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Given two polynomials p(x), q(x) of degree d, we give a combinatorial formula for the finite free cumulants of p(x)\boxtimes _d q(x). We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that \boxtimes _d converges to \boxtimes as d goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building on our results we give a new short and conceptual proof of a recent result (see J. Hoskins and Z. Kabluchko [Exp. Math. (2021), pp. 1–27]; S. Steinerberger [Exp. Math. (2021), pp. 1–6]) that connects root distributions of polynomial derivatives with free fractional convolution powers. [ABSTRACT FROM AUTHOR]

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0. Bounding the rank of this object, we prove that a conjecture by Campana [Ann. Inst. Fourier (Grenoble) 54 (2004), pp. 499–630] and Corvaja–Zannier [Math. Z. 286 (2017), pp. 579–602] holds for Enriques surfaces, as well as K3 surfaces of Picard rank \geq 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 satisfy the weak Hilbert Property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded. [ABSTRACT FROM AUTHOR]

Let X be a quotient of the modular curve X_0(N) whose Jacobian J_X is a simple factor of J_0(N)^{new} over \mathbf {Q}. Let f be the newform of level N and weight 2 associated with J_X; assume f has analytic rank 1. We give analytic methods for determining the rational points of X using quadratic Chabauty by computing two p-adic Gross–Zagier formulas for f. Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotomic p-adic L-function of f constructed by Bertolini, Darmon, and Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148], which lies outside of the range of interpolation. [ABSTRACT FROM AUTHOR]

We construct pointed Prym–Brill–Noether varieties parametrizing line bundles assigned to an irreducible étale double covering of a curve with a prescribed minimal vanishing at a fixed point. We realize them as degeneracy loci in type D and deduce their classes in case of expected dimension. Thus, we determine a pointed Prym–Petri map and prove a pointed version of the Prym–Petri theorem implying that the expected dimension holds in the general case. These results build on work of Welters [Ann. Sci. Ëcole Norm. Sup. (4) 18 (1985), pp. 671–683] and De Concini–Pragacz [Math. Ann. 302 (1995), pp. 687–697] on the unpointed case. Finally, we show that Prym varieties are Prym–Tyurin varieties for Prym–Brill–Noether curves of exponent enumerating standard shifted tableaux times a factor of 2, extending to the Prym setting work of Ortega [Math. Ann. 356 (2013), pp. 809–817]. [ABSTRACT FROM AUTHOR]

Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some technical conditions, that they exhibit Weibull asymptotics with respect to different natural measures on the space of unimodular lattices in \mathbb {R}^d. This follows from very general Poisson approximation results for shrinking targets which should be of independent interest. Furthermore, we show in the appendix that the logarithm laws of Kleinbock-Margulis [Invent. Math. 138 (1999), pp. 451–494], Khinchin and Gallagher [J. London Math. Soc. 37 (1962), pp. 387–390] can be deduced from our distributional results. [ABSTRACT FROM AUTHOR]

We show that the number of non-isotopic commutative semifields of odd order p^{n} is exponential in n when n = 4t and t is not a power of 2. We introduce a new family of commutative semifields and a method for proving isotopy results on commutative semifields that we use to deduce the aforementioned bound. The previous best bound on the number of non-isotopic commutative semifields of odd order was quadratic in n and given by Zhou and Pott [Adv. Math. 234 (2013), pp. 43–60]. Similar bounds in the case of even order were given in Kantor [J. Algebra 270 (2003), pp. 96–114] and Kantor and Williams [Trans. Amer. Math. Soc. 356 (2004), pp. 895–938]. [ABSTRACT FROM AUTHOR]

We study the category of representations of \mathfrak {sl}_{m+2n} in positive characteristic, with p-character given by a nilpotent with Jordan type (m+n,n). Recent work of Bezrukavnikov-Mirkovic [Ann. of Math. (2) 178 (2013), pp. 835–919] implies that this representation category is equivalent to \mathcal {D}_{m,n}^0, the heart of the exotic t-structure on the derived category of coherent sheaves on a Springer fibre for that nilpotent. Using work of Cautis and Kamnitzer [Duke Math. J. 142 (2008), pp. 511–588], we construct functors indexed by affine tangles, between these categories \mathcal {D}_{m,n} (i.e. for different values of n). This allows us to describe the irreducible objects in \mathcal {D}_{m,n}^0 and enumerate them by crossingless (m,m+2n) matchings. We compute the \mathrm {Ext} spaces between the irreducible objects, and conjecture that the resulting Ext algebra is an annular variant of Khovanov's arc algebra. In subsequent work, we use these results to give combinatorial dimension formulae for the irreducible representations. These results may be viewed as a positive characteristic analogue of results about two-block parabolic category \mathcal {O} due to Lascoux-Schutzenberger [Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266], Bernstein-Frenkel-Khovanov [Selecta Math. (N.S.) 5 (1999), pp. 199–241], Brundan-Stroppel [Represent. Theory 15 (2011), pp. 170–243], et al. [ABSTRACT FROM AUTHOR]

