数学前沿

We prove that for any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups. This equality was conjectured by John McKay in 1971. The conjecture was reduced by Isaacs–Malle–Navarro (2007) to a conjecture on representations, linear and projective, of finite simple groups that we finish proving here using the classification of those groups. We study mainly characters of normalizers $\mathrm{N}_{\mathbf{G}}(\mathbf{S})^F$ of Sylow $d$-tori $\mathbf{S} (d\ge 3)$ in a simply-connected algebraic group $\mathbf{G}$ of type $D_l (l\ge 4$) for which $F$ is a Frobenius endomorphism. We also introduce a certain class of $F$-stable reductive subgroups $\mathbf{M}\le \mathbf{G}$ of maximal rank where $\mathbf{M}^\circ$ is of type $\mathrm{D}_k\times \mathrm{D}_{l-k}$. The finite groups $\mathbf{M}^F$ are an efficient substitute for $\mathrm{N}_{\mathbf{G}}(\mathbf{S})^F$ or the $\ell$-local subgroups of $\mathbf{G}^F$ relevant to McKay’s abstract statement. For a general class of those subgroups $\mathbf{M}^F$, we describe their characters and the action of $\mathrm{Aut}(\mathbf{G}^F)_{\mathrm{M}^F}$ on them, showing in particular that $\mathrm{Irr(}\mathbf{M}^F)$ and $\mathrm{Irr}(\mathbf{G}^F)$ share some key features in that regard.

We show that Lubin–Tate theories attached to algebraically closed fields are characterized among $T(n)$-local $\mathbb{E}_{\infty}$-rings as those that satisfy an analogue of Hilbert’s Nullstellensatz. Furthermore, we show that for every $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$, the collection of $E_\infty$-ring maps from $R$ to such Lubin–Tate theories jointly detect nilpotence. In particular, we deduce that every non-zero $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$ admits an $E_\infty$-ring map to such a Lubin–Tate theory. As consequences, we construct $\mathbb{E}_{\infty}$ complex orientations of algebraically closed Lubin–Tate theories, compute the strict Picard spectra of such Lubin–Tate theories, and prove redshift for the algebraic $\mathrm{K}$-theory of arbitrary $\mathbb{E}_{\infty}$-rings.

We show that there is a type-preserving homomorphism from the fundamental group of the figure-eight knot complement to the mapping class group of the three-punctured torus. As a corollary, we obtain infinitely many commensurability classes of purely pseudo-Anosov surface subgroups of mapping class groups of closed surfaces. This gives the first examples of compact aspherical atoroidal surface bundles over surfaces

The question of the total measure of invariant tori in analytic, nearly integrable Hamiltonian systems is considered. In 1985, Arnol’d, Kozlov and Neishtadt, in the Encyclopaedia of Mathematical Sciences, [4] and in subsequent editions, conjectured that in $n = 2$ degrees of freedom the measure of the non-torus set of general analytic nearly integrable systems away from critical points is exponentially small with the size $\varepsilon$ of the perturbation, and that for $n\ge 3$ the measure is, in general, of order $\varepsilon$ (rather than $\sqrt{\varepsilon}$ as predicted by classical KAM Theory). In the case of generic analytic mechanical Hamiltonian systems, we prove lower bounds on the measure of primary and secondary invariant tori, which are in agreement, up to a logarithmic correction, with the above conjectures.

