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We establish a lower bound for the Donaldson-Futaki invariant of optimal degenerations produced by the Kähler-Ricci flow in terms of the greatest Ricci lower bound on arbitrary Fano manifolds. As an application, we can generalize the finiteness of the Futaki invariants on Kähler-Ricci solitons obtained by Guo-Phong-Song-Sturm to the space of all Fano manifolds. Also, we discuss the relation to Hisamoto's inequality for the infimum of the H-functional. [ABSTRACT FROM AUTHOR]

We define invariants of oriented surface-links by enhancing the biquandle counting invariant using biquandle modules, algebraic structures defined in terms of biquandle actions on commutative rings analogous to Alexander biquandles. We show that bead colorings of marked graph diagrams are preserved by Yoshikawa moves and hence define enhancements of the biquandle counting invariant for surface links. We provide examples illustrating the computation of the invariant and demonstrate that these invariants are not determined by the first and second Alexander elementary ideals and characteristic polynomials. [ABSTRACT FROM AUTHOR]

Let ν be either the Ozsváth–Szabó τ–invariant or the Rasmussen s–invariant, suitably normalized. For a knot K, Livingston and Naik defined the invariant tν(K) to be the minimum of k for which ν of the k–twisted positive Whitehead double of K vanishes. They proved that tν(K) is bounded above by −TB(−K), where TB is the maximal Thurston–Bennequin number. We use a blowing-up process to find a crossing change formula and a new upper bound for tν in terms of the unknotting number. As an application, we present infinitely many knots K such that the difference between Livingston–Naik’s upper bound −TB(−K) and tν(K) can be arbitrarily large. [ABSTRACT FROM AUTHOR]

Given an orientation-preserving and area-preserving homeomorphism f of the sphere, we prove that every point which is in the common boundary of three pairwise disjoint invariant open topological disks must be a fixed point. As an application, if K is an invariant Wada-type continuum, then fn|K is the identity for some n > 0. Another application is an elementary proof of the fact that invariant disks for a nonwandering homeomorphism homotopic to the identity in an arbitrary surface are homotopically bounded if the fixed point set is inessential. The main results in this article are selfcontained. [ABSTRACT FROM AUTHOR]

Given a field F of char(F) = 2, we define un(F) to be the maximal dimension of an anisotropic form in Inq F. For n = 1 it recaptures the definition of u(F). We study the relations between this value and the symbol length of Hn2 (F), denoted by sln2(F). We show for any n ≥ 2 that if 2n ≤ un(F) ≤ u²(F) < ∞, then sln2 (F) ≤ Πni=2(ui(F)/2 + 1 - 2i-1). As a result, if u(F) is finite, then sln2 (F) is finite for any n, a fact which was previously proven when char(F) ≠ 2 by Saltman and Krashen. We also show that if sln2(F) = 1, then un(F) is either 2n or 2n+1. [ABSTRACT FROM AUTHOR]

We show a residues formula for maps generically transversal to regular holomorphic distributions. [ABSTRACT FROM AUTHOR]

We resolve a conjecture of Ramos and Li that relates the regularity of an FI-module to its local cohomology groups. This is an analogue of the familiar relationship between regularity and local cohomology in commutative algebra. [ABSTRACT FROM AUTHOR]

The signature function of a knot is an integer-valued step function on the unit circle in the complex plane. Necessary and sufficient conditions for a function to be the signature function of a knot are presented. [ABSTRACT FROM AUTHOR]

The goal of this article is to give a simple arithmetic application of the enhanced homotopy (Lie) theory for algebraic varieties developed by the second and third authors. Namely, we compute an inverse value of the modular j-invariant by using a deformation theory for period matrices of elliptic curves based on homotopy Lie theory. Another key ingredient in our approach is J. Carlson and P. Griffiths' explicit computation regarding infinitesimal variations of Hodge structures. [ABSTRACT FROM AUTHOR]

Grothendieck conjectured in the sixties that the even Künneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck’s conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck’s original conjectures in the case of intersections of quadrics and linear sections of determinantal varieties. Along the way, we also prove the case of quadric fibrations and intersections of bilinear divisors. [ABSTRACT FROM AUTHOR]

