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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. $\mathsf {Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing.

The $\mathsf {Largest\ Suslin\ Axiom}$ ($\mathsf {LSA}$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let $\mathsf {LSA-over-uB}$ be the statement that in all (set) generic extensions there is a model of $\mathsf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets.

We show that over some mild large cardinal theory, $\mathsf {Sealing}$ is equiconsistent with $\mathsf {LSA-over-uB}$. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that $\mathsf {Sealing}$ is weaker than the theory ‘$\mathsf {ZFC} +$ there is a Woodin cardinal which is a limit of Woodin cardinals’.

A variation of $\mathsf {Sealing}$, called $\mathsf {Tower\ Sealing}$, is also shown to be equiconsistent with $\mathsf {Sealing}$ over the same large cardinal theory.

The result is proven via Woodin’s $\mathsf {Core\ Model\ Induction}$ technique and is essentially the ultimate equiconsistency that can be proven via the current interpretation of $\mathsf {CMI}$ as explained in the paper.

We establish the asymptotic expansion in $\beta $ matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a $\frac {1}{N}$ expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ($\beta = 2$) as well as orthogonal ($\beta = 1$) and skew-orthogonal ($\beta = 4$) polynomials outside the bulk.

We consider manifold-knot pairs $(Y,K)$, where Y is a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface $\Sigma $ in a homology ball X, such that $\partial (X, \Sigma ) = (Y, K)$ can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from $(Y, K)$ to any knot in $S^3$ can be arbitrarily large. The proof relies on Heegaard Floer homology.

We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on $\mathbb {N}$ obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc. and represent the canonical way of speaking about large finite numbers. The second one is ordinal collapsing functions, which represent the best-known method of describing large computable ordinals.

We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. The isomorphism theorem asserts that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by cardinal collapsing functions. As an application of this fact, we obtain a restatement of the subsystem $\Pi ^1_1$-${\mathsf {CA_0}}$ of analysis as a higher-type well-ordering principle asserting that binary fast-growing hierarchies preserve well-foundedness.

This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.

The following summarizes the main results proved under suitable partition hypotheses.

• • If $\kappa $ is a cardinal, $\epsilon < \kappa $, ${\mathrm {cof}}(\epsilon ) = \omega $, $\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ and $\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then $\Phi $ satisfies the almost everywhere short length continuity property: There is a club $C \subseteq \kappa $ and a $\delta < \epsilon $ so that for all $f,g \in [C]^\epsilon _*$, if $f \upharpoonright \delta = g \upharpoonright \delta $ and $\sup (f) = \sup (g)$, then $\Phi (f) = \Phi (g)$.

• • If $\kappa $ is a cardinal, $\epsilon $ is countable, $\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ holds and $\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then $\Phi $ satisfies the strong almost everywhere short length continuity property: There is a club $C \subseteq \kappa $ and finitely many ordinals $\delta _0, ..., \delta _k \leq \epsilon $ so that for all $f,g \in [C]^\epsilon _*$, if for all $0 \leq i \leq k$, $\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$, then $\Phi (f) = \Phi (g)$.

• • If $\kappa $ satisfies $\kappa \rightarrow _* (\kappa )^\kappa _2$, $\epsilon \leq \kappa $ and $\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then $\Phi $ satisfies the almost everywhere monotonicity property: There is a club $C \subseteq \kappa $ so that for all $f,g \in [C]^\epsilon _*$, if for all $\alpha < \epsilon $, $f(\alpha ) \leq g(\alpha )$, then $\Phi (f) \leq \Phi (g)$.

• • Suppose dependent choice ($\mathsf {DC}$), ${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$ and the almost everywhere short length club uniformization principle for ${\omega _1}$ hold. Then every function $\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$ satisfies a finite continuity property with respect to closure points: Let $\mathfrak {C}_f$ be the club of $\alpha < {\omega _1}$ so that $\sup (f \upharpoonright \alpha ) = \alpha $. There is a club $C \subseteq {\omega _1}$ and finitely many functions $\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$ so that for all $f \in [C]^{\omega _1}_*$, for all $g \in [C]^{\omega _1}_*$, if $\mathfrak {C}_g = \mathfrak {C}_f$ and for all $i < n$, $\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$, then $\Phi (g) = \Phi (f)$.

• • Suppose $\kappa $ satisfies $\kappa \rightarrow _* (\kappa )^\epsilon _2$ for all $\epsilon < \kappa $. For all $\chi < \kappa $, $[\kappa ]^{<\kappa }$ does not inject into ${}^\chi \mathrm {ON}$, the class of $\chi $-length sequences of ordinals, and therefore, $|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$. As a consequence, under the axiom of determinacy $(\mathsf {AD})$, these two cardinality results hold when $\kappa $ is one of the following weak or strong partition cardinals of determinacy: ${\omega _1}$, $\omega _2$, $\boldsymbol {\delta }_n^1$ (for all $1 \leq n < \omega $) and $\boldsymbol {\delta }^2_1$ (assuming in addition $\mathsf {DC}_{\mathbb {R}}$).

We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of Gao–Huang between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.

We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction functors for Cherednik algebras, but their definition uses different tools.

After this general definition, we focus on quiver gauge theories attached to a quiver $\Gamma $. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak {g}_{\Gamma }$ on category $ \mathcal {O}$ for these Coulomb branch algebras. When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence.

To establish this categorical action, we define a new class of ‘flavoured’ KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.

The $\Delta $-Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $\Delta $-Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $\mathbb {F}_q$ and partitioning the $\Delta $-Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades and Shimozono for the Hall–Littlewood expansion of the symmetric function in the Delta Conjecture at $t=0$.

We give a variant of Artin algebraization along closed subschemes and closed substacks. Our main application is the existence of étale, smooth or syntomic neighborhoods of closed subschemes and closed substacks. In particular, we prove local structure theorems for stacks and their derived counterparts and the existence of henselizations along linearly fundamental closed substacks. These results establish the existence of Ferrand pushouts, which answers positively a question of Temkin–Tyomkin.