# Exercises in Materials Geometry and Topology

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Shape Up 2015
Exercises in Materials Geometry and Topology

Berlin (Germany), 14-18 September 2015
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Dear Scientists and Mathematicians,

We wish to announce our upcoming conference:

Shape Up 2015

Exercises in Materials Geometry and Topology

The conference will be an interdisciplinary discussion meeting on
patterns and geometry, and their role in biological and synthetic
microstructured materials and tissue.. We invite contributions
from biology, chemistry, materials science, mathematics, physics
and related fields addressing the genesis, properties and function
of complex nano-scale geometries, as well as underlying geometric
and topological concepts for the study of complex structure and shape..

The key areas of interest include:

* Geometric and topological concepts in soft condensed matter
* Characterization of structure and function
* Topological data analysis and image reconstruction
* Self-assembly of bi- and polycontinuous phases
* Negatively curved surfaces and hyperbolic geometry
* Structure enumeration & network-like phases
* Function of complex nanostructures in biology
* Bio-inspired design of materials
* Knots and entanglements in physics
* Emergence of chirality, order and the role of disorder
* Symmetry, graphs and discrete geometry
* Tessellations, area-minimising foams and packings
* Three-dimensional topology of cellular microstructures
* Pattern formation and complex structures in soft materials

The conference will run from Monday morning 14 September 2015 to the
late afternoon of Friday 18 September and will feature an extended
poster session, 15-20 contributed talks as well as invited lectures
by

* Simon Copar (U Ljubljana)
* Herbert Edelsbrunner (IST Austria)
* Karsten Grosse-Brauckmann (TU Darmstadt)
* Jemal Guven (U Nacional Autonoma de Mexico)
* Stephen Hyde (Australian National University)
* Motoko Kotani (Tohoku U)
* Rob Kusner (UMass Amherst)
* Jeremy Mason (Bogazici U)
* Elisabetta Matsumoto (Harvard U)
* Konstantin Mischaikow (Rutgers U)
* Piotr Pieranski (U Poznan)
* James Sethian (UC Berkeley)
* Ullrich Steiner (U Fribourg)
* John M. Sullivan (TU Berlin)
* Salvatore Torquato (U Princeton)
* Silvia Vignolini (U Cambridge)

The conference is hosted at the Technical University of Berlin on
Strasse des 17. Juni that leads right up to the famous Brandenburg Gate,
next to Reichstag, the parliament of Germany. Museum Island, Berliner
Philharmonie, Opera Houses, Checkpoint Charlie, Berlin Television Tower
and other famous sites are within a few kilometers walking distance and
are easily accessible via public transport that connects all of inner
Berlin - from vibrant Prenzlauer Berg to Gendarmenmarkt at the heart
of Berlin. All lectures and the poster session will take place at the
Institute of Mathematics, Strasse des 17. Juni 136.

We are inviting abstracts for contributed oral presentations and for
posters. We encourage you to use the latex-template on the website
(http://www.shape-up.academy/) and to include both attractive images
and references in your abstract. The deadline for abstract submission
is

***  31 May 2015  ***

Registration will open 15 June 2015.

We are hopeful of encouraging people from a wide variety of scientific
and mathematical backgrounds to attend. Any queries can be addressed
to the organising committee at shape-up@math.tu-berlin.de.

Best regards, and we hope to see you in Berlin,

The Organising Committee

Myfanwy Evans (TU Berlin), Andrew Kraynik (Sandia, (Ret.)), Frank Lutz
(TU Berlin), Gerd Schroeder-Turk (Murdoch) and Bodo Wilts (Fribourg)

# New book! Elementary Applied Topology

Rob Ghrist has released his new book!

You can find it on amazon!

# Tenure Track Positions, New College of Florida

[On behalf of Vic Reiner, UMN]

The New College of Florida now has several openings in statistics and computer science/data science.   These positions may be of interest to applied topologists.

Details are here.

(The New College is a public honors college; it's where Bill Thurston did his undergraduate studies.)

