The purpose of this article is to give a purely equivariant definition of orbifold Chow rings of quotient Deligne-Mumford stacks. This completes a program begun in [JKK] for quotients by finite groups. The key to our construction is the definition (Section 6.1) of a twisted pullback in equivariant $K$-theory, $K_G(X)\to K_G({\mathbb{I}}_G^2(X))$ taking nonnegative elements to nonnegative elements. (Here ${\mathbb{I}}_G^2(X)=\{(g_1,g_2,x)|g_{1}x=g_{2}x=x\}\subset G\times G\times X$.) The twisted pullback is defined using data about fixed loci of elements of finite order in $G$ but depends only on the underlying quotient stack (Theorem 6.3). In our theory, the twisted pullback of the class $\mathbb{T}\in K_G(X)$, corresponding to the tangent bundle to $[X/G]$, replaces the obstruction bundle of the corresponding moduli space of twisted stable maps. When $G$ is finite, the twisted pullback of the tangent bundle agrees with the class $R(m)$ given in [JKK, Definition 1.5]. However, unlike in [JKK] we need not compare our class to the class of the obstruction bundle of Fantechi and Göttsche [FG] in order to prove that it is a nonnegative integral element of $K_G({\mathbb{I}}_G^2(X))$.

We also give an equivariant description of the product on the orbifold $K$-theory of $[X/G]$. Our orbifold Riemann-Roch theorem (Theorem 7.3) states that there is an orbifold Chern character homomorphism which induces an isomorphism of a canonical summand in the orbifold Grothendieck ring with the orbifold Chow ring. As an application we show (see Theorem 8.7) that if $\mathcal{X}=[X/G]$, then there is an associative orbifold product structure on $K(\mathcal{X})\otimes \mathbb{C}$ distinct from the usual tensor product

The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on $G(n,d)$, a random $d$-regular graph on $n$ vertices. It is well known that almost every such graph for $d\ge 3$ is an expander, and even essentially Ramanujan, implying a mixing time of $O(\log n)$. According to a conjecture of Peres, the simple random walk on $G(n,d)$ for such $d$ should then exhibit cutoff with high probability (whp). As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random $3$-regular graph is whp $(6+o(1)){\log }_{2}n$. In this work we confirm the above conjectures and establish cutoff in total-variation, its location, and its optimal window, both for simple and for non-backtracking random walks on $G(n,d)$. Namely, for any fixed $d\ge 3$, the simple random walk on $G(n,d)$ whp has cutoff at $(d/(d-2)){\log }_{d-1}n$ with window order $\sqrt{\log n}$. Surprisingly, the non-backtracking random walk on $G(n,d)$ whp has cutoff already at ${\log }_{d-1}n$ with constant window order. We further extend these results to $G(n,d)$ for any $d=n^{o(1)}$ that grows with $n$ (beyond which the mixing time is $O(1)$), where we establish concentration of the mixing time on one of two consecutive integers

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-$(p,q)$ compact manifold $M$ supports a conformal action of a connected nilpotent group $H$, then the degree of nilpotence of $H$ is at most $2p+1$, assuming $p\le q$; further, if this maximal degree is attained, then $M$ is conformally equivalent to the universal type-$(p,q)$, compact, conformally flat space, up to finite or cyclic covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure

We prove the multiplicity one case of Lusztig's conjecture on the irreducible characters of reductive algebraic groups for all fields with characteristic above the Coxeter number

In 1980, I. Morrison proved that the slope stability of a vector bundle of rank $2$ over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Zhang, Wang, and Phong-Sturm, we show that the statement holds for higher-rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group