A complete classification of unimodular valuations on the set of lattice polygons with values in the spaces of polynomials and formal power series, respectively, is established. The valuations are classified in terms of their behavior with respect to dilation using extensions to unbounded polyhedra and basic invariant theory.

We extend the theoretical framework of proof mining by establishing general logical metatheorems that allow for the extraction of the computational content of theorems with prima facie “noncomputational” proofs from probability theory, thereby unlocking a major branch of mathematics as a new area of application for these methods. Concretely, we first devise proof-theoretically tame logical systems that allow for the formalization of proofs involving algebras of sets together with probability contents, that is probability measures which are only assumed to be finitely additive. Based on these systems, we provide extensions for the tame treatment of Lebesgue integrals on probability contents as well as $\sigma $-algebras and associated probability measures, all via intensional approaches. All these systems are then shown to be amenable to proof-theoretic metatheorems in the style of proof mining which guarantee the extractability of effective and tame bounds from large classes of ineffective existence proofs in probability theory. Moreover, these extractable bounds are guaranteed to be highly uniform in the sense that they will be independent of all parameters relating to the underlying probability space, particularly regarding events or measures of them. As such, these results in particular provide the first logical explanation for the success and the observed uniformities of the previous ad hoc case studies of proof mining in these areas and further illustrate their extent. Lastly, we establish a general proof-theoretic transfer principle that allows for the lift of quantitative information on a relationship between different modes of convergence for sequences of real numbers to sequences of random variables.

We construct pathological examples of MMP singularities in every positive characteristic using quotients by $\alpha _p$-actions. In particular, we obtain non-$S_3$ terminal singularities, as well as locally stable (respectively stable) families whose general fibers are smooth (respectively klt, Cohen–Macaulay, and F-injective) and whose special fibers are non-$S_2$. The dimensions of these examples are bounded below by a linear function of the characteristic.

We introduce a framework to prove integral rigidity results for the Seiberg–Witten invariants of a closed $4$-manifold X containing a nonseparating hypersurface Y satisfying suitable (chain-level) Floer theoretic conditions. As a concrete application, we show that if X has the homology of a four-torus, and it contains a nonseparating three-torus, then the sum of all Seiberg–Witten invariants of X is determined in purely cohomological terms.

Our results can be interpreted as $(3+1)$-dimensional versions of Donaldson’s TQFT approach to the formula of Meng–Taubes, and build upon a subtle interplay between irreducible solutions to the Seiberg–Witten equations on X and reducible ones on Y and its complement. Along the way, we provide a concrete description of the associated graded map (for a suitable filtration) of the map on $\overline {\mathit {HM}}_*$ induced by a negative-definite cobordism between three-manifolds, which might be of independent interest.

We consider the Cauchy problem to the 3D fractional Schrödinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty is that we combine the normal form methods and the space-time resonance methods. Using the normal form transform enables us to have more flexibility in designing the resolution spaces so that we can control various interactions. It is also convenient for the final data problem.

In this paper we unveil novel monotonicity conditions applicable to Mean Field Games through the exploration of finite dimensional canonical transformations. Our findings contribute to establishing new global well-posedness results for the associated master equations, also in the case of potentially degenerate idiosyncratic noise. Additionally, we show that recent advancements in global well-posedness results, specifically those related to displacement semi-monotone and anti-monotone data, can be easily obtained as a consequence of our main results.

The purpose of this paper is to expose and investigate natural phase-space formulations of two longstanding problems in the restriction theory of the Fourier transform. These problems, often referred to as the Stein and Mizohata–Takeuchi conjectures, assert that Fourier extension operators associated with rather general (codimension 1) submanifolds of Euclidean space may be effectively controlled by the classical X-ray transform via weighted $L^2$ inequalities. Our phase-space formulations, which have their origins in recent work of Dendrinos, Mustata and Vitturi expose close connections with a conjecture of Flandrin from time-frequency analysis, and rest on the identification of an explicit ‘geometric’ Wigner transform associated with an arbitrary (smooth strictly convex) submanifold S of $\mathbb {R}^n$. Our main results are certain natural ‘Sobolev variants’ of the Stein and Mizohata–Takeuchi conjectures and involve estimating the Sobolev norms of such Wigner transforms by geometric forms of classical bilinear fractional integrals. Our broad geometric framework allows us to explore the role of the curvature of the submanifold in these problems, and in particular we obtain bounds that are independent of any lower bound on the curvature; a feature that is uncommon in the wider restriction theory of the Fourier transform. Finally, we provide a further illustration of the effectiveness of our analysis by establishing a form of Flandrin’s conjecture in the plane with an $\varepsilon $-loss. While our perspective comes primarily from Euclidean harmonic analysis, the procedure used for constructing phase-space representations of extension operators is well-known in optics.

