in the 1980s, richard hamilton invented the ricci flow and initiated the geometric-analytic approach towards the poincar\'e conjecture, which was eventually solved by grisha perelman. in this talk, we will briefly discuss the formation of singularities in the ricci flow and the relations with ricci solitons. we will introduce some recent advances and main conjectures in ricci solitons, especially in steady gradient ricci solitons. we end the talk by discussing some recent results (joint work with professor chow and yuxing deng) about asymptotic behaviors and a rigidity theorem of steady solitons.

a calabi-yau manifold is a ricci-flat kahler manifold. it is a longstanding question to understand how the ricci-flat metrics develop singularities when the complex structure degenerates. an especially intriguing phenomenon is that these metrics can collapse to lower dimensions and exhibit very non-algebraic features, and it is challenging to describe the corresponding geometric behavior. in this talk i will discuss an answer to this question for a special class of degenerations. based on joint works with h. hein-j. viaclovsky-r. zhang 2018 and with r. zhang 2019.

in this talk we will describe some recent works on drinfeld modular forms with values in tate algebras (in `equal positive characteristic'). in particular, we will discuss some remarkable identities (proved or conjectural) for eisenstein and poincar\'e series, and the problem of analytically interpolate families of drinfeld modular forms for congruence subgroups at the infinity place.

over the complex numbers, a famous theorem of siu states that the plurigenera $p_m$ of projective manifolds are invariant under deformations. we give examples of families of elliptic surfaces over a dvr of positive or mixed characteristic such that $p_m$ fails to be constant for all sufficiently divisible $m\geq 0$. time permitting, we will show that (asymptotic) invariance of plurigenera holds for families of quasi-elliptic surfaces.

with the motivation of generalizing the corresponding
geometrization of tamagawa-iwasawa theory after kedlaya-pottharst, and
with motivation of establishing the corresponding equivariant version of
the relative p-adic hodge theory after kedlaya-liu aiming at the
deformation of representations of profinite fundamental groups and the
family of \'etale local systems, we initiate the corresponding
hodge-iwasawa theory with deep point of view and philosophy in mind from
early work of kato and fukaya-kato. in this talk, we are going to
discuss some foundational results on the hodge-iwasawa modules and
hodge-iwasawa sheaves, as well as some interesting investigation towards
the goal in our mind, which are taken from our first paper in this
series project.

this talk will be about refined curve counting on local $p^2$, the noncompact calabi-yau 3-fold total space of the canonical line bundle of the projective plane. i will explain how to construct quasimodular forms starting from betti numbers of moduli spaces of one-dimensional coherent sheaves on $p^2$. this gives a proof of some stringy predictions about the refined topological string theory of local $p^2$ in the nekrasov-shatashvili limit. this work is in part joint with honglu fan, shuai guo, and longting wu.