数学前沿

We prove the Relative Manin–Mumford Conjecture for families of abelian varieties in characteristic 0. We follow the Pila–Zannier method to study special point problems, and we use the Betti map which goes back to work of Masser and Zannier in the case of curves. The key new ingredients compared to previous applications of this approach are a height inequality proved by both authors of the current paper and Dimitrov, and the first-named author’s study of certain degeneracy loci in subvarieties of abelian schemes. We also strengthen this result and prove a criterion for torsion points to be dense in a subvariety of an abelian scheme over ℂ. The Uniform Manin–Mumford Conjecture for curves embedded in their Jacobians was first proved by Kühne. We give a new proof, as a corollary to our main theorem, that does not use equidistribution.

We show that the cylindrical tangent cone $C\times \mathbf{R}$ for an area-minimizing hypersurface is unique, where $C$ is the Simons cone $C_{S}= C(S^{3}\times S^{3})$. Previously Simon proved a uniqueness result for cylindrical tangent cones that applies to a large class of cones $C$, however not to the Simons cone. The main new difficulty is that the cylindrical cone $C_{S}\times \mathbf{R}$ is not integrable, and we need to develop a suitable replacement for Simon’s infinite dimensional Łojasiewicz inequality in the setting of tangent cones with non-isolated singularities.

We prove that if $u : B_{1} \subset \mathbb{R}^{2} \rightarrow \mathbb{R}^{n}$ is a Lipschitz critical point of the area functional with respect to outer variations, then $u$ is smooth. This solves a conjecture of Lawson and Osserman from 1977 in the planar case.

We prove a Witt vector version of the usual Grauert–Riemenschneider vanishing theorem over perfect fields of positive characteristic, solving a question raised by Blickle, Esnault, Chatzistamatiou and Rülling. We then deduce some rationality consequences for $F$–rational singularities.

We study the Fröhlich polaron in the regime of strong coupling and prove the asymptotically sharp lower bound on the effective mass $m_{\mathrm{eff}}(\alpha )\geq \alpha ^{4} m_{\mathrm{LP}}-C\alpha ^{4- \epsilon }$, where $m_{\mathrm{LP}}$ is the constant predicted by L. D. Landau and S. I. Pekar [13] in 1948 for the effective mass of the classical polaron. Together with the corresponding upper bound, recently verified by R. Seiringer and the author, we confirm the validity of the celebrated Landau-Pekar formula for the effective mass of the Fröhlich polaron $\underset{\alpha \rightarrow \infty }{\lim }\alpha ^{-4}m_{ \mathrm{eff}}(\alpha )=m_{\mathrm{LP}}$, which has been conjectured by H. Spohn [27] in 1987.

This is the first part of a two-paper sequence establishing the global existence of $3D$ relativistic Vlasov-Maxwell system (RVM) for _arbitrarily large_ smooth localized initial data with cylindrical symmetry. This paper employs Part II’s pointwise estimates–developed independently of Part I–as fundamental tools to demonstrate the iterative smoothing scheme (ISS), which is originated and inspired by   the work of Klainerman-Staffilani in 2002.  Using ISS and a novel singular weighted space-time estimate for the distribution function, we prove upper bounds for the projection and full velocity characteristics via a standard bootstrap argument.  Using the classic momentum method, we find that   the high order momentum at most grows polynomially in time. This further  implies  that the $L^\infty_{x}$-norm of the electromagnetic field, $|(E(t),B(t)|_{L^\infty_x}$, also grows at most polynomially over time. Consequently, this verifies the continuation criteria obtained by Luk-Strain in 2014. As a result, the energy function of $3D$ RVM system doesn’t blow up in finite time,  thereby establishing  global existence.”

We extend Falconer’s 1988 landmark result on the dimensions of self-affine fractals to encompass the dimensions of their projections, showing furthermore that their families of exceptional projections contain algebraic varieties which are preserved by the underlying linear algebraic group. The techniques which we develop allow us to construct examples of additional new phenomena: firstly, we give general examples of equilibrium measures on self-affine fractals which admit non-exact-dimensional projections. Secondly, we construct strongly irreducible self-affine sets which have small sumsets without any arithmetic resonance in their construction.