# Special issue on "Algorithms and Software for Computational Topology"

...in the Journal of Symbolic Computation

Aim and Scope:

The interest in algorithms on topological problems and their implementation has rapidly grown during the last decade. One driving force is the emergence of topological data analysis'' which connects topological concepts like Morse theory and homology to the investigation of real-world data. Another recent track of research substantially expands the realm of possibility for computational approaches in 3-manifold and knot theory. Common to these and other developments is the ability to handle large data collections through an efficient algorithmic framework as well as mature software implementations of those. A workshop session at the International Congress of Mathematical Software (ICMS) in August 2014 was dedicated to this topic (http://icms14.appliedtopology.org/).

The Journal of Symbolic Computation (JSC) invites high-quality contributions from researchers in the area of Computational Topology reporting on original research achievements towards algorithms, software, and applications. The list of topics includes, but is not limited to

• (Persistent) homology
• Topological data analysis
• 3-manifold topology and knot theory
• (Discrete) Morse theory

Researchers which are unsure whether their contribution is suitable are encouraged to contact the guest editor.

Submission instructions:

It is recommended to prepare submission in the same format as regular submission to JSC (see the "Guide for Authors" at http://www.journals.elsevier.com/journal-of-symbolic-computation/).

The paper must start with a introduction that

• clearly states the considered problem
• discusses its relevance and related work
• explains the main contribution of the paper
• explains why the contribution is original and non-trivial

There is no page limit on submitted manuscripts. It is required, however, that

• all related work is completely and carefully discussed
• all theorems are rigorously proved
• important definitions/theorems/algorithms are illustrated by well-chosen examples.

All submitted papers will be refereed according to the high standards of JSC.

Guest editor:

Michael Kerber (Max Planck Institute for Informatics) - mkerber@mpi-inf.mpg.de

The submission deadline is January 31 2015. The special issue is planned to appear in Fall 2015

# AATRN Seminar: Robert Ghrist

Today, the promised AATRN seminar series got started with Robert Ghrist as the inaugural speaker. His lecture, through WebEx, builds up the cellular sheaf perspective on networks with capacity, Max Flow / Min Cut, and the work done by Ghrist, Yasu Hiraoka, and Sanjeevi Krishnan on categorifying and sheafifying MF/MC.

Among the novel insights coming from this talk even if one has been following the UPenn developments for a while was the connection to Poincare Duality: “Flow duality is a form of Poincare duality” — S. Krishnan

The approach detailed by Krishnan in his preprint (on http://www.math.upenn.edu/~sanjeevi/papers/mfmc.pdf ) encodes a flow network as a sheaf of semi-modules of capacities over a semi-ring over the directed graph of the network. Flows correspond to homology, cuts to cohomology.

MF/MC translates to:

Theorem (S. Krishnan)

Given a network X, a distinguished (virtual) edge e and a capacity sheaf F, (all) flowvalues at e of flows on F correspond to the homotopy limit over all cuts of cutvalues of cuts on F.

This approach, and connecting flows and cuts to a lattice structure on the constraints produces a setting where multi-kind flows can easily be analyzed with the same tools as ordinary flows, and where the algebra fixes the duality gaps that show up when naively searching for minima and maxima.

The talk culminated in a primer on the homology and cohomology of directed sheaves, to set us up to read Krishnan's paper, constructing compact support cohomology and Borel-Moore homology for sheaves over directed spaces.

The added abstraction levels seem to enable MF/MC theorems for, for instance, probability distributions. It also carries a promise for insights into duality gaps in MF/MC type problems.

# Applied Algebraic Topology Research Network

Peter Bubenik writes:

[…] Robert Ghrist, Konstantin
Mischaikow, Fadil Santosa and I are starting a Research Network in Applied Algebraic Topology. To start, the main activity of the network will be a weekly interactive online seminar. We have plans to expand our activities in the future.

Rob Ghrist will give the first talk on Tue Sept 23.

