An invariant random subgroup of the countable group $\Gamma$ is a random subgroup of $\Gamma$ whose distribution is invariant under conjugation by all elements of $\Gamma$. We prove that for a nonamenable invariant random subgroup $H$, the spectral radius of every finitely supported random walk on $\Gamma$ is strictly less than the spectral radius of the corresponding random walk on $\Gamma /H$. This generalizes a result of Kesten who proved this for normal subgroups. As a byproduct, we show that, for a Cayley graph $G$ of a linear group with no amenable normal subgroups, any sequence of finite quotients of $G$ that spectrally approximates $G$ converges to $G$ in Benjamini–Schramm convergence. In particular, this implies that infinite sequences of finite $d$-regular Ramanujan–Schreier graphs have essentially large girth.

We prove that if $A$ is a subset of the primes, and the lower density of $A$ in the primes is larger than $5/8$, then all sufficiently large odd positive integers can be written as the sum of three primes in $A$. The constant $5/8$ in this statement is the best possible.

We establish a fundamental connection between the geometric Robinson–Schensted–Knuth (RSK) correspondence and $\mathrm{GL}(N,\mathbb{R})$-Whittaker functions, analogous to the well-known relationship between the RSK correspondence and Schur functions. This gives rise to a natural family of measures associated with $\mathrm{GL}(N,\mathbb{R})$-Whittaker functions which are the analogues in this setting of the Schur measures on integer partitions. The corresponding analogue of the Cauchy–Littlewood identity can be seen as a generalization of an integral identity for $\mathrm{GL}(N,\mathbb{R})$-Whittaker functions due to Bump and Stade. As an application, we obtain an explicit integral formula for the Laplace transform of the law of the partition function associated with a $1$-dimensional directed polymer model with log-gamma weights recently introduced by one of the authors.

We prove that, for a certain class of closed monotone symplectic manifolds, any Hamiltonian diffeomorphism with a hyperbolic fixed point must necessarily have infinitely many periodic orbits. Among the manifolds in this class are complex projective spaces, some Grassmannians, and also certain product manifolds such as the product of a projective space with a symplectically aspherical manifold of low dimension. A key to the proof of this theorem is the fact that the energy required for a Floer connecting trajectory to approach an iterated hyperbolic orbit and cross its fixed neighborhood is bounded away from zero by a constant independent of the order of iteration. This result, combined with certain properties of the quantum product specific to the above class of manifolds, implies the existence of infinitely many periodic orbits.

We give a sufficient condition on a pair of (primitive) polynomials that the associated hypergeometric group (monodromy group of the corresponding hypergeometric differential equation) is an arithmetic subgroup of the integral symplectic group.

We generalize Lusztig’s geometric construction of the Poincaré–Birkhoff–Witt (PBW) bases of finite quantum groups of type $\mathsf{ADE}$ under the framework of Varagnolo and Vasserot. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the Khovanov–Lauda–Rouquier (KLR) algebras. This enables us to prove Lusztig’s conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases. In addition, we verify Kashiwara’s problem on the finiteness of the global dimensions of the KLR algebras of type $\mathsf{ADE}$.