We correct some errors in The class number one problem for the normal CM-fields of degree 32 in Trans. Amer. Math. Soc., vol. 359 no. 10 (2007). [ABSTRACT FROM AUTHOR]

We obtain almost-sure scattering for the Schrödinger equation with a defocusing cubic nonlinearity in the Euclidean space \mathbb {R}^3, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of Bényi, Oh and Pocovnicu [Trans. Amer. Math. Soc. Ser. B 2 (2015), pp. 1–50]. It also extends the results of Dodson, Lührmann and Mendelson [Adv. Math. 347 (2019), pp. 619–676] on the energy-critical equation in \mathbb {R}^4, to the energy-subcritical equation in \mathbb {R}^3. In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect which makes it subcritical, it still has infinite energy. In the present work, we first develop a stability theory from the deterministic scattering results below the energy space, due to Colliander, Keel, Staffilani, Takaoka and Tao. Then, we propose a globalization argument in which we set up the I-method with a Morawetz bootstrap in a stochastic setting. To our knowledge, this is the first almost-sure scattering result for an energy-subcritical Schrödinger equation outside the small data regime. [ABSTRACT FROM AUTHOR]

We study certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman [J. Algebraic Geom. 10 (2001), pp. 623–694], Yoshioka [J. Reine Angew.Math. 515 (1999), pp. 97–123], and Nakajima [ Convolution on homology groups of moduli spaces of sheaves on K3 surfaces , Contemp. Math., vol. 322, Amer. Math. Soc., Providence, RI, 2003, pp. 75–87]. We show that these sequences can be given the structure of a geometric categorical \mathfrak {sl}_2 action in the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an equivalence between derived categories of some moduli spaces that are birational via stratified Mukai flops. [ABSTRACT FROM AUTHOR]

Using the theory of Macdonald [ Symmetric functions and Hall polynomials , The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group \mathfrak {S}_n are given by plethystically transformed Macdonald polynomials specialized at q=t. We generalize this to restricted rational Cherednik algebras of wreath product groups C_\ell \wr \mathfrak {S}_n and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [ Combinatorics, symmetric functions, and Hilbert schemes , Int. Press, Somerville, MA, 2003, pp. 39–111]. [ABSTRACT FROM AUTHOR]

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a "convex combination" of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni [Comm. Pure Appl. Math. 74 (2021), pp. 1100–1126] concerning manifolds with negative k-Ricci curvature and generalize a result of Wu-Yau [Comm. Anal. Geom. 24 (2016), pp. 901–912] and Diverio-Trapani [J. Differential Geom. 111 (2019), pp. 303–314] to the conformally Kähler case. We also show that the compact Kähler manifold is projective and simply connected if the mixed curvature is positive. [ABSTRACT FROM AUTHOR]

We prove sharp L^{12} estimates for exponential sums associated with nondegenerate curves in \mathbb {R}^4. We place Bourgain's seminal result [J. Amer. Math. Soc. 30 (2017), pp. 205–224] in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves. [ABSTRACT FROM AUTHOR]

Let \mathcal {C}(n,k) be the set of k-dimensional simplicial complexes C over a fixed set of n vertices such that: C has a complete k-1-skeleton; C has precisely {{n-1}\choose {k}} k-faces; the homology group H_{k-1}(C) is finite. Consider the probability measure on \mathcal {C}(n,k) where the probability of a simplicial complex C is proportional to |H_{k-1}(C)|^2. For any fixed k, we determine the local weak limit of these random simplicial complexes as n tends to infinity. This local weak limit turns out to be the same as the local weak limit of the 1-out k-complexes investigated by Linial and Peled [Ann. of Math. (2) 184 (2016), pp. 745-773]. [ABSTRACT FROM AUTHOR]