Let G be a finite transitive permutation group on $\Omega $. The G-invariant partitions form a sublattice of the lattice of all partitions of $\Omega $, having the further property that all its elements are uniform (that is, have all parts of the same size). If, in addition, all the equivalence relations defining the partitions commute, then the relations form an orthogonal block structure, a concept from statistics; in this case the lattice is modular. If it is distributive, then we have a poset block structure, whose automorphism group is a generalised wreath product. We examine permutation groups with these properties, which we call the OB property and PB property respectively, and in particular investigate when direct and wreath products of groups with these properties also have these properties.

A famous theorem on permutation groups asserts that a transitive imprimitive group G is embeddable in the wreath product of two factors obtained from the group (the group induced on a block by its setwise stabiliser, and the group induced on the set of blocks by G). We extend this theorem to groups with the PB property, embedding them into generalised wreath products. We show that the map from posets to generalised wreath products preserves intersections and inclusions.

We have included background and historical material on these concepts.

We prove that on any transitive graph G with infinitely many ends, a self-avoiding walk of length n is ballistic with extremely high probability, in the sense that there exist constants $c,t>0$ such that $\mathbb {P}_n(d_G(w_0,w_n)\geq cn)\geq 1-e^{-tn}$ for every $n\geq 1$. Furthermore, we show that the number of self-avoiding walks of length n grows asymptotically like $\mu _w^n$, in the sense that there exists $C>0$ such that $\mu _w^n\leq c_n\leq C\mu _w^n$ for every $n\geq 1$. These results generalise earlier work by Li (J. Comb. Theory Ser. A, 2020). The key to this greater generality is that in contrast to Li’s approach, our proof does not require the existence of a special structure that enables the construction of separating patterns. Our results also extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of G which does not fix an end of G.

We study the hypersimplex under the action of the symmetric group $S_n$ by coordinate permutation. We prove that its equivariant volume, given by the evaluation of its equivariant $H^*$-series at $1$, is the permutation character of decorated ordered set partitions under the natural action of $S_n$. This verifies a conjecture of Stapledon for the hypersimplex. To prove this result, we give a formula for the coefficients of the $H^*$-polynomial. Additionally, for the $(2,n)$-hypersimplex, we use this formula to show that trivial character need not appear as a direct summand of a coefficient of the $H^*$-polynomial, which gives a family of counterexamples to a different conjecture of Stapledon.

A system of linear equations in $\mathbb {F}_p^n$ is Sidorenko if any subset of $\mathbb {F}_p^n$ contains at least as many solutions to the system as a random set of the same density, asymptotically as $n\to \infty $. A system of linear equations is common if any two-colouring of $\mathbb {F}_p^n$ yields at least as many monochromatic solutions to the system of equations as a random 2-colouring, asymptotically as $n\to \infty $. Both classification problems remain wide open despite recent attention.

We show that a certain generic family of systems of two linear equations is not Sidorenko. In fact, we show that systems in this family are not locally Sidorenko, and that systems in this family which do not contain additive tuples are not weakly locally Sidorenko. This endeavour answers a conjecture and question of Kamčev–Liebenau–Morrison. Insofar as methods, we observe that the true complexity of a linear system is not maintained under Fourier inversion; our main novelty is the use of higher-order methods in the frequency space of systems which have complexity one. We also give a shorter proof of the recent result of Kamčev–Liebenau–Morrison and independently Versteegen that any linear system containing a four-term arithmetic progression is uncommon.

We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only modify the tape at a bounded distance around the head, change the state, and move the head in a bounded way. We study three natural subgroups arising in this model: the group of finite-state automata, which generalizes the topological full groups studied in topological dynamics and the theory of orbit-equivalence; the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates, and the group of elementary Turing machines, which are the machines which are obtained by composing finite-state automata and oblivious Turing machines.

We show that both the group of oblivious Turing machines and that of elementary Turing machines are finitely generated, while the group of finite-state automata and the group of reversible Turing machines are not. We show that the group of elementary Turing machines has undecidable torsion problem. From this, we also obtain that the group of cellular automata (more generally, the automorphism group of any uncountable one-dimensional sofic subshift) contains a finitely generated subgroup with undecidable torsion problem. We also show that the torsion problem is undecidable for the topological full group of a full $\mathbb {Z}^d$-shift on a nontrivial alphabet if and only if $d \geq 2$.

Using the $\infty $-categorical enhancement of mixed Hodge modules constructed by the author in a previous paper, we explain how mixed Hodge modules canonically extend to algebraic stacks, together with all the six operations and weights. We also prove that Drew’s approach to motivic Hodge modules gives an $\infty $-category that embeds fully faithfully in mixed Hodge modules, and we identify the image as mixed Hodge modules of geometric origin.