We prove that the rank of a non-trivial co-doubly commuting submodule is 2. More precisely, let ϕ, ψ ∈ H∞(D) be two inner functions. If Qϕ = H²(D)/ϕH²(D) and Qψ = H²(D)/ψH²(D), then rank (Qϕ × Qψ)⊥ = 2. An immediate consequence is the following: Let S be a co-doubly commuting submodule of H²(D2). Then rank S = 1 if and only if S = ΦH²(D2) for some one variable inner function Φ ∈ H∞(D²). This answers a question posed by R. G. Douglas and R. Yang [Integral Equations Operator Theory 38(2000), pp207-221]. [ABSTRACT FROM AUTHOR]

Let X be a nonempty real variety that is invariant under the action of a reflection group G. We conjecture that if X is defined in terms of the first k basic invariants of G (ordered by degree), then X meets a k-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most 3, and F4 and we give computational evidence for H4. This is a generalization of Timofte's degree principle to reflection groups. For general reflection groups, we compute nontrivial upper bounds on the minimal dimension of flats of the reflection arrangement meeting X from the combinatorics of parabolic subgroups. We also give generalizations to real varieties invariant under Lie groups. [ABSTRACT FROM AUTHOR]

Motivic Serre invariants defined by Loeser and Sebag are elements of the Grothendieck ring of varieties modulo L - 1. In this paper, we show that we can lift these invariants to modulo the square of L - 1 after tensoring the Grothendieck ring with ℚ under certain assumptions. [ABSTRACT FROM AUTHOR]

We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group G and admitting equivariant desingularizations with only finitely many G-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties. [ABSTRACT FROM AUTHOR]

We construct an infinite family of topologically slice 2–component boundary links 𝓁i, none of which is smoothly concordant to a split link, such that g4(𝓁i) = i. [ABSTRACT FROM AUTHOR]

In this note, we study the stability of a critical point of a conformally invariant functional Ƒ. For n ≥ 3, by use of the variational formulas, we prove that the Fubini-Study metric on ℂℙn is a strictly stable critical point of Ƒ. [ABSTRACT FROM AUTHOR]

In this note, we study the stability of a critical point of a conformally invariant functional Ƒ. For n ≥ 3, by use of the variational formulas, we prove that the Fubini-Study metric on ℂℙn is a strictly stable critical point of Ƒ. [ABSTRACT FROM AUTHOR]

We give a characterization of sets K of probability measures on a Cantor space X with the property that there exists a minimal homeomorphism g of X such that the set of g-invariant probability measures on X coincides with K. This extends theorems of Akin (corresponding to the case when K is a singleton) and Dahl (when K is finite-dimensional). Our argument is elementary and different from both Akin's and Dahl's. [ABSTRACT FROM AUTHOR]

We describe a minimal global coordinate system of order 30 on the SL(4,C)-character variety of a rank 2 free group. Using symmetry within this system, we obtain a smaller collection of 22 coordinates subject to 5 further real relations that determine conjugation classes of generic pairs of matrices in SU(3, 1). [ABSTRACT FROM AUTHOR]

Let G be a locally compact group, and denote by WAP(M(G)) and AP(M(G)) the spaces of weakly almost periodic, respectively, almost periodic functionals on the measure algebra M(G). Problem 3 in [H.G. Dales, A.T.- M. Lau, D. Strauss, Second duals of measure algebras, Dissertationes Math. (Rozprawy Mat.) 481 (2012), 1-121] asks if WAP(M(G)) and AP(M(G)) admit topological invariant means, and if yes, whether they are unique. The questions regarding existence had already been raised in [M. Daws, Characterising weakly almost periodic functionals on the measure algebra, Studia Math. 204 (2011), no. 3, 213-234]. We answer all these problems in the affirmative. [ABSTRACT FROM AUTHOR]

In this paper, we introduce splitting numbers of subvarieties in a smooth complex variety for a Galois cover, and prove that the splitting numbers are invariant under certain homeomorphisms. In particular cases, we show that splitting numbers enable us to distinguish the topology of complex plane curves even if the fundamental groups of the complements of plane curves are isomorphic. Consequently, we prove that there are π1-equivalent Zariski k-plets for any integer k ≥ 2. [ABSTRACT FROM AUTHOR]

Let G be a discrete countable infinite group that does not have Kazhdan's property (T) and let κ be a generating probability measure on G. Then for each t > 0, there is a type III1 ergodic free nonsingular G-action whose ?-entropy (or the Furstenberg entropy) is t. [ABSTRACT FROM AUTHOR]

