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This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of M ̄ 3 , the moduli stack of stable curves of genus 3. In this paper, we finally compute the Chow ring of M ̄ 3 with Z ⁢ [ 1 / 6 ] -coefficients. [ABSTRACT FROM AUTHOR]

This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of M ̄ 3 , the moduli stack of stable curves of genus 3. In this paper, we finally compute the Chow ring of M ̄ 3 with Z ⁢ [ 1 / 6 ] -coefficients. [ABSTRACT FROM AUTHOR]

We construct minimal surfaces by gluing singly periodic Karcher–Scherk saddle towers along their wings. Such constructions were previously implemented assuming a horizontal reflection plane. We break this symmetry by prescribing phase differences between the saddle towers. It turns out that, in addition to the previously known horizontal balancing condition, the saddle towers must also be balanced under a subtle vertical interaction. This interaction vanishes in the presence of a horizontal reflection plane, hence was not perceived in previous works. Our construction will be presented in a series of papers. In this first paper of the series, we will explain the background of the project and establish the graph theoretical setup that will be useful for all papers in the series. The main task of the current paper is to glue saddle towers into triply periodic minimal surfaces (TPMSs). Our construction expands many previously known TPMSs into new 5-parameter families, therefore significantly advances our knowledge on the space of TPMSs. [ABSTRACT FROM AUTHOR]

The main result of this paper states the subpolynomial growth of the number of integral points with bounded height of a variety over a number field whose fundamental group is large. This generalizes a recent paper of Ellenberg, Lawrence and Venkatesh and replies to two questions asked therein. [ABSTRACT FROM AUTHOR]

We give a parametrization of square roots of the ideal class of the inverse different of rings defined by binary forms in terms of the orbits of a coregular representation. This parametrization, which can be construed as a new integral model of a "higher composition law" discovered by Bhargava and generalized by Wood, was the missing ingredient needed to solve a range of previously intractable open problems concerning distributions of class groups, Selmer groups, and related objects. For instance, in this paper, we apply the parametrization to bound the average size of the 2-class group in families of number fields defined by binary n-ic forms, where n ≥ 3 is an arbitrary integer, odd or even; in the paper [A. Swaminathan, Most integral odd-degree binary forms fail to properly represent a square, preprint 2020], we applied it to prove that most integral odd-degree binary forms fail to primitively represent a square; and in the paper [M. Bhargava, A. Shankar and A. Swaminathan, The second moment of the size of the 2-Selmer group of elliptic curves, preprint 2021], joint with Bhargava and Shankar, we applied it to bound the second moment of the size of the 2-Selmer group of elliptic curves. [ABSTRACT FROM AUTHOR]

In a series of papers, Bamler [5, 4, 6] further developed the high-dimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity theory analogous to Cheeger–Colding theory. In this paper we give an application of his theory to lower bounds for the scalar curvatures of singularity models for Ricci flow. In the case of 4-dimensional non-Ricci-flat steady soliton singularity models, we obtain as a consequence a quadratic decay lower bound for the scalar curvature. [ABSTRACT FROM AUTHOR]

Let 𝕍 be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for (S, 𝕍), which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures. [ABSTRACT FROM AUTHOR]

In this paper, we use a regularized theta lifting to construct harmonic Maass forms corresponding to binary theta functions of weight k ≥ 2 under the 휉-operator. As a result, we show that their holomorphic parts have algebraic Fourier coefficients, with compatible Galois action. As an application, we prove rationality properties of coefficients of harmonic Maass forms corresponding to CM newforms, answering a question of Bruinier, Ono and Rhoades. [ABSTRACT FROM AUTHOR]

In this paper, we compute the first and second variation of the normalized Einstein–Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater than or equal to five, in three dimensions, we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure. [ABSTRACT FROM AUTHOR]

In this paper, we study 푝-adic endoscopy on eigenvarieties for SL 2 over totally real fields, taking a geometric perspective. We show that non-automorphic members of endoscopic 퐿-packets of regular weight contribute eigenvectors to overconvergent cohomology at critically refined endoscopic points on the eigenvariety, and we precisely quantify this contribution. This gives a new perspective on and generalizes previous work of the second author. Our methods are geometric, and are based on showing that the SL 2 -eigenvariety is locally a quotient of an eigenvariety for GL 2 , which allows us to explicitly describe the local geometry of the SL 2 -eigenvariety. In particular, we show that it often fails to be Gorenstein. [ABSTRACT FROM AUTHOR]

