"Groups are typical

algebraic

objects. Traditional approach to groups is a

combinatorial

approach."

Case study from current mathematics: Geometric Group Theory

Mathematical representations:

Representation of one mathematical object by another.

Groups -> generated groups

Graphs of groups

Metric spaces

Cognitive representations:

Notation

Formulas

Tables

Diagrams

Mathematical vs. Cognitive Representations

Algebraic notation:

Letters:

a,b,r,f

Signs: (,),<,>, *

Words:

abbab

Sentences:

ab=ba=e

manipulation with symbols

Groups can be represented by graphs:

Geometric approach:

Visualise!

Graphs are still

algebraic

objects!!!!

So what?

Z(1): 1 2 3 4 5 6 7 8 9 10 11 ...

Zooming out!

Word metric is boring.

Introduce metric on Cayley graphs?

Groups as metric spaces: Hyperbolic groups

Visualising

+ zooming in and out

the visualisations

Results so far

**The Role of Cognitive Representations in Formation of Mathematical Concepts**

Hyperbolic triangles

Groups can be studied as

geometric

objects !(hyperbolic metric spaces)

Crossing borders and linking domains

Making new concepts available (unavailable in combinatorial context)

Solving problems

Starting a new program!

R____________________________

Metric, hyperbolicity, triangle

can be applied to algebraic groups !

OK, we can use groups to study geometry:

Symmetries of geometric objects

Fundamental groups - F. Klein

A group G is a

set

of elements with a binary operation, which satisfies the group

axioms

:

closure: a • b, is also in G

associativity: (a • b) • c = a • (b • c)

identity: e • a = a • e = a

inverse: a • b = b • a = e

"Groups can be represented as

geometric

objects and studied by

geometric

methods!"

Focus

:

pre-formal philosophy

examples from current mathematics

the role of

cognitive

representations

vs.

Combinatorial group theory

Why this case study:

sharp contrasts

advanced mathematics

accessible

Contrast in two dimensions:

1. Combinatorial/algebraic and geometric approaches

2. Mathematical and cognitive representations

Mikhail Gromov, "Hyperbolic groups."

Essays in group theory

, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.

Algebraic vs. geometric

Algebra/Combinatorics

discrete

sets

out of space

distribution of elements and operations

countable and finite

Geometry

continues

figures

in space

measure

spatial relations

curvature

lists (rows and columns)

tables

matrix-like displays

+ diagrams

Mathematical

representations/objects

Cognitive representations

A group

A

generated

group

A Cayley graph

New!

'delta

-hyperbolicity'

'hyperbolic

group'

'triangle'

'quasi-isometry'

By the way,

e.g. adjacency matrix

wouldn't help!

Concluding Message

:

Changes in cognitive representations can play a significant epistemic role by making new concepts and new resources available.

Thank you!

Triangles?! - Yes!

Angles - ?

"0-slimness"

Large-scale geometry

Classification of groups

Tarski monster group

Isomorphism problem (are the two presented groups isomorphic?)

Word problem (are two arbitrary words equal?)

Tarski monster group

Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by A. Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 10^75. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.

metric space

generated group

__________________

sets

groups

generated groups

presented groups

lines

figures

symmetries

transformations of geometric figures

spaces

Combinatorics/Algebra

Geometry

Cognitive representation

Mathematical

representation

Mathematically

point on a plane can be represented by an ordered pair of numbers.

Cognitively

it can be represented by a dot.

Claim

:

Mathematical representations play a significant epistemic role. But sometimes it is the

cognitive

representations which play a significant epistemic role: they lead us to

new resources

and concepts

.

A group? What is it?

A metric space

New!

Quasi-isometry

equivalence

New instruments:

Cayley

graphs,

quasi-

isometry and

delta

-hyperbolicity

Abel Prize "for his

revolutionary

contributions to geometry".

Group elements can be represented by a multiplication table

{..., -2, -1, 0, 1, 2, 3, ...}

{...,-1+

1

, 0+

1

, 1+

1

, ...}

Starikova, I. “From Practice to New Concepts:

Geometric Properties of Groups”,

Philosophia Scientiæ

, 16 (1), 2012, pp.129-151.

Review

:

contrasted mathematical-cognitive representations in an combinatorial and geometric approaches

observed how they function in the case study

now to conclusions

http://www.ihes.fr/~gromov/index.html

**Irina Starikova**

University of San Paulo

Supported by CAPES

University of San Paulo

Supported by CAPES

Cognitive representations in algebra are symbolic and based on

How to see geometry in groups? By changing representation.

Let's look at applications:

natural sciences (chemistry) - mathematics

mathematics (graphs) - computer

mathematics - mathematics (combinatorics/algebra - geometry)

How the contact is possible?

Graphs are good mediators:

allow for intuitive grasp (chemical structure)

computable

beautiful

Possibilities:

graphs (visualised)

computers (limits; data storage, classification of finite simple groups)

more elaborate mathematics, nice properties (cognitive)

A graph can be represented by an

adjacency

matrix

https://www.math.ubc.ca/~anstee/math223/223Petersen.pdf

Problem: What is the role of cognitive representations in mathematics?

"

hidden arguments

" J. Norton

"picture-

proofs

" J.Brown

Is a justificatory role

the only

important epistemic role ?

Concept formation?

Communication between domains?

More questions:

Mathematical objects -what are they?

Foundations of mathematics?

Mathematical educational strategies?

Graph Theory?

Cayley graph

Structure of the talk:

Problem

Example

Discussion

Representations

:

mathematical - abstract

cognitive - e.g. diagrams, formulas