We investigate the additivity A and lineability L cardinal coefficients for the following classes of functions: ES \ SES of everywhere surjective functions that are not strongly everywhere surjective, Darboux-like, Sierpiński-Zygmund, surjective, and their corresponding intersections. The classes SES and ES have been shown to be 2c-lineable. In contrast, although we prove here that ES \ SES is c+-lineable, it is still unclear whether it can be proved in ZFC that ES \ SES is 2c-lineable. Moreover, we prove that if c is a regular cardinal number, then A(ES \ SES) ≤ c. This shows that, for the class ES \ SES, there is an unusually large gap between the numbers A and L. [ABSTRACT FROM AUTHOR]

Let G be a group acting by homeomorphisms on a dendrite X. First, we show that any minimal set M of G is either a finite orbit or a Cantor set (resp. a finite orbit) when the set of endpoints of X is closed (resp. countable). Furthermore, we prove, regardless of the type of the dendrite X, that if the action of G on X has at least two minimal sets, then necessarily it has a finite orbit (and even an orbit consisting of one or two points). Second, we explore the topological and geometrical properties of infinite minimal sets when the action of G has a finite orbit. We show in this case that any infinite minimal set M is a Cantor set which is the set of endpoints of its convex hull [M] and there is no other infinite minimal set in [M]. On the other hand, we consider the familyMof all minimal sets in the hyperspace 2X (endowed with the Hausdorff metric). We prove that M is closed in 2X and that the family F of all finite orbits (when it is non-empty) is dense in M. As a consequence, the union of all minimal sets of G is closed in X. [ABSTRACT FROM AUTHOR]

Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since large classes of L-spaces can be produced from Dehn surgery on knots in S3, it is natural to ask what conditions on the knot group are sufficient to imply that the quotient associated to Dehn surgery is not left-orderable. Clay and Watson developed a criterion for determining the left-orderability of this quotient group and used it to verify the conjecture for surgeries on certain L-space twisted torus knots. We generalize a recent theorem of Ichihara and Temma to provide another such criterion. We then use this new criterion to generalize the results of Clay and Watson and to verify the conjecture for a much broader class of L-spaces obtained by surgery on twisted torus knots. [ABSTRACT FROM AUTHOR]

It is well known that for a line bundle over a closed framed manifold, its sphere bundle can also be given the structure of a framed manifold, usually referred to as a transfer. Given a pair of lines, the procedure can be generalized to obtain a double transfer. We study the quaternionic case and derive a simple formula for the f-invariant of the underlying bordism class, enabling us to investigate its status in the Adams-Novikov spectral sequence. As an application, we treat the situation of quaternionic flag manifolds. [ABSTRACT FROM AUTHOR]

We give a short proof of a conjecture of Stipsicz on the minimality of fiber sums of Lefschetz fibrations, which was proved earlier by Usher. We then construct the first examples of genus g ≥ 2 Lefschetz fibrations on minimal symplectic 4-manifolds which, up to diffeomorphisms of the summands, admit unique decompositions as fiber sums. [ABSTRACT FROM AUTHOR]

For the potential function of a link diagram induced by the optimistic limit of the colored Jones polynomial, we show the existence of a solution of the hyperbolicity equations by directly constructing it. This construction is based on the shadow-coloring of the conjugation quandle induced by a boundary-parabolic representation ρ : π1(L) → PSL(2,C). This gives us a very simple and combinatorial method to calculate the complex volume of ρ. [ABSTRACT FROM AUTHOR]

Félix, Halperin, and Lemaire have shown that the rational module category and the rational Toomer invariant coincide for simply connected Poincaré duality complexes. We establish an analogue of this result for the sectional category of a fibration. [ABSTRACT FROM AUTHOR]

Ozsváth-Stipsicz-Szabó recently defined a one-parameter family ϒK(t) of concordance invariants associated to the knot Floer complex. We compare their invariant to the { -- 1,0, 1}-valued concordance invariant ∈(K), which is also associated to the knot Floer complex. In particular, we give an example of a knot K withϒK(t) = 0 but ∈(K) ≠ 0. [ABSTRACT FROM AUTHOR]

Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these parameters are elements of the invariant trace field of the manifold, providing a connection between the intrinsic geometry of a 3- manifold and its number-theoretic invariants. Further, we explore the question of choosing a minimal collection of arcs and associated parameters to generate the field. We prove that for a tunnel number k manifold it is enough to choose 3k specific parameters. For many hyperbolic link complements, this approach allows one to compute the field from a link diagram. We also give examples of infinite families of links where a single parameter can be chosen to generate the field, and the polynomial for it can be constructed from the link diagram as well. [ABSTRACT FROM AUTHOR]

