Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category ${\mathcal {C}}$ and ${\mathcal {C}}$-module category ${\mathcal {M}}$, we introduce a version of the Drinfeld center ${\mathcal {Z}}({\mathcal {C}})$ of ${\mathcal {C}}$ adapted for ${\mathcal {M}}$; we refer to this category as the reflective center ${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$ of ${\mathcal {M}}$. Just like ${\mathcal {Z}}({\mathcal {C}})$ is a canonical braided monoidal category attached to ${\mathcal {C}}$, we show that ${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$ is a canonical braided module category attached to ${\mathcal {M}}$; its properties are investigated in detail.

Our second goal pertains to when ${\mathcal {C}}$ is the category of modules over a quasitriangular Hopf algebra H, and ${\mathcal {M}}$ is the category of modules over an H-comodule algebra A. We show that the reflective center ${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$ here is equivalent to a category of modules over an explicit algebra, denoted by $R_H(A)$, which we call the reflective algebra of A. This result is akin to ${\mathcal {Z}}({\mathcal {C}})$ being represented by the Drinfeld double ${\operatorname {Drin}}(H)$ of H. We also study the properties of reflective algebras.

Our third set of results is also in the Hopf setting above. We show that reflective algebras are quasitriangular H-comodule algebras, and we examine their corresponding quantum K-matrices; this yields solutions to the qRE. We also establish that the reflective algebra $R_H(\mathbb {k})$ is an initial object in the category of quasitriangular H-comodule algebras, where $\mathbb {k}$ is the ground field. The case when H is the Drinfeld double of a finite group is illustrated.

The consistency of the theory $\mathsf {ZF} + \mathsf {AD}_{\mathbb {R}} + {}$‘every set of reals is universally Baire’ is proved relative to $\mathsf {ZFC} + {}$‘there is a cardinal that is a limit of Woodin cardinals and of strong cardinals’. The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf {ZF} + \mathsf {AD}_{\mathbb {R}} + {}$‘every set of reals is Suslin’ is consistent relative to $\mathsf {ZFC} + {}$‘there is a cardinal $\lambda $ that is a limit of Woodin cardinals and of $\mathord {<}\lambda $-strong cardinals’. The $\Sigma ^2_1$ reflection property of our model is proved using genericity iterations as in Neeman [18] and Steel [22].

We develop a theory of generalized characters of local systems in $\infty $-categories, which extends classical character theory for group representations and, in particular, the induced character formula. A key aspect of our approach is that we utilize the interaction between traces and their categorifications. We apply this theory to reprove and refine various results on the composability of Becker-Gottlieb transfers, the Hochschild homology of Thom spectra, and the additivity of traces in stable $\infty $-categories.

We prove that the minimal exponent for local complete intersections satisfies an Inversion-of-Adjunction property. As a result, we also obtain the Inversion of Adjunction for higher Du Bois and higher rational singularities for local complete intersections.

In 1967, Gerencsér and Gyárfás [16] proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of $K_n$, there is a monochromatic path on $\lceil (2n+1)/3\rceil $ vertices, and this is best possible. There have since been hundreds of papers on graph-Ramsey theory with some of the most important results being motivated by a series of conjectures of Burr and Erdős [2, 3] regarding the Ramsey numbers of trees (settled in [31]), graphs with bounded maximum degree (settled in [5]), and graphs with bounded degeneracy (settled in [23]).

In 1993, Erdős and Galvin [13] began the investigation of a countably infinite analogue of the Gerencsér and Gyárfás result: What is the largest d such that in every $2$-coloring of $K_{\mathbb {N}}$ there is a monochromatic infinite path with upper density at least d? Erdős and Galvin showed that $2/3\leq d\leq 8/9$, and after a series of recent improvements, this problem was finally solved in [7] where it was shown that $d={(12+\sqrt {8})}/{17}$.

This paper begins a systematic study of quantitative countably infinite graph-Ramsey theory, focusing on infinite analogues of the Burr-Erdős conjectures. We obtain some results which are analogous to what is known in finite case, and other (unexpected) results which have no analogue in the finite case.

The goal of this note is to prove that Hodge-de Rham degeneration holds for smooth and proper $\mathbf {F}_p$-schemes X with $\dim (\text{X})<p^n$ assuming that two conditions hold: its category of quasicoherent sheaves admits a lift to the truncated Brown-Peterson spectrum $\mathrm {BP}\langle {n-1}\rangle $, and the Hochschild-Kostant-Rosenberg spectral sequence for X degenerates at the $\mathrm{E}_2$-page. This result is obtained from a noncommutative version thereof, whose proof is essentially the same as Mathew’s argument in [Mat20].