In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says that if a complete n-manifold M of Ricci curvature, Ric M ≥ (n - 1) , has the maximal diameter π, then M is isometric to the unit sphere S 1 n . The main result in this paper is a quantitative maximal diameter rigidity: if M satisfies that Ric M ≥ n - 1 , diam ⁡ (M) ≈ π , and the Riemannian universal cover of every metric ball in M of a definite radius satisfies a Reifenberg condition, then M is diffeomorphic and bi-Hölder close to S 1 n . [ABSTRACT FROM AUTHOR]

In this paper, we study the analytic properties of solutions to the Vafa–Witten equation over a compact Kähler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the C ∗ \mathbb{C}^{*} invariant locus of the moduli space is shown to behave similarly to the Hermitian Yang–Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes’ results on rank two solutions over Kähler surfaces together with a new complex geometric interpretation. The moduli space of SU ⁢ ( 2 ) \mathsf{SU}(2) monopoles and some related examples are also discussed in the final section. [ABSTRACT FROM AUTHOR]

In the present paper, we reformulate Mœglin's explicit construction of local A-packets of split odd special orthogonal groups and symplectic groups. By this reformulation together with results of the previous paper with Mínguez, we can compute the A-packets explicitly. Also, we give a non-vanishing criterion of our parametrization, and an algorithm to compute certain derivatives. Finally, we prove a formula for the Aubert duals of irreducible representations of Arthur type. [ABSTRACT FROM AUTHOR]

The first part of the paper studies the boundary behavior of holomorphic isometric mappings F = (F 1 , ... , F m) from the complex unit ball 픹 n , n ≥ 2 , to a bounded symmetric domain Ω = Ω 1 × ⋯ × Ω m up to constant conformal factors, where Ω i ′ s are irreducible factors of Ω. We prove every non-constant component F i must map generic boundary points of 픹 n to the boundary of Ω i . In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to a product of unit balls and Lie balls. [ABSTRACT FROM AUTHOR]

The article presents an obituary for author Ernst Steinitz.

We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A "quantitative" refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories. [ABSTRACT FROM AUTHOR]

In this paper, we study the hypercritical deformed Hermitian-Yang–Mills equation on compact Kähler manifolds and resolve two conjectures of Collins–Yau [Moment maps, nonlinear PDE, and stability in mirror symmetry, preprint (2018), https://arxiv.org/abs/1811.04824]. [ABSTRACT FROM AUTHOR]

We continue the study of the Chow ring of the moduli stack 픐 g , n of prestable curves begun in [Y. Bae and J. Schmitt, Chow rings of stacks of prestable curves I, Forum Math. Sigma 10 2022, Paper No. e28]. In genus 0, we show that the Chow ring of 픐 0 , n coincides with the tautological ring and give a complete description in terms of (additive) generators and relations. This generalizes earlier results by Keel and by Kontsevich and Manin for the spaces of stable curves. Our argument uses the boundary stratification of the moduli stack together with the study of the first higher Chow groups of the strata, in particular providing a new proof of the results of Kontsevich and Manin. [ABSTRACT FROM AUTHOR]

This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety of examples. The second paper proves that Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries; it also gives some first applications. [ABSTRACT FROM AUTHOR]

In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen's result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition. [ABSTRACT FROM AUTHOR]

We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies Λ ⁢ (X m , n) ⊂ E of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for ∇ m s λ which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one. [ABSTRACT FROM AUTHOR]

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate: any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension. [ABSTRACT FROM AUTHOR]

Since the early days of Tomita–Takesaki theory, it is known that a von Neumann algebra 푀 that admits a state 휑 with trivial centralizer M φ must be a type III1 factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense G δ set among all faithful normal states on any III1 factor with separable predual. Through Connes' Radon–Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper. [ABSTRACT FROM AUTHOR]

We use Hitchin's twistor transform for two-dimensional projective structures to obtain normal coordinates in a pseudoconcave neighbourhood of an O ⁢ (1) rational curve; in the construction, we present every such neighbourhood as Q D / F for some holomorphic foliation ℱ, where Q D is an open neighbourhood in the standard quadric Q ⊂ P 2 × P 2 . As a consequence of the normal coordinates, we obtain a new normal form for Cauchy Riemann structures on the three-sphere that are isotopic to the standard one. We end the paper with explicit calculations for the cases arising from deformations of the normal isolated singularities X ⁢ Y = Z n . [ABSTRACT FROM AUTHOR]