Let X be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel σ-additive measure μ which is good in the sense that for any clopen subsets U, V ⊂ X with μ(U) < μ(V) there is a clopen set W ⊂ V with μ(W) = μ(U). We study σ-ideals with Borel base on X which are invariant under the action of the group Hμ(X) of measurepreserving homeomorphisms of (X, μ), and show that any such σ-ideal I is equal to one of seven σ-ideals: {∅}, [X] ≤ω, E, M∩N, M, N, or [X] ≤c. Here [X] ≤κ is the ideal consisting of subsets of cardinality ≤ κ in X, M is the ideal of meager subsets of X, N = {A ⊂ X : μ(A)=0} is the ideal of null subsets of (X, μ), and E is the σ-ideal generated by closed null subsets of (X, μ). [ABSTRACT FROM AUTHOR]

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces. [ABSTRACT FROM AUTHOR]

Let F be a non-Archimedean local field of characteristic 0 and G = SL(n,F). Let (π,W) be an irreducible smooth self-dual representation G. The space W of π carries a non-degenerate G-invariant bilinear form (,) which is unique up to scaling. The form (,) is easily seen to be symmetric or skew-symmetric and we set ε(π) = ±1 accordingly. In this article, we show that ε(π) = 1 when π is an Iwahori spherical representation of G. [ABSTRACT FROM AUTHOR]

Autonomous systems depend only on the elapsed time, so their attractors and limit sets exist in current time. Similarly, the pullback limit defines a component set of a nonautonomous pullback attractor at each instant of current time. The forward limit defining a nonautonomous forward attractor is different as it is the limit to the asymptotically distant future. In particular, the limiting objects forward in time do not have the same dynamical meaning in current time as in the autonomous or pullback cases. Nevertheless, the pull-back limit taken within a positively invariant family of compact subsets allows the component set of a forward attractor to be constructed at each instant of current time. Every forward attractor has such a positively invariant family of compact subsets, which ensures that the component sets of a forward attractor can be constructed in this way. It is, however, only a necessary condition and not sufficient for the constructed family of subsets to be a forward attractor. The analysis here is presented in the state space ℝd to focus on the dynamical essentials rather than on functional analytical technicalities; in particular, those concerning asymptotic compactness properties. [ABSTRACT FROM AUTHOR]

We prove that the universal enveloping algebra of a Poisson-Ore extension is a length two iterated Ore extension of the original universal enveloping algebra. As a consequence, we observe certain ring-theoretic invariants of the universal enveloping algebras that are preserved under iterated Poisson-Ore extensions. We apply our results to iterated quadratic Poisson algebras arising from semiclassical limits of quantized coordinate rings and a family of graded Poisson algebras of Poisson structures of rank at most two. [ABSTRACT FROM AUTHOR]

This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant bΔ of a simplicial d-manifold Δ taking values in the range 0 ≤ bΔ ≤ d - 1. The main result is that bΔ influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold Δ is (2d - bΔ)-connected. The parameter bΔ has the property that bΔ = 0 if the complex Δ is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d + 1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of bΔ involves the concept of banner triangulations of manifolds, a generalization of flag triangulations. [ABSTRACT FROM AUTHOR]

We consider the problem where Ω is a smooth bounded open domain in ℝ² which contains the point ξ. We prove that if λ > 8π, problem (P) has a solutions u_ such that u∈(x) 8π + λ/2 G(x, ξ) uniformly on compact sets of Ω \ {ξ} as∈ goes to zero. Here G denotes Green's function of Dirichlet Laplacian in Ω. If λ ... 8πℕ we will not make any symmetry assumptions on Ω, while if λ ∈ 8πℕ we will assume that Ω is invariant under a rotation through an angle 8π2 λ around the point ξ. [ABSTRACT FROM AUTHOR]

Based on the orthogonal Labastida-Mariño-Ooguri-Vafa conjecture made by L. Chen and Q. Chen (2012), we derive an infinite product formula for Chern-Simons partition functions, which generalizes Liu and Peng's recent results to the orthogonal case. Symmetry property of this new infinite product structure is also discussed. [ABSTRACT FROM AUTHOR]

We prove that the discriminant of a nonsingular space curve of genus g ≥ 2 is stable with respect to the standard action of the special linear group. [ABSTRACT FROM AUTHOR]

We investigate two "categorified" braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer homology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group. In particular, our proof in the Khovanov case is completely combinatorial. [ABSTRACT FROM AUTHOR]