Let $F$ be a non-archimedean local field and let $G$ be a connected reductive group defined over $F$. We assume that $G$ splits over a tame extension of $F$ and that the residual characteristic $p$ does not divide the order of the Weyl group. To each discrete Langlands parameter of the Weil group of $F$ into the complex $L$-group of $G$ we associate explicitly a finite set of irreducible supercuspidal representations of $G(F)$, and relate its internal structure to the centralizer of the parameter. We give evidence that this assignment is an explicit realization of the local Langlands correspondence

We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is $\mathrm{SO}(2,1)$ or $\mathrm{SO}(3,1)$ and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense subgroup. We show that for every starting point $x\in G/\Lambda$ which is not trapped in a finite $\mu$-invariant set, the $n$-step distribution $\mu^{*n}*\delta_x$ of the walk equidistributes toward the Haar measure. Moreover, under arithmetic assumptions on the pair $(\Lambda,\mu)$, we show the convergence occurs at an exponential rate, tempered by the obstructions that $x$ may be high in a cusp or close to a finite orbit. Our approach is substantially different from that of Benoist–Quint [13], whose equidistribution statements only hold in Cesàro average and are not quantitative, that of Bourgain-Furman-Lindenstrauss-Mozes [21] concerning the torus case, and that of Lindenstrauss-Mohammadi-Wang and Yang [57, 77, 59] about the analogous problem for unipotent flows. A key new feature of our proof is the use of a new phenomenon with we call multi slicing. The latter is a generalization of the discretized project theorems à la Bourgain and we believe it presents independent interest.

The Ramsey number $R(k)$ is the minimum $n \in \mathbb {N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that R(k) \leqslant (4 – \varepsilon )^k for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erdös and Szekeres, proved in 1935.

We determine the asymptotic quantum variance of microlocal lifts of Hecke–Maass cusp forms on the arithmetic compact hyperbolic surfaces attached to maximal orders in quaternion algebras. Our result extends those for the non-compact modular surface obtained by Luo–Sarnak–Zhao, whose method required a cusp. The global arguments in our proof involve an analytic study of the theta correspondence, the interplay between additive and multiplicative harmonic analysis on quaternion algebras, the equidistribution of translates of elementary theta functions, and the Rallis inner product formula. These reduce the proof to local problems involving the construction and analysis of microlocal lifts via integral operators on the group, addressed using an analytic incarnation of the method of coadjoint orbits.

We study intrinsic diophantine approximation on Grassmann varieties. Using a new correspondence between the diophantine properties of a linear subspace $x$ in $\mathbb{R}^d$ and certain diagonal orbits in the space of lattices, we are able to solve some problems suggested by Schmidt in 1967. In particular, we obtain a version of Dirichlet’s principle in this setting with an optimal exponent.

We prove that there exists a $G_\delta$ dense set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument, from showing that for any translation surface the product of the flows in almost every pair of directions is ergodic with respect to Lebesgue measure. This in turn is proven by showing that for every translation surface the flows in almost every pair of directions do not share non-trivial common eigenvalues.

The theory of $\Theta$-stratifications generalizes a classical stratification of the moduli of vector bundles on a smooth curve, the Harder-Narasimhan-Shatz-Nitsure stratification, to any moduli problem that can be represented by an algebraic stack. Using derived algebraic geometry, we develop a structure theory, which is a refinement of the theory of local cohomology, for the derived category of quasi-coherent complexes on an algebraic stack equipped with a $\Theta$-stratification. We then apply this to the $D$-equivalence conjecture, which predicts that birationally equiavlent Calabi-Yau manifolds have equivalent derived categories of coherent sheaves. We prove that any two projective Calabi-Yau manifolds that are biratonally equivalent to a smooth moduli space of Gieseker semistable coherent sheaves on a $K3$ surface have equivalent derived categories. These are the first known cases of the $D$-equivalence conjecture for a birational equivalence class in dimension greater than three.

We give the first examples of knots $K_1$, $K_2$ in the $3$-sphere for which their unknotting numbers satisfy $u(K_1 \# K_2) \lt u(K_1)+ u(K_2)$. This answers question 1.69(B) from Kirby’s problem list, in the negative.

In this paper we show that iterating Nash blowups or normalized Nash blowups does not resolve the singularities of algebraic varieties of dimension four or higher over an algebraically closed field of arbitrary characteristic.

We prove that there exist closed, aspherical, smooth 4-manifolds that are homeomorphic but not diffeomorphic. These provide counterexamples to a smooth analog of the Borel conjecture in dimension four.

