In this paper, we study the linear systems $|-mK_X|$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $|-mK_X|$ is non-empty and contains an element with ``good singularities" for some natural number m depending only on $d$; if in addition $X$ is $\epsilon$-lc for some $\epsilon > 0$, then we show that we can choose $m$ depending only on $d$ and $\epsilon$ so that $|-mK_X|$ defines a birational map. Further, we prove Shokurov's conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.

We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short).

In the first part of this paper, we prove new curvature pinching estimates that ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n\ge 12$, we show that blow-up limits are weakly PIC2. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension $3$.

In the second part, we develop a theory of ancient solutions that have bounded curvature, are $\kappa$-noncollapsed, are weakly PIC2, and are uniformly PIC. This is an extension of Perelman's work; the additional ingredients needed in the higher dimensional setting are the differential Harnack inequality for solutions to the Ricci flow satisfying the PIC2 condition, and a rigidity result due to Brendle-Huisken-Sinestrari for ancient solutions that are uniformly PIC1.

In the third part of this paper, we prove a Canonical Neighborhood Theorem for the Ricci flow with initial data with positive isotropic curvature, which holds in dimension $n\ge 12$. This relies on the curvature pinching estimates together with the structure theory for ancient solutions. This allows us to adapt Perelman's surgery procedure to this situation. As a corollary, we obtain a topological classification of all compact manifolds with positive isotropic curvature of dimension $n \ge 12$ that do not contain nontrivial incompressible $(n-1)$-dimensional space forms.

The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, where $w_1$ and $w_2$ are non-trivial words in disjoint sets of variables, induces almost uniform distributions on finite simple groups with respect to the $L^1$ norm. Our first main result provides a positive solution to this problem.

We also provide a geometric characterization of words inducing almost uniform distributions on finite simple groups of Lie type of bounded rank, and study related random walks.

Our second main result concerns the probabilistic $L^\infty$ Waring problem for finite simple groups. We show that for every $l\ge 1$, there exists (an explicit) $N = N(l)=O(l^4)$, such that if $w_1,\ldots,w_N$ are non-trivial words of length at most $l$ in pairwise disjoint sets of variables, then their product $w_1 \cdots w_N$ is almost uniform on finite simple groups with respect to the $L^\infty$ norm. The dependence of $N$ on $l$ is genuine. This result implies that, for every word $w = w_1 \cdots w_N$ as above, the word map induced by $w$ on a semisimple algebraic group over an arbitrary field is a flat morphism.

Applications to representation varieties, subgroup growth, and random generation are also presented. In particular, we show that, for certain one-relator groups $\Gamma$, a random homomorphism from $\Gamma$ to a finite simple group $G$ is surjective with probability tending to $1$ as $|G|\to \infty$.

We prove that K-polystable degenerations of $\mathbb{Q}$-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable $\mathbb{Q}$-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable $\mathbb{Q}$-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable $\mathbb{Q}$-Fano variety is finite.

A left-order on a group $G$ is a total order $\lt$ on $G$ such that for any $f$, $g$ and $h$ in $G$ we have $f \lt g \Leftrightarrow hf \lt hg$. We construct a finitely generated subgroup $H$ of $\mathrm{Homeo} (I^2;\delta I^2)$, the group of those homeomorphisms of the disc that fix the boundary pointwise, and show $H$ does not admit a left-order. Since any left-order on $\mathrm{Homeo} (I^2;\delta I^2)$ would restrict to a left-order on $H$, this shows that $\mathrm{Homeo} (I^2;\delta I^2)$ does not admit a left-order. Since $\mathrm{Homeo}(I;\delta I)$ admits a left-order, it follows that neither $H$ nor $\mathrm{Homeo} (I^2;\delta I^2)$ embed in $\mathrm{Homeo}(I;\delta I)$.

The chromatic number of $G\times H$ can be less than the minimum of the chromatic numbers of finite simple graphs $G$ and $H$.