The mathematician M. Gromov stands among geometers as one of the most original and productive researcher, with unique contributions to the field of geometric group theory. In the recent years, he turned his attention towards the applications of mathematics to neuroscience. His ideas have been collected in a series of articles that form a kind of mathematical diary. In this introductory talk, we will provide some pointers to these texts and work out one example of application of geometric group theory to the large scale structure of neural pathways.

1. Gromov and the ”ergo-brain”
Sylvain Bonnot
IME-USP
NeuroMat 03/10/2018
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2. Introduction
Goal:
M. Gromov’s work in geometry: some recurrent themes
(geometric group theory, finitely presented groups, metric spaces
and their convergence, isometric embeddings)
Work in neuroscience: Ergo-Brain, ergosystems, the example of
Language
Back to geometry: concrete worked-out example of application of
finitely presented groups to neuroscience.
Website: https://www.ihes.fr/˜gromov
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3. Sequence of papers
”It is unimaginable that a viable representation of the brain/mind interface is
possible without being incorporated into a broader mathematical framework”,
M.Gromov ”Structures, Learning and ergosystems”.
1 In a Search for a Structure, Part 1: On Entropy, (2013), 27 pages.
2 Ergostructures, Ergologic and the Universal Learning Problem:
Chapters 1, 2, (2103), 37 pages.
3 Math Currents in the Brain, 2014, 10 pages.
4 Structures, Learning and Ergosystems: Chapters 1-4, 6., 159 pages.
5 Great Circle of Mysteries, Birkh¨auser, 209 pages, 2018.
6 Learning and Understanding in the Mirror of Mathematics,
Chapters 1 and 2, 2018, 49 pages.
The Ergo Project: The ultimate aim of the ergo project is designing a
universal learning program that upon encountering an interesting flow of
signals, e.g., representing a natural language, starts spontaneously
interacting with this flow and will eventually arrive at understanding of the
meaning of messages carried by this flow.
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4. Gromov and his texts
Exercise 1 in
”Manifolds of
Nonpositive
Curvature, Birkh
¨auser, 1985: was
open problem...took
20 years to be solved!
”Ergo-writing”: 60
pages of quotations
+ free form writing +
a sci-fi story...
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5. Some themes in Gromov’s research: geometric group
theory
Groups are geometric objects:
Groups (i.e sets with an
associative product, with an
inverse for each element) are
geometric objects.
Free group: use two letters and
form all possible words:
a3.b.a2.b−1.a
”Word metric”: distance in the
tree.
Gromov boundary of the
group: the leaves of this tree
Figure: Free group on two generators
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6. Some themes in Gromov’s research: ”Groups are
geometric objects”
Finitely presented groups:
a, b, c|R, S, T
we form finite words from a finite
alphabet A = {a, b, c, . . .} , with
certain simplification rules
R, S, T, . . ..
Example: alphabet A = {a, b}
Word example: ab3.a−1b. Example
of simplification a2.a−1 = a.
Cyclic groups: c|cN = 1
Product group Z × Z:
a, b|ab = ba .
Figure: Mattress group
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7. Gromov Hyperbolic spaces
Hyperbolic spaces: the Poincar´e model.
Huge generalization:
Definition
A space is (Gromov-)δ−hyperbolic if triangles (with sides α, β, γ)are δ−thin,
i.e α ⊂ Nδ(β ∪ γ).
Examples: the space H2(δ = ln(1 +
√
2))
Non-examples: the plane R2 (Hint: use a dilation!).
What happens when δ → 0? Trees are 0-hyperbolic spaces!
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8. Space of metric spaces with a metric on it:
Gromov-Hausdorff convergence
Gromov-Hausdorff metric: ”I knew it for a long time, but it seemed too
trivial to write. Sometimes, you just have to say it”, M. G
Definition (Hausdorff distance)
If X, Y are compact subsets of a metric space Z the Hausdorff distance
dH(X, Y) is the infimal such that: X is contained in the −neighborhood of
Y and Y is contained in the −neighborhood of X.
Remark: topological structure not preserved
Definition (Gromov-Hausdorff distance)
The Gromov-Hausdorff distance is the infimum of the collection of Hausdorff
distances obtained from pairs of isometric embeddings of X and Y into the
same metric space Z.
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9. Gromov-Hausdorff convergence
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10. Convex integration I: the h-principle
Making a torus from a square:
Isometric embedding of the torus: unfortunately the two curves,
green and red do not have the same length! (First solved by J. Nash)
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11. Convex integration II (the Hevea Project)
Making waves (within waves, within waves...) to increase the
length:
Smooth fractals: the embedding is C1 (versus Von Koch curve or Julia
sets)
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12. Convex integration III: green and black have equal
lengths!
the end result:
All curves drawn are smooth!
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13. Convex integration IV: view from the inside
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14. Ego-Brain, Ergo brain
“An essential ingredient of ergo-learning strategy is a search for symmetry–
repetitive patterns–in flows of signals. Even more signicantly, an ergo system
creates/identies such patterns by reducing/compressing ”information” and by
structuralizing ”redundancies” in these flows.” –Gromov
Definition
Ergosystem: a formal (hypothetical) structure implementing the
transformation of incoming flows of signals, and the extraction of structure
from those flows, through a process of goal-free learning (as opposed to
reinforcement learning).
Definition
Ego brain: all mental processes associated to survival needs.
The learning process follows a universal set of simple rules for
extracting structural information from these signals.
Universality ⇒ non-pragmatic character of learning.
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15. Category theory
”Probably, the full concept of ”ergo-brain“ can be expressed in a language
similar to that of n-categories”, M. G.
Category C: objects X, Y, . . . ∈ O(C), and sets of morphisms (arrows)
between them, with a composition law
h ◦ (g ◦ f) = (h ◦ g) ◦ f
Examples: Vec, Top, Man, Banach, Groups
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16. Understanding and dictionaries
”Most of what we know about the structure of the human ergo-brain is what
we see through the window of human language.”, M.G. ”In either case, the
process of what we call “understanding” is interpreted as assembling an
ergo-dictionary EDI – a kind of “concentrated extract” of the combinatorial
structure(s) that are present (but not immediately visible) in flows/arrays of
linguistic signals.”, M. G.
1 Discretization and Formation of Units
2 Analogy, Similarity, Equivalence, Equality, Sameness
3 Classification, Reduction, Clustering, Compression
Folding a (large) library L: make L into a metric space X as follows:
points in X: fixed length (10 ≤ l ≤ 30) strings of letters from L.
distance: d(x, y) := k where ∈ (0, 1) and k is the maximal length
so that the initial segments are equal.
Summarizing the metric space: at scale δ > 0 take a δ−dense net
Xδ. Its minimal size N(δ) represents the size of X at scale δ.
Folding a library of images: work with slightly overlapping patches
of images. 16