We prove Zagier’s conjecture on the value at $s=4$ of the Dedekind $\zeta$-function of a number field $\mathrm{F}$: $$\zeta_{\mathrm{F}}(4) = \pi^{4(r_1+r_2)} |d_{\mathrm{F}}|^{-1/2} \cdot \mathrm{det}\Bigl(\mathcal{L}_4(\sigma_i(y_j))\Bigr), \quad 1\le i,j \le r_2.$$ Here $\{\sigma_2\}$ is the set of all complex embeddings of $\mathrm{F}$, up to the conjugation. Namely, up to a standard factor $\zeta_{\mathrm{F}}(4)$ is equal to a $r_2\times r_2$ determinant, whose entries are $\mathbb{Q}$-linear combinations of the values of a single-valued version of $\mathcal{L}_4(z)$ of the classical $4$-logarithm at some numbers in $\sigma_j(\mathrm{F})$. These linear combinations are subject to a specific condition discovered by Zagier. For an infinite field $\mathrm{F}$, we define a map of $K$-groups $K_{8-i}(\mathrm{F})$, $i=1,\ldots,4$, to the $i$-th cohomology of the weight $4$ polylogarithmic motivic complex $\mathcal{B}^\bullet(\mathrm{F},4)$. When $\mathrm{F}$ is the function field of a complex variety, composing the map with the regulator map on the polylogarithmic complex to the Deligne cohomology, we get a $\mathbb{Q}^\times$-multiple of Beilinson’s regulator. This implies that the comdposition $K_7(\mathbb{C}) \to H^1\mathcal{B}^\bullet(\mathbb{C},4) \to \mathbb{R}$, where the second map is given by $\mathcal{L}_4(z)$, is a $\mathbb{Q}^\times$-multiple of Borel’s regulator. This plus Borel’s theorem on the ranks of algebraic $K$-groups of number fields implies Zagier’s conjecture. We get a strong evidence for a part of Freenss Conjecture describing the weight four part $\mathcal{L}_4(\mathrm{F})$ of the motivic Lie coalgebra of $\mathrm{F}$ via higher Bloch groups $\mathcal{B}_j(\mathrm{F})$ as an extension: $$0\to \mathcal{B}_4(\mathrm{F}) \to \mathcal{L}_4(\mathrm{F} \to \Lambda^2\mathcal{B}_2(\mathrm{F})\to 0.$$ The main tools are motivic correlators and a new link of cluster varieties to polylogarithms.

We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of uniform hypergraphs that satisfy a certain pseudorandomness condition. As a further generalisation, we obtain the same conclusion only assuming an extendability property and the existence of a robust fractional clique decomposition

We compute the rational homotopy groups in degrees up to approximately $\tfrac{3}{2}d$ of the group of diffeomorphisms of a closed $d$-dimensional disc fixing the boundary. Based on this we determine the optimal rational concordance stable range for high-dimensional discs, describe the rational homotopy type of $B\mathrm {Top}(d)$ in a range, and calculate the second rational derivative of the functor $B\mathrm {Top}(-)$ in the sense of Weiss’ orthogonal calculus.

Take a closed monotone symplectic manifold containing a smooth anticanonical divisor. The quantum connection on its cohomology has singularities at zero and infinity (in the quantum parameter). At zero it has a regular singular point, by definition. We show that the singularity at infinity is of unramified exponential type. The argument involves: realizing cohomology as a deformation of the symplectic cohomology of the divisor complement; the corresponding deformation of the wrapped Fukaya category; a new categorical interpretation of the Fourier-Laplace transform of $D$-modules; and the regularity theorem of Petrov-Vaintrob-Vologodsky in noncommutative geometry.

We prove that an irreducible lattice in a connected real semi-simple Lie group of real rank at least two and finite center is not left-orderable.

We show that the Khovanov-Rozansky $\mathfrak{gl}_2$ skein lasagna module distinguishes the exotic pair of knot traces $X_{-1}(-5_2)$ and $X_{-1}(P(3,-3,-8))$, an example first discovered by Akbulut. This gives the first analysis-free proof of the existence of exotic compact orientable $4$-manifolds. We also present a family of exotic knot traces that seem not directly recoverable from gauge/Floer-theoretic methods. Along the way, we present new explicit calculations of the Khovanov skein lasagna modules, and we define lasagna generalizations of the Lee homology and Rasmussen $s$-invariant, which are of independent interest. Other consequences of our work include a slice obstruction of knots in $4$-manifolds with nonvanishing skein lasagna module, a sharp shake genus bound for some knots from the lasagna $s$-invariant, and a construction of induced maps on Khovanov homology for cobordisms in $k\mathbb{C}\mathbb{P}^2$.

We give an infinite family of embeddings of $\mathbb{R}P^2$ into $S^4$ such that they are mutually topologically isotopic but are not smoothly isotopic to each other. Moreover, they are topologically isotopic to the standard $P^2$-knot. To prove that these $P^2$-knots are not smoothly isotopic to each other, we construct a gauge theoretic invariant of embedded surfaces in $4$-manifolds using a variant of the Seiberg–Witten theory, which is called the Real Seiberg–Witten theory.

