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<HTML><HEAD><TITLE>Symplectic geometry seminar</TITLE>

<H1>University of Toronto's Symplectic Geometry Seminar </H1>

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<H2> Aravind Asok </H2></CENTER>

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<H2>GIT, solvable groups, and $A^1$-homotopy theor</H2>

<BLOCKQUOTE><B>Abstract: </B>Geometric invariant theory mainly concerns itself with

formation and study of quotients of varieties by actions of reductive

groups. Recently, there has been much interest in studying quotients

of varieties by the actions of non-reductive groups (e.g., the group

of translations of affine space, automorphism groups of weighted

projective spaces), and in particular their cohomological or homotopic

invariants. In this talk, we will illustrate this study by focusing

on GIT quotients of (open subsets of) affine space by free actions of

solvable groups; familiar examples include all smooth projective toric

varieties.

Morel and Voevodsky's $A^1$-homotopy theory provides a firm foundation

for the study of ``generalized algebraic cohomology theories" on the

category of smooth varieties. In particular, it can be used to study

algebraic K-theory and (higher) Chow groups of such varieties. We

will explain some of the rich interplay between constructions of GIT

and $A^1$-homotopy theory and indicate how extremely refined

$A^1$-homotopic invariants are actually ``computable" in an appropriate

sense. Along the way, we will observe that i) quotients by free

actions of tori are often ``covering spaces" from the standpoint of

$A^1$-homotopy theory, ii) quotients of affine space by free actions of

unipotent groups are often contractible varieties from the standpoint

of $A^1$-homotopy theory. We will explain how these and related

observations/computations can be used to test many classical

conjectures in algebraic geometry.

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