recently we developed a general framework to compute asymptotic expansions of certain quantities coming from random matrix theory. more precisely if one considers the expectation of the trace of a sufficiently smooth function evaluated in a random matrix, one can compute a taylor expansion (in the dimension of our random matrix) of this quantity. this method relies notably on free stochastic calculus whom i will briefly talk about. in a previous work we studied the case of gue random matrices, in this talk we consider polynomials in independent haar unitary matrices. i will explain the additional difficulties that this model brings then give a few applications of this result to random matrix theory as well as links with weingarten calculus.

given a finite set, the collection of partitions of this set forms a poset category under the coarsening relation. this category is directly related to a space of trees, which in turn has interesting connections to operads. but what if the finite set comes equipped with a group action? what is an "equivariant partition"? and what connection is there to equivariant trees? we will explore possible answers to these questions in this talk, based on joint work with j. bergner, p. bonventre,d. chan, and m. sarazola.

this talk will survey several results concerning the topology at infinity of ricci solitons, with an emphasis on splitting theorems for solitons that have more than one end or, more generally, counting the number of ends geometrically.

shrinking kahler-ricci solitons model finite-time singularities of the kahler-ricci flow, hence the need for their classification. i will talk about the classification of such solitons in 4 real dimensions. this is joint work with deruelle-sun, cifarelli-deruelle, and bamler-cifarelli-deruelle.

zeta and l-functions are ubiquitous in modern number theory. while some work in the past has brought homotopical methods into the theory of zeta functions, there is in fact a lesser-known zeta function that is native to homotopy theory. namely, every suitably finite decomposition space (aka 2-segal space) admits an abstract zeta function as an element of its incidence algebra. in this talk, i will show how many 'classical' zeta functions in number theory and algebraic geometry can be realized in this homotopical framework. i will also discuss work in progress towards a categorification of motivic zeta and l-functions.

 in dimensions four and higher, the ricci flow may encounter singularities modelled on cones with nonnegative scalar curvature. it may be possible to resolve such singularities and continue the flow using expanding ricci solitons asymptotic to these cones, if they exist. i will discuss joint work with richard bamler in which we develop a degree theory for four-dimensional asymptotically conical expanding ricci solitons, which in particular implies the existence of expanders asymptotic to a large class of cones.
 

 we adapt some carleman estimates from earlier joint work with l. wang to asymptotically conical ricci flows and prove a general backward uniqueness theorem for sections of mixed parabolic inequalities along these flows. as an application, we prove that a solution which flows smoothly into a cone on an end must be a shrinking soliton. we will also discuss other related uniqueness results for asymptotically conical shrinking solitons.

this year's symposium will celebrate the return to in-person convening and usher in the next 50 years of the symposium.  the symposium will be held on the weekend of may 20-21, 2023 at uc san diego. the inaugural ronald getoor distinguished lecture will be delivered on sunday afternoon at the symposium. registration, which is required, for the symposium and/or the getoor lecture, is available via the webpage listed below. 

more information is available at the webpage here: https://sites.google.com/view/scps2023/home

ricci flow is proved to be a powerful tool in the field of differential geometry. to obtain geometric or topological applications via continuing the ricci flow by surgeries, it is of central importance to understand at least qualitatively the (finite-time) singularity models. in this talk, we present some recent developments mainly regarding the qualitative descriptions of the singularity models. we present two notions of blow-downs and their relations. we show an optimal scalar curvature estimate for singularity models. we then introduce some optimal qualitative and asymptotic descriptions for steady ricci solitons, which are self-similar solutions of the ricci flow and may arise as singularity models.

 in this talk, we will discuss some additional structure in the kahler setting for bamler’s limit spaces of noncollapsed ricci flows. we will review various notions of singular set stratification, and then state an improved dimension estimate for odd dimensional strata of limits of kahler-ricci flows. we will also see that tangent flows of kahler metric flows admit natural isometric actions, which are locally free away from the vertex in the case that the tangent flows are static.