Please check out our web site at https://www.ima.umn.edu/topology/ and become a member.

# Efficiently implementing the persistence algorithm

Around the time of my recent submission Multicore homology via Mayer Vietoris an interesting optimization was made which seems to have a nice impact on the running time of the persistence algorithm.

When computing persistence  the boundary matrix is usually stored sparsely. In particular, over $\mathbb{Z}_2$ each column stores only the indices of the rows of the matrix which are nonzero in it's column.

In the parlance of [ZC '05]  each column is identified as:

$x = \partial(\textrm{Cascade}[\sigma]).$

The column $x$ is laid out in memory as a [dynamic] array of computer words  corresponding to row indices, with the natural order of these words being the filtration order.

The persistence algorithm proceeds iteratively, column by column, adding previous columns to the current one. If $x$ is the current column being operated on the next column we will add to it

can be identified as:

$y = \textrm{Cascade[Youngest[}\partial(\textrm{Cascade}[\sigma])\textrm{]]]}$

Until either the column is zeroed out or a unique pairing occurs, e.g. $y = 0$.

Now let me restate the definition of $y$ as a procedure:

Step 1) The youngest entry of $x$ is identified.

Step 2) The address of the column $y$ is computed.

Step 3) The address of the column $y$ is fetched into CPU registers.

Usually, this sequence of three steps occurs only after the previous addition completed. Chain addition is generally are implemented as a set_symmetric_difference from begin() to end() on each chain, e.g. oldest entry of each column to the youngest.

In recent versions of both CTL as well as PHAT a change was made to how these column additions  are carried out. The columns are added now added from youngest to oldest elements. This should have the advantage of better locality of reference.

This is because when inspecting the youngest entry we are loading it, along with some amount of the end of the chain into memory. If we then proceeded to add front to back we might end up retiring this cache line in favor of the beginning of the array.

However, adding backwards has even more benefits. Once we add backwards we know that the first pair of elements we inspect will agree, necessarily. However, it is possible,  that the two vectors we add together agree at their tail significantly more. We may delay the allocation of the output buffer for the sum until we find the first mismatch between these two chains. This saves us a small amount of memory. But their is even more.

Supposing that $x$ is not a positive column, e.g. after a call to mismatch we will find a pair of elements in disagreement, precisely the younger of these two elements is the new youngest element.  We may now compute the address of the next column to add to $x$ and request that the computer prefetches the column at that address.

In other words we get some added low level parallelism: as we add the pair of current columns, we are simultaneously fetching the next column to add.

How well does all of this work? In my opinion, quite well. Here is a  plot of the running time for the persistence algorithm showing the original running time, the running time when adding backwards, and the running time when adding backwards with prefetching, on the various datasets from the multicore paper

Feel free to comment and discuss on the results of this experiments. I'll answer questions the best I can.

# Wasserstein barycenters and Frechet means

At this past year at the IMA, there has been some attention spent on a number of interesting aspects in bringing persistent homology closer to statistical methods.

One core step in this process has been to figure out what we mean by an average persistence diagram, a question that has an answer proposed by Munch, Bendich, Turner, Mukherjee, Mattingly, Harer and Mileyko in various constellations.

The details on Frechet means for persistent homology is not what this post is about, however. Instead, I want to bring up something I just saw presented today at ICML. Marco Cuturi and Arnaud Doucet got a paper entitled Fast Computation of Wasserstein Barycenters accepted to this large machine learning conference.

In their paper, Cuturi-Doucet present work on computing the barycenter (or average, or mean) of N probability measures using the Wasserstein distance: they articulate the transport optimization problem of finding a measure minimizing Wasserstein distance to all the N given measures, and present a smoothing approach that allows them to bring the problem onto a convex optimization shape. In the talk — though not present in the paper — the authors argue that they can achieve quadratic running times through these approximation steps. Not only that, but their approach ends up amenable for GPGPU computation and significant parallelization and vectorization benefits.