We introduce the notion of echeloned spaces – an order-theoretic abstraction of metric spaces. The first step is to characterize metrizable echeloned spaces. It turns out that morphisms between metrizable echeloned spaces are uniformly continuous or have a uniformly discrete image. In particular, every automorphism of a metrizable echeloned space is uniformly continuous, and for every metric space with midpoints, the automorphisms of the induced echeloned space are precisely the dilations.

Next, we focus on finite echeloned spaces. They form a Fraïssé class, and we describe its Fraïssé-limit both as the echeloned space induced by a certain homogeneous metric space and as the result of a random construction. Building on this, we show that the class of finite ordered echeloned spaces is Ramsey. The proof of this result combines a combinatorial argument by Nešetřil and Hubička with a topological-dynamical point of view due to Kechris, Pestov and Todorčević. Finally, using the method of Katětov functors due to Kubiś and Mašulović, we prove that the full symmetric group on a countable set topologically embeds into the automorphism group of the countable universal homogeneous echeloned space.

We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show the following: (1) If the extremal geometry contains lines, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a cubic norm structure; (2) If there exists a field extension of degree at most $2$ such that the extremal geometry over that field extension contains lines, and in addition, there exist symplectic pairs of extremal elements, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a quadrangular algebra.

One of our key tools is a new definition of exponential maps that makes sense even over fields of characteristic $2$ and $3$, which ought to be interesting in its own right.

Not only was Jacques Tits a constant source of inspiration through his work, he also had a direct personal influence, notably through his threat to speak evil of our work if it did not include the characteristic 2 case.

—The Book of Involutions [KMRT98, p. xv]

We describe the modulo $2$ de Rham-Witt complex of a field of characteristic $2$, in terms of the powers of the augmentation ideal of the $\mathbb {Z}/2$-geometric fixed points of real topological restriction homology ${\mathrm {TRR}}$. This is analogous to the conjecture of Milnor, proved in [Kat82] for fields of characteristic $2$, which describes the modulo $2$ Milnor K-theory in terms of the powers of the augmentation ideal of the Witt group of symmetric forms. Our proof provides a somewhat explicit description of these objects, as well as a calculation of the homotopy groups of the geometric fixed points of ${\mathrm {TRR}}$ and of real topological cyclic homology, for all fields.

This article presents new rationality results for the ratios of critical values of Rankin–Selberg L-functions of $\mathrm {GL}(n) \times \mathrm {GL}(n')$ over a totally imaginary field $F.$ The proof is based on a cohomological interpretation of Langlands’s contant term theorem via rank-one Eisenstein cohomology for the group $\mathrm {GL}(N)/F,$ where $N = n+n'.$ The internal structure of the totally imaginary base field has a delicate effect on the Galois equivariance properties of the critical values.

Let R be a commutative ring. One may ask when a general R-module P that satisfies $P \oplus R \cong R^n$ has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: if $V_r(\mathbb {A}^n)$ denotes the variety $\operatorname {GL}(n) / \operatorname {GL}(n-r)$ over a field k, then the projection $V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$ has a section if and only if the following holds: any module P over any k-algebra R with the property that $P \oplus R \cong R^n$ has a free summand of rank $r-1$. Using techniques from $\mathbb {A}^1$-homotopy theory, we characterize those n for which the map $V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$ has a section in the cases $r=3,4$ under some assumptions on the base field.

We conclude that if $P \oplus R \cong R^{24m}$ and R contains the field of rational numbers, then P contains a free summand of rank $2$. If R contains a quadratically closed field of characteristic $0$, or the field of real numbers, then P contains a free summand of rank $3$. The analogous results hold for schemes and vector bundles over them.

Let S be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration $f:\,S \to B$ of genus g. We prove a quadratic upper bound on the genus g (i.e., $g\leq h\big (\chi (\mathcal {O}_S)\big )$, where h is a quadratic function). We also construct examples showing that the quadratic upper bounds cannot be improved to linear ones. In the special case when $p_g(S)=q(S)=1$, we show that $g\leq 14$.

We prove an asymptotic formula for the second moment of central values of Dirichlet L-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulo d inside the full group of characters modulo q. Suppose that $\nu _p(d) \geq \nu _p(q)/2$ for all primes p dividing q. In this range, we obtain an asymptotic formula with a power-saving error term; curiously, there is a secondary main term of rough size $q^{1/2}$ here which is not predicted by the integral moments conjecture of Conrey, Farmer, Keating, Rubinstein, and Snaith. The lower-order main term does not appear in the second moment of the Riemann zeta function, so this feature is not anticipated from the analogous archimedean moment problem.