In this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger–Colding theory. Let N i be a sequence of smooth manifolds with Ricci curvature ≥ - n ⁢ κ 2 on B 1 + κ ′ ⁢ (p i) for constants κ ≥ 0 , κ ′ > 0 , and volume of B 1 ⁢ (p i) has a positive uniformly lower bound. Assume B 1 ⁢ (p i) converges to a metric ball B 1 ⁢ (p ∞) in the Gromov–Hausdorff sense. For a sequence of area-minimizing hypersurfaces M i in B 1 ⁢ (p i) with ∂ ⁡ M i ⊂ ∂ ⁡ B 1 ⁢ (p i) , we prove the continuity for the volume function of area-minimizing hypersurfaces equipped with the induced Hausdorff topology. In particular, each limit M ∞ of M i is area-minimizing in B 1 ⁢ (p ∞) provided B 1 ⁢ (p ∞) is a smooth Riemannian manifold. By blowing up argument, we get sharp dimensional estimates for the singular set of M ∞ in ℛ , and 풮 ∩ M ∞ . Here, ℛ and 풮 are the regular and singular parts of B 1 ⁢ (p ∞) , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Yau–Tian–Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles, thus for toric manifolds and homogeneous toric bundles, M admits an extremal metric in c 1 ⁢ (L) if and only if (M , L) is relatively K ^ -stable. [ABSTRACT FROM AUTHOR]

In this paper we study two functions F ⁢ (x) and J ⁢ (x) , originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Stark's conjecture. We also discuss connections with 1-cocycles for the modular group PSL ⁡ (2 , ℤ) . [ABSTRACT FROM AUTHOR]

In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. Applying the arithmetic degree and canonical height in positive characteristic, we prove the Dynamical Mordell–Lang Conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit Conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell–Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces. [ABSTRACT FROM AUTHOR]

In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting. [ABSTRACT FROM AUTHOR]

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional κ-solutions. In this paper, we present an alternative proof for this fact and show that compact κ-solutions are rotationally symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular Ricci flows. [ABSTRACT FROM AUTHOR]

In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q-series which satisfy some remarkable modularity properties and recover SU ⁢ (2) quantum invariants of 3-manifolds at roots of unity. In particular, our work gives rise to a 2-variable refinement of the Z ^ -invariant. [ABSTRACT FROM AUTHOR]

In this paper, we prove a conjecture raised by Angella, Otal, Ugarte and Villacampa recently, which states that if the Strominger connection (also known as Bismut connection) of a compact Hermitian manifold is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold, then the metric must be pluriclosed. What we actually showed is a bit more: for any given Hermitian manifold, the Strominger Kähler-like condition is equivalent to the pluriclosedness of the metric plus the parallelness of the torsion. [ABSTRACT FROM AUTHOR]

In this paper we study the uniqueness of graphical mean curvature flow with locally Lipschitz initial data. We first prove that rotationally symmetric entire graphs are unique, without any further assumptions. Our methods also give an alternative simple proof of uniqueness in the one-dimensional case. In the general case, we establish the uniqueness of entire proper graphs that satisfy a uniform lower bound on the second fundamental form. The latter result extends to initial conditions that are proper graphs over subdomains of ℝ n . A consequence of our result is the uniqueness of convex entire graphs, which allow us to prove that Hamilton's Harnack estimate holds for mean curvature flow solutions that are convex entire graphs. [ABSTRACT FROM AUTHOR]

In this paper we study the cohomology of PEL-type Rapoport–Zink spaces associated to unramified unitary similitude groups over ℚ p in an odd number of variables. We extend the results of Kaletha–Minguez–Shin–White and Mok to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapoport–Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider. [ABSTRACT FROM AUTHOR]

Unique continuation of harmonic functions on RCD space is a long-standing open problem, with little known even in the setting of Alexandrov spaces. In this paper, we establish the weak unique continuation theorem for harmonic functions on RCD ⁡ (K , 2) spaces and give a counterexample for strong unique continuation in the setting of RCD ⁡ (K , N) space for any N ≥ 4 and any K ∈ ℝ . [ABSTRACT FROM AUTHOR]