We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length\nonbreakingspace $N$ taking values of size close to $N^{3/4}$, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate $N(\sigma ,T)\le T^{30(1-\sigma )/13+o(1)}$ and asymptotics for primes in short intervals of length $x^{17/30+o(1)}$.

We present a counterexample to Viterbo’s volume-capacity conjecture. This implies, in particular, that in contrast with a well-known conjecture, symplectic capacities do not coincide on the class of convex domains in the classical phase space.

Unconditionally, we prove the Iwaniec–Sarnak conjecture for $L^4$-norms of the Hecke–Maass cusp forms. From this result, we can justify that for even Maass cusp form $\phi$ with the eigenvalue $\lambda_{\phi}=\frac{1}{4}+t_{\phi}^2$, for $a>0$, a sufficiently large $h>0$ and for any $0<\epsilon_1<\epsilon/10^7$ ($\epsilon>0$), for almost all $1\le k < t_{\phi}^{1-\epsilon}$, we are able to find $\beta_k = \{X_k +yi : a < y < a+h\}$ with $-\frac{1}{2} + \frac{k-1}{t_{\phi}^{1-\epsilon}} \le X_k \le -\frac{1}{2} +\frac{k}{t_{\phi}^{1-\epsilon}}$ such that the number of sign changes of $\phi$ along the segment $\beta_k$ is $\gg_{\epsilon} t_{\phi}^{1-\epsilon_1}$ as $t_\phi\to \infty$. Also, we obtain the similar result for horizontal lines. On the other hand, we conditionally prove that for a sufficiently large segment $\beta$ on $\mathrm{Re}(z) = 0$ and $\mathrm{Im}(z) > 0$, the number of sign changes of $\phi$ along $\beta$ is $\gg_\epsilon t_{\phi}^{1-\epsilon}$ and consequently, the number of inert nodal domains meeting any compact vertical segment on the imaginary axis is $\gg_{\epsilon} t_{\phi}^{1-\epsilon}$ as $t_{\phi}\to\infty$.

A family of random matrices $\mathbf{X}^N=(X_1^N,\dots,X_d^N)$ is said to converge strongly to a family of bounded operators $\mathbf{x}=(x_1,\dots,x_d)$ when $\Vert P(\mathbf{X}^N,\mathbf{X}^{N\ast})\Vert \to \Vert P(\mathbf{x},\mathbf{x}^*)\Vert$ for every noncommutative polynomial $P$. This phenomenon plays a key role in several recent breakthroughs on random graphs, geometry, and operator algebras. However, proofs of strong convergence are notoriously delicate and have relied largely on problem-specific methods. In this paper, we develop a new approach to strong convergence that uses only soft arguments. Our method exploits the fact that for many natural models, the expected trace of $P(\mathbf{X}^N,\mathbf{X}^{N*})$ is a rational function of $\frac{1}{N}$ whose lowest order asymptotic are easily understood. We develop a general technique to deduce strong convergence directly from these inputs using the inequality of A. and V. Markov for univariate polynomials and elementary Fourier analysis. To illustrate the method, we develop the following applications: 1. We give a short proof of the result of Friedman that random regular graphs have a near-optimal spectral gap, and obtain a sharp understanding of the large deviations probabilities of the second eigenvalue. 2. We prove a strong quantitative form of the strong convergence property of random permutation matrices due to Bordenave and Collins. 3. We extend the above to any stable representation of the symmetric group, providing many new examples of the strong convergence phenomenon.

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\mathrm{SL}_2(\mathbb{R})$ in arithmetic quotients of $\mathrm{SL}_2(\mathbb{C})$ and $\mathrm{SL}_2(\mathrm{R}) \times \mathrm{SL}_2(\mathbb{R})$. The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.

An explicit construction of locally testable codes of constant rate, constant distance and constant number of queries is given. Hence answering affirmatively the $c^3$-problem. roblem.