binary, and so for each non-leaf vertex we have an option to flip left and right branches, leading to different plane embeddings (at least if all the critical values are distinct) of the same merge tree. Each of them will produce its own height walk, corresponding to the same (up to embedding) merge tree.

Thus, the number of different functions, up to a reparametrization of the domain, resulting in the same merge tree, in univariate case, is \(2^M\), where \(M\) is the number of local maxima, i.e. the number of branching vertices in the merge tree. This was observed by Justin Curry.

So, what would be the situation in \(\Real^n\)? It is, actually, very close to the univariate case. Fix a merge tree \(T\) (a binary tree with distinct real numbers, heights, assigned to its vertices), and denote by \(F(T)\) the space of Morse functions with merge tree \(T\). The critical values of these functions correspond to the heights of the merge tree, with exception of the largest one, the root. The critical points of index \(0\) correspond to the leaves (with exception of the root), and the branching vertices correspond to the critical points of index \(1\).

Theorem: The space \(F(T)\) is homotopy equivalent to the product of \(I\) spheres \(S^{n-1}\).

The mapping from \(F(T)\) to \((S^{n-1})^I\) is given by the collections of unit vector parallel to the unstable direction of the critical points of index 1, oriented towards the component with deeper minimum.

]]>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.…

]]>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. the total output, split by the rough areas as defined by the Mathematical Subject Classification scheme.

Here are the results. The left display shows the *fraction* of the world’s math output that folks at University of Illinois produced since (rather arbitrarily chosen cutoff) 2013… The right display shows the total number of published works in each area across all institutions.

The usual caveats comparing publication numbers in different areas apply: some publish at a much higher rate than others; the disparities somewhat reflected in the right table. Still, the tables below invite some deeper level of introspection.

Say, the number of works in *K*-theory – worldwide – is small. It is small not just because it is harder to write a competent paper there than, say, in game theory, but also because the sum total of people working in the area is low. So a department hosting a high fraction of the world output is akin to being home to an endangered species. Shall we protect our *K*-theorists from extinction? Or let them being taken over by operator algebra and algebraic topology? To exacerbate these existential worries, are our stats correct in the first place? – when typical number of papers in a subject area is *globally* in below a hundred a year, fluctuations start to be very pronounced.

ON the other end, at the highest output areas in math proper (PDEs, Combinatorics, Probability, Number theory…) it is important to remember that within each of them there are broad subareas with *very* different rates of typical output. A lot of publishable papers in, say, combinatorics could have been written pretty fast, but many are deep and use highly technical tools, and would clearly require years of gestation. Perhaps the top level rubrics of MSC provide too coarse a partition to use these numbers algorithmically.

Still, these numbers do tell us something. Judge yourself.

]]>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.…

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.

We will assume that the mutual interactions between the agents in the flock are governed by the interaction term of the energy

$$

R(\xx)=\sum_{k\neq l} R(x_k, x_l).

$$

Here $latex R$ is a repulsive term forcing the agents to avoid each other. For the case of planar terrain, Coulomb interactions are especially attractive options, thanks to connections to potential theory and related holomorphic parameterizations:

$$

R(x_k,x_l)=-\log |x_k-x_l|.

$$

To keep the agent confined, we impose a global term,

$$

U_0(\xx)=\sum_k U_0(x_k).

$$

Here $latex U$ is a coercive function ($latex U(x_k):=\alpha |x_k|^2$ is an obvious choice).

Last element of the setup, is the controller term, – or, rather, an explicit dependence of the global energy sector on some control parameters.

We consider here two versions:

- The steering agent is herself embedded into the terrain, and her contribution to the potential also has the repulsive force structure:

$$

U_p(x_k)=U_0(x_k)-K\log|p-x_k), p\in X.

$$

We will be referring to this setting as the “shepherd” setting. - The confining potential function $latex U_p$ is made explicitly dependent on the parameter $latex p\in P$, some space of parameters. As a non-example, we can consider $latex U_p(x)=U_0(x)+\langle p, x\rangle$. More interesting would be to make the quadratic form $latex U$ dependent on the parameter running through positive definite forms.

As we indicated above, the overall dynamics is assumed to be fast on the agents, settling to their equilibrium state corresponding to the fixed values of the steering parameter $latex p$, and slow for the parameter $latex p$. This leads one to the following formulation:

$$

\dot{x}_k=-\frac{1}{\epsilon}\frac{\partial U_p(\xx)}{\partial x_k},\\

U_p(\xx)=\sum_k \left( U_p(x_k)+\sum_{l\neq k} R(x_k,x_l)\right),\\

\dot{p}=u, u\in U.

$$

Intuitively, the model corresponds to Ginibre ensemble (eigenvalues of Gaussian complex matrices), with a perturbative term (corresponding to “shepherd”).

