The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.

We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$-fold delooping of the filtered loop space $E_2$-groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle $S^1_{fil}$ of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an $E_2$-cogroupoid object in the $\infty $-category of spaces. We relate this cogroupoid structure with the more commonly studied ‘pinch map’ on $S^1$, as well as the Todd class of the Lie algebroid $\mathbb {T}_{X}$; this is an invariant of a smooth and proper scheme X that arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontrivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally, we record some consequences of this bit of structure at the level of Hochschild cohomology.

In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.

Let $\boldsymbol {k} := (k_1,\ldots ,k_s)$ be a sequence of natural numbers. For a graph G, let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ such that, for every $c \in \{1,\dots ,s\}$, the edges of colour c contain no clique of order $k_c$. Write $F(n;\boldsymbol {k})$ to denote the maximum of $F(G;\boldsymbol {k})$ over all graphs G on n vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for $n \to \infty $:

1. (i) A sufficient condition on $\boldsymbol {k}$ which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results.

2. (ii) Addressing the original question of Erdős and Rothschild, in the case $\boldsymbol {k}=(3,\ldots ,3)$ of length $7$, the unique extremal graph is the complete balanced $8$-partite graph, with colourings coming from Hadamard matrices of order $8$.

3. (iii) In the case $\boldsymbol {k}=(k+1,k)$, for which the sufficient condition in (i) does not hold, for $3 \leq k \leq 10$, the unique extremal graph is complete k-partite with one part of size less than k and the other parts as equal in size as possible.

We introduce Chern classes in $U(m)$-equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the $\mathbf {MU}$-cohomology of $B U(m)$. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the $\mathbf {MU}$-completion theorem of Greenlees–May and La Vecchia.

The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains.

We consider the symmetric group, $S_n$, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of global functions on $S_n$, which are functions whose $2$-norm remains small when restricting $O(1)$ coordinates of the input, and assert that low-degree, global functions have small q-norms, for $q>2$.

As applications, we show the following:

1. 1. An analog of the level-d inequality on the hypercube, asserting that the mass of a global function on low degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group $A_n$.

2. 2. Isoperimetric inequalities on the transposition Cayley graph of $S_n$ for global functions that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube.

3. 3. Hypercontractive inequalities on the multi-slice and stability versions of the Kruskal–Katona Theorem in some regimes of parameters.

For a finite-type surface $\mathfrak {S}$, we study a preferred basis for the commutative algebra $\mathbb {C}[\mathscr {R}_{\mathrm {SL}_3(\mathbb {C})}(\mathfrak {S})]$ of regular functions on the $\mathrm {SL}_3(\mathbb {C})$-character variety, introduced by Sikora–Westbury. These basis elements come from the trace functions associated to certain trivalent graphs embedded in the surface $\mathfrak {S}$. We show that this basis can be naturally indexed by nonnegative integer coordinates, defined by Knutson–Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety.

We show that the measure of maximal entropy of every complex Hénon map is exponentially mixing of all orders for Hölder observables. As a consequence, the Central Limit Theorem holds for all Hölder observables.

We use the theory of trianguline $(\varphi ,\Gamma )$-modules over pseudorigid spaces to prove a modularity lifting theorem for certain Galois representations which are trianguline at p, including those with characteristic p coefficients. The use of pseudorigid spaces lets us construct integral models of the trianguline varieties of [BHS17], [Che13] after bounding the slope, and we carry out a Taylor–Wiles patching argument for families of overconvergent modular forms. This permits us to construct a patched quaternionic eigenvariety and deduce our modularity results.

Let G be a permutation group on a finite set $\Omega $. The base size of G is the minimal size of a subset of $\Omega $ with trivial pointwise stabiliser in G. In this paper, we extend earlier work of Fawcett by determining the precise base size of every finite primitive permutation group of diagonal type. In particular, this is the first family of primitive groups arising in the O’Nan–Scott theorem for which the exact base size has been computed in all cases. Our methods also allow us to determine all the primitive groups of diagonal type with a unique regular suborbit.

We identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of ${\mathrm {S}^{k}} \times {\mathrm {S}^{l}}$, where $3 \le k < l \le 2k - 2$. The result is expressed in terms of Lie graph complex homology.