Their approach works anywhere that the Wasserstein metric is defined, and so in particular should work (and most likely give the same results) on the persistence diagram setting studied by the persistent statistician constellations mentioned above.

I for one would be excited to hear anything about the relations between these (mostly) parallel developments.

# ATMCS 6: Day 5

Ryan H. Lewis summarizes the morning talks, and Andrew Cooper the afternoon talks:

Christopher Hoffman gives the first talk about recent advanced in random topology. He began by discussing the stopping time for monotone graph properties. An example is for a sequence of graphs $\langle G_j \rangle$ we define:
$\tau_{\textrm{connected}} = \min j \textrm{ such that } G_j \textrm{ is connected}$
For example in the following example we have $\tau_{\textrm{connected}} = 4$

He is interested in generalizing two results about monotone Erdos-Renyi random graphs to facts about erdos-renyi random simplicial complexes.

The Linial-Meshulam model is a collection $Y_i$ of 2-dimensional simplicial complexes where $Y_0$ is a complete graph on $n$ vertices and $Y_i = Y_{i-1} \cup \{\textrm{a } 2 \textrm{ cell} \}.$

It turns out that the first 2-cycle either has 4 faces with probability converging to $c_0 = .909$ or it is larger than $\frac{n}{\log{(n)}}$ with probability converging to 1 - $c_0$.

To generalize the second result to studying isolated edges one can study when the  $H_1(Y, C)=0$ for  $\mathbb{Z}, \mathbb{Z}_2,$ and $\mathbb{Q}$ coefficients, as well as the $\pi_1(Y) = 0$. He presents a series of results relating the stopping times for these events.

In the future they want to use probabilistic methods to demonstrate the existence of complexes desired but unobtained by classical methods in topology.

Paul Villeuotrox (sp?)

Talks about using persistent homology on a wide range of epithelial cells.

By viewing such a structure as a cover of the plane, it's nerve is a 2D topological space. A filtration of this space is given by assigning to a vertex it's degree, and each cell the maximum filtration value of it's boundary.

He has found that persistent homology has proven useful for studying the structure of these cell networks.

He finds that by comparing the barcodes produced from these pipelines to the barcodes produced by complexes built on a random complexes whose underlying graph is endowed with the degree distribution that has been observed empirically, that while persistent $H_0$ appears to be similar between these two types of complexes, persistent $H_1$ seems to be very different.

Raul Rabadan talked about The Topology of Evolution.

The only figure in Darwin’s Origin of Species is a (mostly-binary) tree. This “Tree of Life” paradigm used starting in the 1970s to analyze genomic sequences. The first major discovery using the tree paradigm was Carl Woese’s 1977 discovery of Domain Archaea.

Woese excluded viruses because they lacked some of the genes he used. But even if he had included them, he would have had trouble: viruses have a high level of horizontal gene transfer, so the choice of tree as a structure to represent the phylogeny is not very good for viruses.

Nor is it very good for bacteria and archaea. Nor is it very food for plants (even Darwin knew this). Nor is it very good for us: when you get gonorrhoea, it gets you! 10% of gonorrhoea genes are human. 8% of human genes are viral. Your genome is something like a “cemetery of past infections” rather than a list of all your ancestors.

If we can’t use a tree to model evolution, what can we use? Mathematically, the problem is:

Given a set of genomes and a way of comparing them, how do we represent their relationships without importing (too many) assumptions from biology?

We would like an answer which is statistical and incorporates the notion of scale. We’d also like to detect when clonal (descent) transfer happens, and when horizontal (non-descent) transfer happens. Answer: use persistent homology to detect the topology of the genetic data!

Persistence detects not just topology, but topology at scales. In 0th homology, scale represents taxa: as we increase the filtration value, we are collecting together more and more distantly related genomes. In 1st homology, a long bar represents a transfer between distantly-related taxa.

For example, though overall flu genes show  a lot of cycles, if we restrict our attention to a particular segment there are almost no long-persisting cycles. HIV, on the other hand, has persistent cycles even when we focus on small suites of genes. Thus persistence detects the fact that gene transfer in flu occurs by trading whole segments, whereas in HIV it occurs by trading much smaller units of genetic material.

