A celebrated result of Rödl and Ruciński states that for every graph $F$, which is not a forest of stars and paths of length 3, and fixed number of colours $r\geqslant 2$ there exist positive constants $c,C$ such that for $p\leqslant cn^{-1/m_{2}(F)}$ the probability that every colouring of the edges of the random graph $G(n,p)$ contains a monochromatic copy of $F$ is $o(1)$ (the ‘0-statement’), while for $p\geqslant Cn^{-1/m_{2}(F)}$ it is $1-o(1)$ (the ‘1-statement’). Here $m_{2}(F)$ denotes the 2-density of $F$. On the other hand, the case where $F$ is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in $G(n,p)$. Recently, the natural extension of the 1-statement of this theorem to $k$-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rödl and Schacht. In particular, they showed an upper bound of order $n^{-1/m_{k}(F)}$ for the 1-statement, where $m_{k}(F)$ denotes the $k$-density of $F$. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of threshold exists if $k\geqslant 4$: there are $k$-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour class. Along the way we obtain a general bound on the 1-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Spöhel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.

We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^{1}$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.

We show that the perfect derived categories of Iyama’s d-dimensional Auslander algebras of type ${\mathbb {A}}$ are equivalent to the partially wrapped Fukaya categories of the d-fold symmetric product of the $2$-dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk and those of its $(n-d)$-fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type ${\mathbb {A}}$. As a by-product of our results, we deduce that the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk organise into a paracyclic object equivalent to the d-dimensional Waldhausen $\text {S}_{\bullet }$-construction, a simplicial space whose geometric realisation provides the d-fold delooping of the connective algebraic K-theory space of the ring of coefficients.

For a split reductive group $G$ over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated $G$-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic $t$-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of $\ell$ in the Langlands correspondence for function fields.

We establish the Bernstein-centre type of results for the category of mod p representations of $\operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p)$ . We treat all the remaining open cases, which occur when p is $2$ or $3$ . Our arguments carry over for all primes p. This allows us to remove the restrictions on the residual representation at p in Lue Pan’s recent proof of the Fontaine–Mazur conjecture for Hodge–Tate representations of $\operatorname {\mathrm {Gal}}(\overline {\mathbb Q}/\mathbb {Q})$ with equal Hodge–Tate weights.

The hyperbolic quadratic eigenvalue problem (HQEP) was shown to admit Courant–Fischer type min–max principles in 1955 by Duffin and Cauchy type interlacing inequalities in 2010 by Veselić. It can be regarded as the closest analog (among all kinds of quadratic eigenvalue problem) to the standard Hermitian eigenvalue problem (among all kinds of standard eigenvalue problem). In this paper, we conduct a systematic study on the HQEP both theoretically and numerically. On the theoretical front, we generalize Wielandt–Lidskii type min–max principles and, as a special case, Fan type trace min/max principles and establish Weyl type and Wielandt–Lidskii–Mirsky type perturbation results when an HQEP is perturbed to another HQEP. On the numerical front, we justify the natural generalization of the Rayleigh–Ritz procedure with existing principles and our new optimization principles, and, as consequences of these principles, we extend various current optimization approaches—steepest descent/ascent and nonlinear conjugate gradient type methods for the Hermitian eigenvalue problem—to calculate a few extreme eigenvalues (of both positive and negative type). A detailed convergence analysis is given for the steepest descent/ascent methods. The analysis reveals the intrinsic quantities that control convergence rates and consequently yields ways of constructing effective preconditioners. Numerical examples are presented to demonstrate the proposed theory and algorithms.

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\,Y/\text{Br}_{1}\,Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron–Severi lattices. Over a field of characteristic $0$, we prove that the existence of a strong uniform bound on the size of the odd torsion of $\text{Br}Y/\text{Br}_{1}Y$ is equivalent to the existence of a strong uniform bound on integers $n$ for which there exist non-CM elliptic curves with abelian $n$-division fields. Using the same methods we show that, for a fixed prime $\ell$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric Néron–Severi lattice, $\#(\text{Br}Y/\text{Br}_{1}Y)[\ell ^{\infty }]$ is bounded by a constant that depends only on $\ell$, $r$, and the discriminant.

We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

We study the Chow ring of the moduli stack $\mathfrak {M}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. This paper provides foundations for the paper [BS21].

In the appendix (jointly with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of $\mathfrak {M}_{g,n}$ and is important in its own right. We also develop operational Chow groups for algebraic stacks.

Let $\mathsf {C}$ be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to $\mathsf {C}$ . We prove that the decorated double Bott–Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras.

We explicitly describe the Donaldson–Thomas transformations on double Bott–Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock–Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson–Thomas transformations on a family of double Bott–Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov’s periodicity conjecture in the cases of $\Delta \square \mathrm {A}_r$ .

When $\mathsf {C}$ is of type $\mathrm {A}$ , the double Bott–Samelson cells are isomorphic to Shende–Treumann–Zaslow’s moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their $\mathbb {F}_q$ -points we obtain rational functions that are Legendrian link invariants.

The jaggedness of an order ideal $I$ in a poset $P$ is the number of maximal elements in $I$ plus the number of minimal elements of $P$ not in $I$. A probability distribution on the set of order ideals of $P$ is toggle-symmetric if for every $p\in P$, the probability that $p$ is maximal in $I$ equals the probability that $p$ is minimal not in $I$. In this paper, we prove a formula for the expected jaggedness of an order ideal of $P$ under any toggle-symmetric probability distribution when $P$ is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.

We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm (WCGA), a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in $L_p$, $2\le p<\infty $.

We consider smooth, complex quasiprojective varieties $U$ that admit a compactification with a boundary, which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.

We consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck’s conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori-double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.

We incorporate the new theory of equivariant moving frames for Lie pseudogroups into Vessiot’s method of group foliation of differential equations. The automorphic system is replaced by a set of reconstruction equations on the pseudogroup jets. The result is a completely algorithmic and symbolic procedure for finding both invariant and noninvariant solutions of differential equations admitting a symmetry group.

We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries.

We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.