Agents positions $latex x(t)=(x_k(t))_{k=1}^M$, “shepherd” (steering agent) trajectory $latex p(t)$.

Potential function:

$$U(x,p) = \frac{\alpha}{2}\|x\|^2 + \frac{1}{2}\sum_{k\neq j} \log\|x_k-x_j\|^2 + \frac{1}{2}\sum_k\log\|x_k-p\|^2.$$

Dog trajectory initial and final condition: $latex p(0)=p(T)=p_0$

Agents initial condition at equilibrium: $latex \nabla U(x(0),p(0))=0$

Temporal evolution:

$$\dot x(t) = -\nabla U(x(t),p(t))$$

The gradient (for each agent agent) is:

$$

\begin{align}

\nabla_{x_k} U(x,p) &= \nabla \frac{\alpha}{2}\|x_k\|^2 + \frac{1}{2}\sum_{j\neq k} \nabla_{x_k}\log\|x_k-x_j\|^2 + \frac{1}{2} \nabla_{x_k}\log\|x_k-p\|^2\\

&=

\alpha x + \sum_{k\neq j} \frac{x_k-x_j}{\|x_k-x_j\|^2}

+ \frac{x_k-p}{\|x_k-p\|^2}

\end{align}

$$

Example of simulation with herd of 12:

One can see starting (dot) and ending (cross) positions of the agents. The trajectory of the steering agent is solid blue. The agents start and end in an equilibrium position; n3o braiding of the agents observed.

- Assume the first, “shepherd”, model. Fix a (smooth) trajectory $latex p:[0,T]\to X\equiv \Real^2$, starting and ending at the same point $latex p_0$ far away from the origin. This loop would generate a movement among the charges $latex x_k, k=1,\ldots,n$.Do the point return to the same positions reshuffled?
- When they do, one generates so called braid, an element in the fundamental group of the configuration space of $latex n$ (indistinguishable) points in plane. What elements can be generated?
- What elements can be generated if the movements of the shepherd are constrained, for example to stay outside of the convex hull of the agents?

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$

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?…

]]>$

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?

It quite easy to see that it is a continuous martingale. What is the quadratic variation of $latex o$?

If the directions orthogonal to the affine span $latex L(\xx)$ of the tuple, the answer is immediate: it is proportional to

$$

\sum_{k=1}^n p^2_k,

$$

where $latex p_k$ are the barycentric coordinates of $latex o$ with respect to $latex x_1,\ldots,x_n$. Indeed, it is immediate that if $latex dx_k^\perp$ are the components of the Brownian increments orthogonal to $latex L(\xx)$, then

$$

do^\perp(\xx)=\sum_k p_k dx_k^\perp.

$$

It follows -as $latex dx_k^\perp$ are independent and uncorrelated) that the component of $latex o$ orthogonal to $latex L(\xx)$ is a martingale with variation equal to (Euclidean norm on $latex L(\xx)^\perp$ times) $latex \sum_k p_k^2$.

Let’s turn to the motion of $latex o(\xx)$ inside $latex L(\xx)$. Let $latex S(\xx)\subset L(\xx)$ be the sphere centered at the centroid and containing all the points of the sample. Let $latex dx_k^\parallel=\xi_k+\eta_k$ be the decomposition of the $latex L(\xx)$ component of $latex dx_k$ into the vector in the tangent space $latex \xi_k\in T_{x_k}S(\xx)$, and the vector orthogonal to it. If we denote by $latex n_k=(x_k-o(\xx))/|x_k-o(\xx)|$ the unit norm vector from the centroid to $latex k$-th point of the sample, then $latex \eta_k=s_k\cdot n_k$, where $latex s_k$ is the increment of a Brownian motion independent of all all other components (as follows from the orthogonalities hardwired in the construction).

Now, the condition that as the points of the sample move, the distances to the centroid remain constant is equivalent to

$$

\langle \eta_k-\nu,n_k\rangle=\langle \eta_l-\nu,n_l\rangle,

$$

for all $latex 1\leq l,k\leq n$. (Here we denote by

$$

\nu:=do(\xx)^\parallel

$$

the $latex L(\xx)$-component of $latex do(\xx)$.)

Equivalently, this means that $latex \langle \nu, n_k-n_l\rangle=s_k-s_l$.

As $latex s_k, s_l$ are independent, with unit quadratic variation, we conclude that

$$

\langle (n_k-n_l),C (n_k-n_l)\rangle=2

$$

for all $latex k,l$, – here

$$

C:=\ex \nu\otimes\nu’

$$

is the quadratic variation of the centroid in the affine plane spanned by the sample.

Geometrically, an equivalent description is that $latex C=AA’$, where $latex A$ is the linear part of the affine transformation taking the points of the sample to the regular $latex n$-simplex with sides $latex \sqrt{2}$.

]]>