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact R3 and the hierarchy of positive, strongly quasi-positive, and quasipositive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every Legendrian knot with an exact, embedded Lagrangian filling is quasi-positive. On the other hand, we show that if a knot type is positive, then it has a Legendrian representative with an exact embedded Lagrangian filling. Further, we produce examples that show that strong quasi-positivity and fillability are independent conditions. [ABSTRACT FROM AUTHOR]

We study a class of smooth torus manifolds whose orbit space has the combinatorial structure of a simple polytope with holes. We construct moment angle manifolds for such polytopes with holes and use them to prove that the associated torus manifolds admit stable almost complex structure. We give a combinatorial formula for the Hirzebruch Xy genus of these torus manifolds. We show that they have (invariant) almost complex structure if they admit positive omniorientation. We give examples of almost complex manifolds that do not admit a complex structure. When the dimension is four, we calculate the homology groups and describe a method for computing the cohomology ring. [ABSTRACT FROM AUTHOR]

Let α be a countable ordinal and ℙ(α) the collection of its subsets isomorphic to α. We show that the separative quotient of the poset ⟨ℙ(α), ⊂⟩ is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form P(ωγ)/Iωγ, where γ ∈ Lim ⋃ {1} and Iωγ is the corresponding ordinal ideal. Moreover, the poset ⟨ℙ(α), ⊂⟨ is forcing equivalent to a two-step iteration of the form (P(ω)/Fin)+ * π, where [ω] ⊩ "π is an ω1 closed separative pre-order" and, if h = ω1, to (P(ω)/Fin)+. Also we analyze the quotients over ordinal ideals P(ωδ)/Iωδ and the corresponding cardinal invariants hωδ and tωδ. [ABSTRACT FROM AUTHOR]

We provide an alternative proof that Koschorke's K-invariant is injective on the set of link homotopy classes of «.-component homotopy Brunnian links BLM(n). The existing proof (by Koschorke in 1997) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lin's string links. We frame the result in the language of Fox's torus homotopy groups and the rational homotopy Lie algebra Π.. of the configuration space. It allows us to express the relevant Milnor's μ--invariants as homotopy periods of Conf(n). [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic p ≥ 0. Let A be a commutative k-algebra with multiplicative identity and let M be an A-module. Let G be a linear algebraic group acting rationally on both A and M. In this paper we study AG-modules of rath order invariants, In(M,G). The In(M,G) are defined inductively by I0(M, G) = {0} and In(M,G) = {m ∈ M:g.m --m ∈ In-1(M, G) for all g ∈ G}. We show that some fundamental problems concerning these modules can be reduced to the case In (k[G], G) where G acts on itself by right translation. We study the questions as to when In(M, G) is a finitely generated AG-module and how the In(M, G) are related to equivariant mappings. For the classical case of Ga acting on binary forms, we describe the In(M, G) and determine when they are Cohen-Macaulay. [ABSTRACT FROM AUTHOR]

In this note, we derive formulas for the F-pure threshold, higher jumping numbers, and test ideals of diagonal and Fermat hypersurfaces. For these hypersurfaces, we answer a question of Schwede regarding the denominators of F-pure thresholds, and obtain tight upper bounds for the number of higher jumping numbers. Our results are valid over all (or all but finitely many) characteristics, and therefore allow us to construct examples in which the characteristic p setting is drastically different than that over C. [ABSTRACT FROM AUTHOR]

For homogeneous metrics on the spaces over the complexes, quaternions and octonions, it is shown that the Ricci flow can move a metric of strictly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with indefinite signature. A variant of the method of Böhm and Wilking is given, proving that one can flow a metric of positive sectional curvature to one with Ricci curvature of indefinite signature in the quaternionic and octonian cases. A proof is given that this cannot occur in the complex case. [ABSTRACT FROM AUTHOR]

We prove upper bounds for the Hilbert-Samuel multiplicity of standard graded Gorenstein algebras. The main tool that we use is Boij- Söderberg theory to obtain a decomposition of the Betti table of a Gorenstein algebra as the sum of rational multiples of symmetrized pure tables. Our bound agrees with the one in the quasi-pure case obtained by Srinivasan [J. Algebra, vol. 208, no. 2 (1998)]. [ABSTRACT FROM AUTHOR]

Let S be a cancellative left-amenable semigroup and let A be a finite set. We prove that every pre-injective cellular automaton τ : AS → AS is surjective. [ABSTRACT FROM AUTHOR]