ELLIPTIC KRV

Emma Qin

Mentor: Florian Naef

PRIMES Conference

May 19th, 2018

GRAPHICAL REPRESENTATIONS

corresponds to

THE ASSOCIATIVE ALGEBRA

xyyxxxy

x x x xy y y

6= xxxxyyy( )

THE ROOTED LIE TREE

corresponds to

GRAPHICAL REPRESENTATIONS

x

[[y, x], y]

y

y

6= [y, [x, y]]

THE LIE TREE, or F(L)

corresponds to

a derivation:

GRAPHICAL REPRESENTATIONS

u(x) = 2[x, [y, x]]

u(y) = �2[[y, x], y]

y

x

xy

GRAPHICAL REPRESENTATIONS

PROPERTIES OF THE LIE TREE

THE LIE TREE

= �

The Lie Brackets [ · , · ] gives a bilinear

operation that satisfies:

1. (Antisymmetry)

2. (Jacobi Identity)

[x, y] = �[y, x]

[x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0

GRAPHICAL REPRESENTATIONS

PROPERTIES OF THE LIE TREE

THE LIE TREE

The Lie brackets [ · , · ] gives a bilinear

operation that satisfies:

1. (Antisymmetry)

2. (Jacobi Identity)

[x, y] = �[y, x]

[x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0

� + = 0

GRAPHICAL REPRESENTATIONS

DEF: krv

We’ll understand the definition in terms of graphs.

krv = krv(1,1) = {u 2 tder(1, 1) = Der+(L(x, y)) | u([x, y]) = 0, div(u) = 0}

krv = {der u | u([x, y]) = 0, div(u) = 0}

GRAPHICAL REPRESENTATIONS

Graphical Interpretation of

corresponds to

a derivation:

Recall:

u([x, y]) = 0

u(x) = 2[x, [y, x]]

u(y) = �2[[y, x], y]

krv = {der u | u([x, y]) = 0, div(u) = 0}

GRAPHICAL REPRESENTATIONS

THE TRACE

Recall that

is a term in the Associative Algebra

GRAPHICAL REPRESENTATIONS

tr( x� y � y � x� x ) =

GRAPHICAL REPRESENTATIONS

Graphical Interpretation of div

krv = {der u | u([x, y]) = 0, div(u) = 0}

x y

y

x

xy

GRAPHICAL REPRESENTATIONS

Graphical Interpretation of div

krv = {der u | u([x, y]) = 0, div(u) = 0}

x y

y

x

xy

y x

y x

y xyx

= �

GRAPHICAL REPRESENTATIONS

Graphical Interpretation of div

krv = {der u | u([x, y]) = 0, div(u) = 0}

x y

y

x

xy

y xyx

= �

= � y x x y

+y x y x

GRAPHICAL REPRESENTATIONS

CURRENT PROGRESS

Def: the birds-on-a-wire graph Claim:

Every lie tree can be rewritten

as birds-on-a-wire graph

GRAPHICAL REPRESENTATIONS

x xx yyyyy

GRAPHICAL REPRESENTATIONS

BASIS FOR

PROGRESS AND GOALS

x� x x� y y � y

F (L)7

x x

x x x x xx

x x x x

GRAPHICAL REPRESENTATIONS

MORE SMALL ELEMENTS

( )

▸ Elements in krv with 3 ’s (?)

PROGRESS AND GOALS

x

▸ Elements in krv with 2 ’sx

�2nx x

GOALS

▸ Study the elements of with small total number of

and ’s.

▸ Possibly using a computer

▸ Study the elements of with small number of ’s.Currently, we are working on elements with 3 ’s.

PROGRESS AND GOALS

krv = {der u | u([x, y]) = 0, div(u) = 0}x

y

x

xkrv = {der u | u([x, y]) = 0, div(u) = 0}

ACKNOWLEDGEMENTS

I would like to thank:

‣ Dr. Florian Naef for proposing this project and for working with

me every week,

‣ Dr. Tanya Khovanova for helping me prepare for the

presentation and offering advice on the research report,

‣ MIT PRIMES-USA for offering this wonderful opportunity,

‣ and my parents for supporting me throughout the process.