In this paper, we prove Deligne’s conjecture on the algebraicity of the critical values of symmetric power $L$-functions associated with modular forms of weight at least $5$. We also establish new cases of Blasius’ conjecture on the algebraicity of the critical values of tensor product $L$-functions associated with modular forms. Additionally, we prove an algebraicity result for the critical values of Rankin–Selberg $L$-functions for $\mathrm{GL}_n \times \mathrm{GL}_2$ in the unbalanced case, which extends the previous results of Furusawa and Morimoto for $\mathrm{SO}(V) \times \mathrm{GL}_2$. These results are applications of our main theorem on the algebraicity of cross ratios of special values of Rankin–Selberg $L$-functions.

Let $k \in \mathbb{Z}_+$ and $(X,\mathcal{B}(X),\mu)$ be a probability space equipped with a family of commuting invertible measure-preserving transformations $T_1,\ldots,T_k: X\to X$. Let $P_1,\ldots,P_k\in \mathbb{Z}[n]$ be polynomials with integer coefficients and distinct degrees. We establish pointwise almost everywhere convergence of the multilinear polynomial ergodic averages $$A_{N;X,T_1,\ldots,T+k}^{P_1,\ldots P_k} (f_1,\ldots,f_k)(x) := \frac{1}{N} \sum_{n=1}^N f_1\bigl(T_1^{P_1(n)}x\bigr) \cdots f_k\bigl(T_k^{P_k(n)}x\bigr), \quad x\in X,$$ as $N\to \infty$ for any functions $f_1,\ldots,f_k \in L^\infty(K)$. Besides a couple of results in the bilinear setting (when $k = 2$ and then only for single transformations), this is the first pointwise result for general polynomial ergodic averages in arbitrary measure-preserving systems. This answers a question of Bergelson from 1996 in the affirmative for any polynomials with distinct degrees, and makes progress on the Furstenberg–Bergelson–Leibman conjecture. In this paper, we build a versatile _multilinear circle method_ by developing the Ionescu–Wainger multiplier theorem for the set of canonical fractions, which gives a positive answer to a question of Ionescu and Wainger from 2005. We also establish multilinear $L^p$-improving bounds and an inverse theorem in higher order Fourier analysis for averages over polynomial corner configurations, which we use to establish a multilinear analogue of Weyl’s inequality and its real counterpart, a Sobolev smoothing estimate.

For all algebraic groups over non-Archimedean local fields, the bounded cohomology vanishes. This follows from the corresponding statement for automorphism groups of Bruhat–Tits buildings, which hinges on the solution to the flatmate conjecture raised in earlier work with Bucher. Vanishing and invariance theorems for arithmetic groups are derived.

Under a strong genericity condition, we prove the local analogue of the ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil–Buzzard–Emerton on the crystalline slopes of Kisin’s crystabelline deformation spaces, (b) Gouvêa’s $\lfloor\frac{k-1}{p+1}\rfloor$-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurves. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and very generic case, (d) Gouvêa–Mazur conjecture, (e) a variant of Gouvêa’s conjecture on slope distributions, and (f) a refined version of Coleman–Mazur–Buzzard–Kilford spectral halo conjecture.

We prove a conjecture of Schmid and the second named author [SV1] that the unitarity of a representation of a real reductive Lie group with real infinitesimal character can be read off from a canonical filtration, the Hodge filtration. Our proof rests on three main ingredients. The first is a wall crossing theory for mixed Hodge modules: the key result is that, in certain natural families, the Hodge filtration varies semi-continuity with jumps controlled by extension functors. The second ingredient is a Hodge-theoretic refinement of Beilinson-Bernstein localization: we show that the Hodge filtration of a mixed Hodge module on the flag variety satisfies the usual cohomology vanishing and global generation properties enjoyed by the underlying $\mathcal{D}$-module. The third ingredient is an explicit calculation of the Hodge filtration on a tempered Hodge module. As byproucts of our work, we obtain a version of Saito’s Kodaira vanishing for twisted mixed Hodge modules, a calculation of the Hodge filtration on a certain object in category $\mathcal{O}$, and a host of new vanishing results for coherent sheaves on flag varieties.

In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures; our general results describe the differential algebraic relations between the solutions of the partial differential equation satisfied by the uniformizers. In particular, we give a proof of the full Ax-Schanuel Theorem with derivatives for some uniformizers of simple projective structure on curves including uniformizers of any Fuchsian group of the first kind and any genus. Combining our techniques with those of Ax, we give a strong Ax-Schanuel result for the combination of the derivatives of the $j$-function and the exponential function. In the general setting of Shimura varieties, we obtain an Ax-Schanuel theorem for the derivatives of uniformizing maps. Our techniques combine tools from differential geometry, differential algebra and the model theory of differentially closed fields.