We also obtain an asymptotic result for smaller d, with $\nu _p(q)/3 \leq \nu _p(d) \leq \nu _p(q)/2$, with a power-saving error term for d larger than $q^{2/5}$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughly d, which is only slightly smaller than the diagonal main term.

Kontsevich ([Kir95, Problem 3.48]) conjectured that $\mathrm {BDiff}(M, \text {rel }\partial )$ has the homotopy type of a finite CW complex for all compact $3$-manifolds with nonempty boundary. Hatcher-McCullough ([HM97]) proved this conjecture when M is irreducible. We prove a homological version of Kontsevich’s conjecture. More precisely, we show that $\mathrm {BDiff}(M, \text {rel }\partial )$ has finitely many nonzero homology groups each finitely generated when M is a connected sum of irreducible $3$-manifolds that each have a nontrivial and non-spherical boundary.

Let $\mathcal {X}\to \mathbb {D}$ be a flat family of projective complex 3-folds over a disc $\mathbb {D}$ with smooth total space $\mathcal {X}$ and smooth general fibre $\mathcal {X}_t,$ and whose special fiber $\mathcal {X}_0$ has double normal crossing singularities, in particular, $\mathcal {X}_0=A\cup B$, with A, B smooth threefolds intersecting transversally along a smooth surface $R=A\cap B.$ In this paper, we first study the limit singularities of a $\delta $-nodal surface in the general fibre $S_t\subset \mathcal {X}_t$, when $S_t$ tends to the central fibre in such a way its $\delta $ nodes tend to distinct points in R. The result is that the limit surface $S_0$ is in general the union $S_0=S_A\cup S_B$, with $S_A\subset A$, $S_B\subset B$ smooth surfaces, intersecting on R along a $\delta $-nodal curve $C=S_A\cap R=S_B\cap B$. Then we prove that, under suitable conditions, a surface $S_0=S_A\cup S_B$ as above indeed deforms to a $\delta $-nodal surface in the general fibre of $\mathcal {X}\to \mathbb {D}$. As applications, we prove that there are regular irreducible components of the Severi variety of degree d surfaces with $\delta $ nodes in $\mathbb {P}^3$, for every $\delta \leqslant {d-1\choose 2}$ and of the Severi variety of complete intersection $\delta $-nodal surfaces of type $(d,h)$, with $d\geqslant h-1$ in $\mathbb {P}^4$, for every $\delta \leqslant {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$

We study the problem of the irreducibility of the Hessian variety ${\mathcal {H}}_f$ associated with a smooth cubic hypersurface $V(f)\subset {\mathbb {P}}^n$. We prove that when $n\leq 5$, ${\mathcal {H}}_f$ is normal and irreducible if and only if f is not of Thom-Sebastiani type (i.e., if one cannot separate its variables by changing coordinates). This also generalizes a result of Beniamino Segre dealing with the case of cubic surfaces. The geometric approach is based on the study of the singular locus of the Hessian variety and on infinitesimal computations arising from a particular description of these singularities.

A conjecture of Jackson from 1981 states that every d-regular oriented graph on n vertices with $n\leq 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large n. In fact we prove a more general result that for all $\alpha>0$, there exists $n_0=n_0(\alpha )$ such that every d-regular digraph on $n\geq n_0$ vertices with $d \ge \alpha n $ can be covered by at most $n/(d+1)$ vertex-disjoint cycles, and moreover that if G is an oriented graph, then at most $n/(2d+1)$ cycles suffice.

We study the generalized Ramsey–Turán function $\mathrm {RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an n-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by Erdős, Sós, Bollobás, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all $s\ge 3$, showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when t is much larger than s. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold.

Let k be an algebraically closed field of characteristic $p>0$. Let X be a normal projective surface over k with canonical singularities whose anticanonical divisor is nef and big. We prove that X is globally F-regular except for the following cases: (1) $K_X^2=4$ and $p=2$, (2) $K_X^2=3$ and $p \in \{2, 3\}$, (3) $K_X^2=2$ and $p \in \{2, 3\}$, (4) $K_X^2=1$ and $p \in \{2, 3, 5\}$. For each degree $K_X^2$, the assumption of p is optimal.

Let X be a toric Calabi-Yau 3-fold and let $L\subset X$ be an Aganagic-Vafa outer brane. We prove two versions of open WDVV equations for the open Gromov-Witten theory of $(X,L)$. The first version of the open WDVV equation leads to the construction of a semi-simple (formal) Frobenius manifold, and the second version leads to the construction of a flat (formal) F-manifold.