a long-standing open question, formalized by blackadar and kirchberg in the mid 90's, asks for an abstract characterization of c$^*$-subalgebras of af-algebras. i will discuss some recent progress on this question: every separable, exact c$^*$-algebra which satisfies the uct and admits a faithful, amenable trace embeds into an af-algebra. moreover, the af-algebra may be chosen to be simple and unital with unique trace and the embedding may be taken to be trace-preserving. modulo the uct, this characterizes c$^*$-subalgebras of simple, unital af-algebras. as an application, for any countable, discrete, amenable group $g$, the reduced c$^*$-algebra of $g$ embeds into a uhf-algebra.

the motivating problems in geometry are classification problems. one tries to understand the isomorphism classes of certain types of geometric objects. a peculiar feature of algebraic geometry is that there is often a variety or scheme, called the moduli space, whose points correspond to isomorphism classes of the objects we want to study. in this talk, i will talk about the history of moduli problems and spaces, and explain some of the mathematics behind the contributions grothendieck made to the subject.

in the celebrated work of bershadsky--cecotti--ooguri--vafa the genus one string partition function in the b-model is identified with certain analytic torsion of the hodge laplacian on a k$\ddot{\text{a}}$hler manifold. in a joint work with shu shen (imj-prg) and jianqing yu (ustc) we study the analogous torsion in landau--ginzburg models. i will explain the corresponding index theorem based on the asymptotic expansion of the heat kernel of the schr$\ddot{\text{o}}$dinger operator. i will also explain the rigorous definition of the bcov torsion for homogeneous polynomials on ${\mathbb c}^n$. lastly i will explain the conjecture stating that in the calabi--yau case the bcov torsion solves the holomorphic anomaly equation for marginal deformations.

we consider $\sigma_k$-curvature equation with $h_k$-curvature condition on a compact manifold with boundary $(x^{n+1}, m^n, g)$. when restricting to the closure of the positive $k$-cone, this is a fully nonlinear elliptic equation with a fully nonlinear robin-type boundary condition. we prove a general bifurcation theorem in order to study nonuniqueness of solutions when 2k is less than n. we explicitly give examples of product manifolds with multiple solutions. it is analogous to schoen example for yamabe problem on $s^1\times s^{n-1}$. this is joint work with jeffrey case and ana claudia moreira.

full approximation scheme (fas) is a widely used multigrid method for nonlinear problems. in this talk, we shall provide a new framework to analyze fas for convex optimization problems and improve the original method. we view fas as an inexact version of nonlinear multigrid methods based on space decomposition and subspace correction. the local problem in each subspace can be simplified to be linear and one gradient decent iteration is enough to ensure a linear convergence.

this is a joint work with steve wise (university of tennessee) and xiaozhe hu (tuffs university).

book proofs for preferential attachment models are found through continuous
time processes with exponential waiting times. in turning these arguments around
we find large deviation results and a description of the process conditional on remaining
stable.
joint work with subhabrata sen and svante janson

today the main emphasis in local number theory (i.e the local
langlands) is on the finite places. in charactacteristic 0 the infinite
place is the ``elephant in the room''. this is especially true in the
whittaker theory in which serious difficulties separate the infinite from
the finite places. whittaker models were developed to help the study of
fourier coefficients at cusps of non-holomophic cusp forms (i.e maass cusp
forms) through representation theory. the first of these lectures will start
with an explanation of the role of whittaker models in the theory of
automorphic forms. it will continue with a description of the main results.
the second lecture will explain the proof of the whittaker plancherel
theorem.

the schottky problem is a classical and fundamental question in arithmetic algebraic geometry,
about the characterization of jacobian varieties among abelian varieties.
this question is equivalent to studying the torelli locus (i.e., the image of moduli of curves under the torelli map) inside siegel modular varieties.
in positive characteristics, a first approximation to this problem is understanding the discrete invariants (e.g., p-rank, newton polygon, ekedahl--oort type) occurring for jacobians of smooth curves. the coleman--oort conjecture predicts that if the genus is large, then up to isomorphism, there are only finitely many smooth projective curves over the field of complex numbers, of genus g and jacobian an abelian variety with complex multiplication. an effective version of the colemann--oort conjecture proposes 8 as an explicit lower bound.

after introducing the framework for these problems, i will discuss recent progress towards the schottky problem in positive characteristics which is inspired by the coleman--oort conjecture, and which relies on our understanding of special subvarieties (a.k.a, shimura subvarieties) of siegel varieties.

birkar and zhang recently introduced the notion of generalized pair. these pairs are closely related to the canonical bundle formula and have been a fruitful tool for recent developments in birational geometry. in this talk, i will introduce a version of the canonical bundle formula for generalized pairs. this machinery allows us to develop a theory of adjunction and inversion thereof for generalized pairs. i will conclude by discussing some applications to a conjecture of prokhorov and shokurov.