For human genomic data, persistence bars are about 2 centiMorgans-per-megabase long.

Michael Robinson talked about Morphisms between Logic Circuits

Logic circuits are described by their truth tables. But computers take time to do computations: fast input switching yields the wrong” output (as evidenced by the flickering screen on Dr. Robinson’s slides). How can we analyze the failures of circuits due to problems of timing? Use sheaves!

Sheaves allow local specifications (we are really good at understanding small circuits) to determine global behavior (what we need to get a handle on). Plus sheaves whose stalks are vector spaces are `just’ linear-algebraic, so we can compute their cohomology using easy, well-known techniques.

As we try to associate a vector space to each logic gate, we encounter various aspects of engineering practice like one-hot encoding.

The zeroth cohomology of the switching sheaf detects the (synchronous) classical logic behavior of the system. The first cohomology of the switching sheaf detects stored information (hence, the possibility of a timing problem in the circuit).

But cohomology of the switching sheaf doesn’t tell us everything. The categorification approach says we should consider morphisms to get more information. Given a circuit we want to understand, we can construct a circuit with the same logical behavior.Then we can ask how many morphisms of the switching sheaves there are which cover the identity on inputs and outputs.

Sometimes there aren’t any such morphisms. Sometimes there are a few. Apparently there are never exactly three.

# ATMCS 6: Day 4

Today's summaries are provided by Isabel Darcy.

Omer Bobrowski talked about Topological Estimation for Super Level Sets.  The goal is to determine the homology of an unknown space from a sample of noisy data points.  Super-level sets of a density function f correspond to dense regions: {x | f(x) > L}.  In general, the density function is not known but can often be estimated.  One can try to reconstruct the homology by looking at $U_n(L, r)$ = the union of balls of fixed radius r around each point in a super level set, {x | f(x) > L}.

But if not enough points are chosen (i.e., L large), then the space may not be adequately covered by $U_n(L, r)$.  If too many points are chosen (i.e., L small), then more noise may be picked up.  However one can obtain the correct homology with high probability by looking at how $U_n(L_1, r)$ includes into $U_n(L_2, r)$ for $L_2 < L_1$.  This induces a map on their homologies.  The image of this map is the homology estimator, which equals the correct homology of the space of interest with high probability.

Elizabeth Munch talked about The Interleaving Distance for Reeb Graphs. Reeb graphs provide an efficient description to understand the properties of a real-valued function on a topological space and are useful in many applications.  Thus it would be very useful to have a method for comparing two Reeb graphs.  Interleavings (and interleaving distances) have been used to compare persistence modules.  Interleavings can be applied to Reeb graphs by defining a generalization of a Reeb graph as a functor.  One consequence is a concrete algorithm for smoothing a Reeb graph in order to remove noise.

Peter Bubenik talked about Generalized Persistence Modules and Stability.  Generalized persistence modules is an abstract formulation of persistence modules using category theory which includes many forms of persistence modules that are currently studied.  One consequence of this formulation is that one can give simpler common proofs for many standard results such as stability.

Yuliy Baryshnikov talked about Integral Operators in Euler World. One can integrate functions that take on a finite number of values using the Euler characteristic as the measure.  For example if f(x) = 4 for x in [0, 1] and 0 elsewhere, then the integral of f with respect to the Euler characteristic = 4 times the Euler characteristic of [0, 1] = 4(-1 + 2) = 4.  In applications, sometimes it is easier to solve a problem by transforming it into a simpler problem using an integral transform.

An example of an integral transform is convolution: Given functions f and g, one can create a new function, f*g, where the value f*g(x) is obtained from f and g by integrating the product f(t)g(x-t) with respect to the euler characteristic.  Given f and f*g, one would like to be able to reconstruct g: that is one would like to calculate the inverse of the Euler integral transform.  Cases where one can calculate the inverse transform were discussed.