No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

Let f be a holomorphic automorphism of a compact Kähler manifold. Assume that f admits a unique maximal dynamic degree d_p with only one eigenvalue of maximal modulus. Let μ be its equilibrium measure. In this paper, we prove that μ is exponentially mixing for all d.s.h. test functions.

We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan-Gross-Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding L-functions. Ratner’s results on measure classification provide an important input to the proof. Our local results include asymptotic expansions for certain special functions arising from representations of higher-rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino-Ikeda conjecture.

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space P^n, in which case Bir(X) is the Cremona group of rank n, or when X⊂P^{n+1} is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank n⩾3 is not generated by linear and Jonquières elements.

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 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 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 C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of 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 SL_6 .

We show that there are 2^{2^{ℵ_0}} different closed ideals in the Banach algebra L(L_p(0,1)),1<p≠2<∞. 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^{ℵ_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<q<2 the space Xq of Rosenthal, which is isomorphic to a complemented subspace of L_q(0,1), admits such an operator.

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 Airy1 and Airy2 processes. The process takes values in real valued functions which look locally like Brownian motion, and is Hölder 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.

For any 2⩽r⩽∞,n⩾2, we prove the existence of an open set U of C^r‑self‑mappings of any n‑manifold so that a generic map f in U displays a fast growth of the number of periodic points: the number of its n‑periodic points grows as fast as asked. This complements the works of Martens–de Melo–van Strien, Kaloshin, Bonatti–Díaz–Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. Furthermore, for any 1⩽r<∞ and any k⩾0, we prove the existence of an open set \hat U of k-parameter families in U such that for a generic (f_p)^p ∈ \hat U, for every ∥p∥⩽1, the map f_p displays a fast growth of the number of periodic points. This gives a negative answer to a problem asked by Arnold in 1992 in the finitely smooth case.

In this paper, we study gravitational instantons (i.e., complete hyperkähler 4‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in the ALG and ALH cases. (3) In the ALF‑D_k case, it must have an O(4)‑multiplet.

We obtain an asymptotic expansion of planar orthogonal polynomials with respect to exponentially varying weights, relevant for random matrix theory. As a consequence we show that the density of states in the random normal matrix model has universal error function transition across smooth interfaces for the limiting eigenvalue density. The key ingredient in the proof of the asymptotic expansion is the construction of the orthogonal foliation flow: a smooth flow of closed curves foliating a neighborhood of the interface, where the weighted L^2-mass of the orthogonal polynomial concentrates. The defining property of the flow is that the orthogonal polynomial of a given degree with respect to the disintegrated weighted area measure on the curves is approximately stationary in the flow parameter. To compute the coefficient functions in the expansion, we develop an algorithm which collapses the orthogonality relations to conditions along the interface, which become inhomogeneous Toeplitz kernel equations.