(joint with Daniele Dona and Jitendra Bajpai): Babai's conjecture states that, for any finite simple non-abelian group G, the diameter of G is bounded by (log |G|)^C for some absolute constant C. By now, the conjecture is known for groups of Lie type of bounded rank; otherwise put, for those groups, we have bounds of the form diam(G)=O((log |G|)^{C_r}) with C_r and the implied constant depending only on the rank r. This is work that started with Helfgott (giving complete proofs for SL_2(F_p) and SL_3(F_p), together with more general material) and culminated with Breuillard-Green-Tao and Pyber-Szabó.

In the work of B-G-T and P-S, the bounds obtained on C_r increase very rapidly with r; in P-S, C_r has exponential-tower dependence on r, while B-G-T -which relies on ultrafilters - would need to be rephrased entirely to even reach that level.

We will discuss two kinds of improvements:

• by means of careful work on controlling exceptional loci of maps, bounding degrees, etc., and changing the inductive procedure first used by Larsen-Pink, one can obtain a bound of type C_r << exp(r^c), c an absolute constant (really 2+epsilon);

• by means of further changes to the overall strategy, plus dimensional bounds specific to tori and conjugacy classes (not completely dissimilarly from Helfgott's work previous to B-G-T and P-S), we can give polynomial bounds on C_r.

In particular, for any classical Chevalley group G=G(F_q) of rank r with q not too small with respect to r,

diam(G) <= (log |G|)^{O(r^4)}

More generally, we show similar bounds hold for reductive linear algebraic groups over finite fields, and the same methods should carry over to all groups of Lie type.