Y u r i B u r d a U n i v e r s i t y o f T o r o n t o

y b u r d a @ m a t h . t o r o n t o . e d u

Topological Methods in Klein’s Resolvent Problem

A M S S e s s i o n o n D i f f e r e n t i a l A l g e b r a i c G e o m e t r y a n d G a l o i s T h e o r y

J a n u a r y 6 , 2 0 1 2

Klein’s resolvent problem

Klein’s resolvent problem

Example

Klein’s resolvent problem

n=3 k=1

n=4 k=2

n=5 k≤2 k≥2

Hermite (1861) Kronecker, Klein (1888)

n=6 k≤3 Joubert (1876) n k ≥[n/2] Buhler, Reichstein,

Serre (1997)

n=7 k=4 Duncan (2010)

We’ll see: k ≥[n/2]

Picture behind the problem

Reducing algebraic problem to topological one

Reducing algebraic problem to topological one

Topological theorem

Example in Klein’s resolvent problem

C o v e r i n g s o v e r T o r i a n d

T o p o l o g i c a l A p p r o a c h t o K l e i n ' s R e s o l v e n t P r o b l e m h t t p : / / a r x i v . o r g / a b s / 1 1 0 7 . 3 4 4 4

( s u b m i t t e d t o “ T r a n s f o r m a t i o n G r o u p s ” )

Y u r i B u r d a y b u r d a @ m a t h . t o r o n t o . e d u

Questions?

Characteristic classes for coverings

Characteristic classes for coverings

Another example

Additional results

�  Can add any algebraic functions with monodromy group of odd order

�  Can allow taking branches �  Estimates for arbitrary algebraic functions �  Generic function of k parameters of degree at least

2k can’t be simplified

Future directions

�  Topological analogue of Hilbert’s 13th problem for simple enough coverings (e.g. coverings over a Riemann surface of genus 2)

�  Hilbert’s 13th problem?! �  Relate this to other work on essential dimension of

algebraic objects