We study log canonical thresholds (also called global log canonicalthreshold or $\alpha$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds; that is, given a natural number $d$ and a positive real number $\epsilon$, the set of Fano varieties of dimension $d$ with $\epsilon$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which, in particular, answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case.

This paper is concerned with the structure of Gromov-Hausdorff limit spaces $(M^n_i,g_i,p_i)\stackrel{d_{GH}}{\longrightarrow} (X^n,d,p)$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $\mathrm{Ric}_{M^n_i}\geq -(n-1)$ as well as the noncollapsing assumption $\mathrm{Vol}(B_1(p_i))>\mathrm{v}>0$. In such cases, there is a filtration of the singular set, $S^0\subset S^1\cdots S^{n-1}:= S$, where $S^k:=${$x\in X:$ no tangent cone at $x$ is $(k+1)$-symmetric}. Equivalently, $S^k$ is the set of points such that no tangent cone splits off a Euclidean factor $\mathbb{R}^{k+1}$. It is classical from Cheeger-Colding that the Hausdorff dimension of $S^k$ satisfies $\mathrm{dim}\, S^k\leq k$ and $S=S^{n-2}$, i.e., $S^{n-1}\setminus S^{n-2}=\emptyset$. However, little else has been understood about the structure of the singular set $S$.

Our first result for such limit spaces $X^n$ states that $S^k$ is $k$-rectifiable for all $k$. In fact, we will show for $\mathcal{H}^k$-a.e.\ $x\in S^k$ that \textit{every} tangent cone $X_x$ at $x$ is $k$-symmetric, i.e., that $X_x= \mathbb{R}^k\times C(Y)$ where $C(Y)$ might depend on the particular $X_x$. Here $\mathcal{H}^k$ denotes the $k$-dimensional Hausdorff measure. As an application we show for all $0\lt \epsilon\lt \epsilon(n,\mathrm{v})$ there exists an $(n-2)$-rectifiable closed set $S^{n-2}_\epsilon$ with $\mathcal{H}^{n-2}(S_{\epsilon}^{n-2}) \lt C(n,\mathrm{v},\epsilon)$, such that $X^n\setminus S^{n-2}_\epsilon$ is $\epsilon$-bi-Hölder equivalent to a smooth Riemannian manifold. Moreover, $S=\bigcup_\epsilon S^{n-2}_\epsilon$. As another application, we show that tangent cones are unique $\mathcal H^{n-2}$-a.e.

In the case of limit spaces $X^n$ satisfying a $2$-sided Ricci curvature bound $|\mathrm{Ric}_{M^n_i}|\leq n-1$, we can use these structural results to give a new proof of a conjecture from Cheeger-Colding stating that $S$ is $(n-4)$-rectifiable with uniformly bounded measure. We can also conclude from this structure that tangent cones are unique $\mathcal H^{n-4}$-a.e.

Our analysis builds on the notion of quantitative stratification introduced by Cheeger-Naber, and the neck region analysis developed by Jiang-Naber-Valtorta. Several new ideas and new estimates are required, including a sharp cone-splitting theorem and a geometric transformation theorem, which will allow us to control the degeneration of harmonic functions on these neck regions.

We prove that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n \ge 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n \ge 5$. We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{S}\mathrm{Aut}(F_n)$ (with $n \ge 6$) and of $\mathrm{SL}_n(\mathbb{Z})$ (with $n \ge 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n > 6$.

We present an algorithm that computes the product of two $n$-bit integers in $O(n \mathrm{log}\, n)$ bit operations, thus confirming a conjecture of Schönhage and Strassen from 1971. Our complexity analysis takes place in the multitape Turing machine model, with integers encoded in the usual binary representation. Central to the new algorithm is a novel ``Gaussian resampling" technique that enables us to reduce the integer multiplication problem to a collection of multidimensional discrete Fourier transforms over the complex numbers, whose dimensions are all powers of two. These transforms may then be evaluated rapidly by means of Nussbaumer's fast polynomial transforms.

A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property, then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 \in G$, there exists $y\in G$ such that $G = \langle x_1y\rangle = \langle x_2,y\rangle$. In other words, $s(G) \ge 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods, and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups with socle an exceptional group of Lie type, and this is the case we treat in this paper.