17. Cech complex
Graph: vertices are points from the cloud, edges are placed whenever
d(x, y) < , triangles are inserted whenever we have triple
intersections.
Vietoris-Rips complex:
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18. Folding: use of Category theory
Objects: Let a library L be represented by a collection of tapes with
strings s of symbols, e.g., letters or words.
Arrows: s1 → s2 is present when there is a substring s1 ⊂ s2 with
s1 s1.
Claim: The resulting category C→ carries full information about L.
Linguistic 2-space: strings from a given library L are represented by
line segments of lengths equal to the number of letters in them.
Rectangular 2-cells s × [0, 1] are attached to string segments S0, S1
whenever there is an arrow between S0, S1. (compare with algebraic
topology).
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19. Back to Finitely generated groups: the formalism of
braids
Geometric braids: paths in R3 joining A1, . . . , An to B1, . . . , Bn
(Equivalence of braids: deforming continuously one into the other
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20. Algebraic description of braids: braids and finitely
presented groups
Generators of the braid group: they are simple crossings
Relations of the braid group:
Definition
The braid group Bn is the group defined
with generators {σ1, . . . , σn−1} and
relations
σiσi+1σi = σi+1σiσi+1 1 ≤ i ≤ n − 2
σiσj = σjσi |i − j| ≥ 2 20

21. Braids as a group
Group operation: the product of
two braids is simply the
concatenation of them
Identity element: the 1 element is the braid with parallel strands.
Braid inverse:
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22. Topology of neural pathways: joint with L. Monteiro de
Moura
Topological description of the
”fornix” The fornix is a small
bundle of nerve fibers in the brain
that acts as the major output tract
of the hippocampus.
Reconstruction from MRI data (tractography):
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23. Braid associated to the tract
Clustering the streamlines (35): Tracts (or clusters approximations of
those) are computed with DiPy (”open source software project for
computational neuroanatomy, focusing mainly on diffusion magnetic
resonance imaging (dMRI) analysis”, lead developer Garyfallidis).
From tracts to trajectories in space: tracts are then fed to Braidlab (J.L.
Thiffeault ) to produce lists of simple crossings. 23

24. Braiding of the tract
Braid as a word in the generators:
B([-8,-11,13,-10,-9,-6,3,-14,-13,-8,-14,30,-3,25,-10,
-12,8,-30,-8,-25,-31,32,12,8,-33,34,31,-11,33,-5,32,33,10,-31,
9,-29,-30,-32,29,10,-31,-32,-30,-28,11,31,-29,-30])
Normalized representation of the tract: after reduction (Dehornoy
handle reduction algorithm, in Python)
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25. Advantages of the braid representation
Concise encoding: Braids encode in a concise form complicated
patterns of crossing (compare to DiPy’s 50 streamlines*(40
samples)*(3 dims)...)
Simplify and compare: Once written in algebraic form, one can
simplify braids, or compare 2 different braids.
Synthetic data: easily generated, just produce random words in
the generators σi.
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