In dimensions n ≥ 4 , an ancient κ-solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is κ-noncollapsed. In this paper, we study the classification of ancient κ-solutions to n-dimensional Ricci flow on S n , extending the result in [S. Brendle, P. Daskalopoulos and N. Sesum, Uniqueness of compact ancient solutions to three-dimensional Ricci flow, Invent. Math. 226 2021, 2, 579–651] to higher dimensions. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman. [ABSTRACT FROM AUTHOR]

In this paper, we describe a certain kind of q-connections on a projective line, namely Z-twisted (G , q) -opers with regular singularities using the language of generalized minors. In part one we explored the correspondence between these q-connections and 푄푄 -systems/Bethe Ansatz equations. Here we associate to a Z-twisted (G , q) -oper a class of meromorphic sections of a G-bundle, satisfying certain difference equations, which we refer to as (G , q) -Wronskians. Among other things, we show that the 푄푄 -systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells. [ABSTRACT FROM AUTHOR]

It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ⁢ (픸 2) of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ⁢ (ℙ 2) up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ⁢ (ℙ 2) admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } and that all Borel subgroups of rank r ∈ { 1 , 2 } are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ . This moduli space is of dimension 2 ⁢ ℊ - 1 . [ABSTRACT FROM AUTHOR]

In this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulae for them. This implies an elementary "double copy" formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Néron–Tate heights on curves, single-valued multiple zeta values and polylogarithms. The results of the present paper are used in [F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, preprint 2019, https://arxiv.org/abs/1910.01107] to prove a recent conjecture of Stieberger which relates the coefficients in a Laurent expansion of two different kinds of periods of twisted cohomology on the moduli spaces of curves ℳ0,n of genus zero with n marked points. We also study a morphism between certain rings of "motivic" periods, called the de Rham projection, which provides a bridge between complex periods and single-valued periods in many situations of interest. [ABSTRACT FROM AUTHOR]

We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 over a field 𝒌 of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : 𝒞 → 𝒞 ⊠ Verp , where Verp is the Verlinde category (the semisimplification of Rep𝐤(ℤ/p); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive characteristic, preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik, On symmetric fusion categories in positive characteristic, Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], where it is used to show that if 𝒞 is finite and semisimple, then it admits a fiber functor to Verp. The main new feature is that when 𝒞 is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor 𝒞 → Verp. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F, and use it to show that for categories with finitely many simple objects F does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory 𝒞ex inside any STC 𝒞 with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by F. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to Verp. This is the strongest currently available characteristic p version of Deligne's theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in 𝒞ex. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra 𝒌 [d]/d2 with d primitive and R-matrix R = 1 ⊗ 1 + d ⊗ d), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki, Exact sequences of tensor categories with respect to a module category, Adv. Math. 308 2017, 1187–1208]. [ABSTRACT FROM AUTHOR]

We show that under natural and quite general assumptions, a large part of a matrix for a bounded linear operator on a Hilbert space can be preassigned. The result is obtained in a more general setting of operator tuples leading to interesting consequences, e.g., when the tuple consists of powers of a single operator. We also prove several variants of this result of independent interest. The paper substantially extends former research on matrix representations in infinite-dimensional spaces dealing mainly with prescribing the main diagonals. [ABSTRACT FROM AUTHOR]

A conjecture of Erdős states that, for any large prime q, every reduced residue class (mod ⁡ q) can be represented as a product p 1 ⁢ p 2 of two primes p 1 , p 2 ≤ q . We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integer q, every reduced residue class (mod ⁡ q) can be written as p 1 ⁢ p 2 ⁢ p 3 with p 1 , p 2 , p 3 ≤ q primes. We also show that, for any ε > 0 and any sufficiently large integer q, at least (2 3 - ε) ⁢ φ ⁢ (q) reduced residue classes (mod ⁡ q) can be represented as a product p 1 ⁢ p 2 of two primes p 1 , p 2 ≤ q . The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of ℤ q × of small index and study in detail the exceptional case that there exists a quadratic character ψ ⁢ (mod ⁡ q) such that ψ ⁢ (p) = - 1 for very many primes p ≤ q . [ABSTRACT FROM AUTHOR]

