Using the geometric Satake correspondence, the Mirković–Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group $G$. We prove that these bases are “perfect”, i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra $\mathfrak{g}$. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra $\mathbb{C}[N]$ of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson–Drinfeld Grassmannian, thus proving a conjecture of Anderson.

In the third part of the paper, we define a map from $\mathbb{C}[N]$ to a convolution algebra of measures on the dual of the Cartan subalgebra of $\mathfrak{g}$. We characterize this map using the universal centralizer space of $G$. We prove that the measure associated with an MV basis element equals the Duistermaat–Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah.

Finally, we use the map to measures to compare the MV basis and Lusztig’s dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in $\operatorname{SL}_6$.

An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process (TASEP) with arbitrary right-finite initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [54], [10]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data.

In the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Brownian motion, for the transition probabilities of the scaling invariant Markov process at the centre of the KPZ universality class. The formula readily reproduces known special self-similar solutions such as the $\operatorname{Airy}_1$ and $\operatorname{Airy}_2$ processes. The process takes values in real valued functions which look locally like Brownian motion, and is Hölder $\frac{1}{3}-$ in time.

Both the KPZ fixed point and TASEP are shown to be stochastic integrable systems in the sense that the time evolution of their transition probabilities can be linearized through a new Brownian scattering transform and its discrete analogue.

We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1)), 1 \lt p \neq 2 \lt \infty$. This solves a problem in A. Pietsch’s 1978 book “Operator Ideals”. The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non-Hilbertian. We give a criterion for a space with an unconditional basis to have $2^{2^{\aleph_0}}$ closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for $1 \lt q \lt 2$ the space $\mathfrak{X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.