We study the local theta correspondence for dual pairs of the form $\mathrm {Aut}(C)\times F_{4}$ over a p-adic field, where C is a composition algebra of dimension $2$ or $4$, by restricting the minimal representation of a group of type E. We investigate this restriction through the computation of maximal parabolic Jacquet modules and the Fourier–Jacobi functor.

As a consequence of our results, we prove a multiplicity one result for the $\mathrm {Spin}(9)$-invariant linear functionals of irreducible representations of $F_{4}$ and classify the $\mathrm {Spin}(9)$-distinguished representations.

In p-adic Hodge theory and the p-adic Langlands program, Banach spaces with $\mathbf {Q}_p$-coefficients and p-adic Lie group actions are central. Studying the subrepresentation of G-locally analytic vectors, $W^{\mathrm {la}}$, is useful because $W^{\mathrm {la}}$ can be studied via the Lie algebra $\mathrm {Lie}(G)$, which simplifies the action of G. Additionally, $W^{\mathrm {la}}$ often behaves as a decompletion of W, making it closer to an algebraic or geometric object.

This article introduces a notion of locally analytic vectors for W in a mixed characteristic setting, specifically for $\mathbf {Z}_p$-Tate algebras. This generalization encompasses the classical definition and also specializes to super-Hölder vectors in characteristic p. Using binomial expansions instead of Taylor series, this new definition bridges locally analytic vectors in characteristic $0$ and characteristic p.

Our main theorem shows that under certain conditions, the map $W \mapsto W^{\mathrm {la}}$ acts as a descent, and the derived locally analytic vectors $\mathrm {R}_{\mathrm {la}}^i(W)$ vanish for $i \geq 1$. This result extends Theorem C of [Por24], providing new tools for propagating information about locally analytic vectors from characteristic $0$ to characteristic p.

We provide three applications: a new proof of Berger-Rozensztajn’s main result using characteristic $0$ methods, the introduction of an integral multivariable ring $\widetilde {\mathbf {A}}_{\mathrm {LT}}^{\dagger ,\mathrm {la}}$ in the Lubin-Tate setting, and a novel interpretation of the classical Cohen ring $\mathbf {{A}}_{\mathbf {Q}_p}$ from the theory of $(\varphi ,\Gamma )$-modules in terms of locally analytic vectors.

We study new properties of generalised Harish-Chandra theory aiming at explaining the inductive local-global conditions for finite groups of Lie type in nondefining characteristic. In particular, we consider a parametrisation of generalised Harish-Chandra series that is compatible with Clifford theory and with the action of automorphisms on irreducible characters and we reduce it to the verification of certain requirements on stabilisers and extendibility of characters. This parametrisation is used by the author in a separate paper to obtain new conjectures for finite reductive groups that can be seen as geometric realisations of the local-global counting conjectures and their inductive conditions. As a by-product, we extend the parametrisation of generalised Harish-Chandra series given by Broué–Malle–Michel to the nonunipotent case by assuming maximal extendibility.

We define a new class of plane billiards – the “pensive billiard” – in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality of the angles of incidence and reflection. This generalizes so-called “puck billiards” proposed by M. Bialy, as well as a “vortex billiard,” that is, the motion of a point vortex dipole in two-dimensional hydrodynamics on domains with boundary. We prove the variational origin and invariance of a symplectic structure for pensive billiards, as well as study their properties including conditions for a twist map, the existence of periodic orbits, etc. We also demonstrate the appearance of both the golden and silver ratios in the corresponding hydrodynamical vortex setting. Finally, we introduce and describe basic properties of pensive outer billiards.

We build on a 1990 paper of Bukovský and Copláková-Hartová. First, we remove the hypothesis of ${\mathsf {CH}}$ from one of their minimality results. Then, using a measurable cardinal, we show that there is a $|\aleph _2^V|=\aleph _1$-minimal extension that is not a $|\aleph _3^V|=\aleph _1$-extension, answering the first of their questions.

In 1967, Klarner proposed a problem concerning the existence of reflecting n-queens configurations. The problem considers the feasibility of placing n mutually nonattacking queens on the reflecting chessboard, an $n\times n$ chessboard with a $1\times n$ “reflecting strip” of squares added along one side of the board. A queen placed on the reflecting chessboard can attack the squares in the same row, column, and diagonal, with the additional feature that its diagonal path can be reflected via the reflecting strip. Klarner noted the equivalence of this problem to a number theory problem proposed by Slater, which asks: for which n is it possible to pair up the integers 1 through n with the integers $n+1$ through $2n$ such that no two of the sums or differences of the n pairs of integers are the same. We prove the existence of reflecting n-queens configurations for all sufficiently large n, thereby resolving both Slater’s and Klarner’s questions for all but a finite number of integers.