We consider smooth locally Hamiltonian flows on compact surfaces of genus g ≥ 1 to prove a deviation formula of Birkhoff integrals for smooth observables. Our work generalizes results of [G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 2002, 1, 1–103] and [A. I. Bufetov, Limit theorems for translation flows, Ann. of Math. (2) 179 2014, 2, 431–499] (and a recent result in [K. Frączek and C. Ulcigrai, On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions, preprint 2021]) which prove the existence of the deviation spectrum of Birkhoff integrals for observables whose jets vanish at sufficiently high order around fixed points of the flow. They showed that ergodic integrals display a power deviation spectrum with exactly g positive exponents related to the positive Lyapunov exponents of the cocycle so-called Kontsevich–Zorich, a renormalization cocycle over the Teichmüller flow on a stratum of the moduli space of translation surfaces. Our paper extends the study of the deviation spectrum of ergodic integrals beyond the case of observables whose jets vanish at sufficiently high order around fixed points. We prove the existence of some extra terms in the deviation spectrum related to the non-vanishing of some derivatives of the observable at fixed points. The proof of this new phenomenon is inspired by tools developed in the recent work of the first author and Ulcigrai for locally Hamiltonian flows having only (simple) non-degenerate saddles. In full generality, due to the occurrence of (multiple) degenerate saddles, we introduce new methods of handling functions with polynomial singularities over a full measure set of interval exchange transformations. [ABSTRACT FROM AUTHOR]

A subvariety of a quasi-projective complex variety X is called "universally irreducible" if its preimage inside the universal cover of X is irreducible. In this paper we investigate sufficient conditions for universal irreducibility. We consider in detail complete intersection subvarieties of small codimension inside Siegel moduli spaces of any finite level. Moreover, we show that, for g ≥ 3 , every Siegel modular form is the product of finitely many irreducible analytic functions on the Siegel upper half-space ℍ g . We also discuss the special case of singular theta series of weight 1 2 and of Schottky forms. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give a formula for the leading coefficient at s = 1 of the L-function of one-motives over function fields in terms of Weil-étale cohomology, generalizing the Weil-étale version of the Birch and Swinnerton-Dyer conjecture in the authors' previous work. As a consequence we express the Tamagawa number of a torus introduced by Ono–Oesterlé in terms of Weil-étale cohomology, and reprove their Tamagawa number formula. [ABSTRACT FROM AUTHOR]

The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a "Type-0" singularity, in contrast to the well known "Type-I" and "Type-II" ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow-up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree–like networks till the first singular time, under the assumption that all the "tangents flows" have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results from [T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, J. Differential Geom. 111 2019, 1, 39–89] and [J. Lira, R. Mazzeo, A. Pluda and M. Saez, Short–time existence for the network flow, preprint 2021] to "restart" the flow after the singularity. [ABSTRACT FROM AUTHOR]

In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals B ^ 픤 ⁢ (∞) for an arbitrary quantum group U q ⁢ (픤) , which is the product of infinite copies of the crystal B ⁢ (∞) . For a complete duality datum 풟 in the Hernandez–Leclerc category 풞 픤 0 of a quantum affine algebra U q ′ ⁢ (픤) , we prove that the set ℬ 풟 ⁢ (픤) of the isomorphism classes of simple modules in 풞 픤 0 has an extended crystal structure isomorphic to B ^ 픤 fin ⁢ (∞) . An explicit combinatorial description of the extended crystal ℬ 풟 ⁢ (픤) for affine type A n (1) is given in terms of affine highest weights. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 and n ≥ 1 be any integers. In this paper, we prove that all Hecke operators act trivially on the space of ordinary Drinfeld cuspforms of level Γ 1 ⁢ (t n) and weight k . [ABSTRACT FROM AUTHOR]

In this paper, we answer in the affirmative the long-standing conjecture that the first cohomology group of the Murray–von Neumann algebra S ⁢ (ℳ) of all operators affiliated with a type II 1 von Neumann algebra ℳ is 0. That is, we show that all derivations of S ⁢ (ℳ) are inner. [ABSTRACT FROM AUTHOR]

In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding's method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C ⁢ Y × ℝ n - k ( 2 ≤ k ≤ n ) in a non-collapsing metric cone C ⁢ X × ℝ n - k obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X. [ABSTRACT FROM AUTHOR]