recently okounkov proved nekrasovs conjecture expressing the
partition function of k-theoretic dt invariants of the hilbert scheme of
points hilb($\mathbb c^3$, points) on affine 3-space as an explicit
plethystic exponential. we generalise nekrasovs formula to higher rank,
where the quot scheme of finite length quotients of the trivial rank $r$
bundle replaces hilb($\mathbb c^3$,points). this proves a conjecture of
awata-kanno. specialising to cohomological invariants, we obtain the
statement of szabos conjecture. we discuss some further applications if
time permits. this is joint work with nadir fasola and sergej monavari.

the iwasawa theory of selmer groups provides a natural way for p-adic approach to the celebrated birch and swinnerton dyer conjecture. over a non-commutative p-adic lie extension, the (dual) selmer group becomes a module over a non-commutative iwasawa algebra and its structure can be understood by analyzing ``(left) reflexive ideals'' in the iwasawa algebra. in this talk, we will start by recalling several existing conjectures in iwasawa theory and then we will give an explicit ring-theoretic presentation, by generators and relations, of such iwasawa algebras and sketch its implications in understanding the (two-sides) reflexive ideals. generalizing clozel's work for $sl(2)$, we will also show that such an explicit presentation of iwasawa algebras can be obtained for a much wider class of p-adic lie groups viz. uniform pro-p groups and the pro-p iwahori of $gl(n,z_p)$. further, if time permits, i will also sketch some of my recent iwasawa theoretic results joint with sujatha ramdorai.

using ideas from geometric group theory we provide a novel approach to green's conjecture on syzygies of canonical curves. via a strong vanishing result for koszul modules we deduce that a general canonical curve of genus $g$ satisfies green's conjecture when the characteristic is zero or at least $(g+2)/2$. our results are new in positive characteristic (and answer positively a conjecture of eisenbud and schreyer), whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by voisin. joint work with aprodu, papadima, raicu and weyman.