We introduce and study a notion of relative 1-homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local Hölder regularity of area minimizing surfaces in a given relative 1-homotopy class in proper geodesic metric spaces admitting a local quadratic isoperimetric inequality. If the underlying space has trivial second homotopy group then relatively 1-homotopic maps are relatively homotopic. We also obtain an analog for closed surfaces in a given 1-homotopy class. Our theorems generalize and strengthen results of Lemaire [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), pp. 91–103], Jost [J. Reine Angew. Math. 359 (1985), pp. 37-54], Schoen–Yau [Ann. of Math. (2) 110 (1979), pp. 127–142], and Sacks–Uhlenbeck [Trans. Amer. Math. Soc. 271 (1982), pp. 639–652]. [ABSTRACT FROM AUTHOR]

Jockusch [J. Combin. Theory Ser. A 72 (1995), pp. 318–321] constructed an infinite family of centrally symmetric (cs, for short) triangulations of 3-spheres that are cs-2-neighborly. Recently, Novik and Zheng [Adv. Math. 370 (2020), 16 pp.] extended Jockusch's construction: for all d and n>d, they constructed a cs triangulation of a d-sphere with 2n vertices, \Delta ^d_n, that is cs-\lceil d/2\rceil-neighborly. Here, several new cs constructions, related to \Delta ^d_n, are provided. It is shown that for all k>2 and a sufficiently large n, there is another cs triangulation of a (2k-1)-sphere with 2n vertices that is cs-k-neighborly, while for k=2 there are \Omega (2^n) such pairwise non-isomorphic triangulations. It is also shown that for all k>2 and a sufficiently large n, there are \Omega (2^n) pairwise non-isomorphic cs triangulations of a (2k-1)-sphere with 2n vertices that are cs-(k-1)-neighborly. The constructions are based on studying facets of \Delta ^d_n, and, in particular, on some necessary and some sufficient conditions similar in spirit to Gale's evenness condition. Along the way, it is proved that Jockusch's spheres \Delta ^3_n are shellable and an affirmative answer to Murai–Nevo's question about 2-stacked shellable balls is given. [ABSTRACT FROM AUTHOR]

Let G be a reductive group, and let X be a smooth quasi-projective complex variety. We prove that any G-irreducible, G-cohomologically rigid local system on X with finite order abelianization and quasi-unipotent local monodromies is integral. This generalizes work of Esnault and Groechenig [Selecta Math. (N. S.) 24 (2018), pp. 4279–4292; Acta Math. 225 (2020), pp. 103–158] when G= \mathrm {GL}_n, and it answers positively a conjecture of Simpson [Inst. Hautes Études Sci. Publ. Math. 75 (1992), pp. 5–95; Inst. Hautes Études Sci. Publ. Math. 80 (1994), pp. 5–79] for G-cohomologically rigid local systems. Along the way we show that the connected component of the Zariski-closure of the monodromy group of any such local system is semisimple; this moreover holds when we relax cohomological rigidity to rigidity. [ABSTRACT FROM AUTHOR]

We show that two-weight L2 bounds for sparse square functions (uniform with respect to sparseness constants, and in both directions) do not imply a two-weight L2 bound for the Hilbert transform. We present an explicit counterexample, making use of the construction due to Reguera–Thiele from [Math. Res. Lett. 19 (2012)]. At the same time, we show that such two-weight bounds for sparse square functions do not imply both separated Orlicz bump conditions on the involved weights for p = 2 (and for Young functions satisfying an appropriate integrability condition). We rely on the domination of L log L bumps by Orlicz bumps observed by Treil–Volberg in [Adv. Math. 301 (2016), pp. 499-548] (for Young functions satisfying an appropriate integrability condition). [ABSTRACT FROM AUTHOR]

We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelski's conjecture for amenable theories (proved by Krupiński, Newelski, and Simon [J. Math. Log. 19 (2019), no. 2, 1950012, p. 55]) cannot be dropped. [ABSTRACT FROM AUTHOR]

The article presents a study to examine the solvability of second-order equations with partially BMO coefficients. It analyses the Lp-solvability of second-order elliptic and parabolic equations in nondivergence form using divergence-form equations. It mentions the leading coefficients has been assumed to be in locally BMO spaces.

The main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of $ *$ [ABSTRACT FROM AUTHOR]

For an $n$-component link, Milnor's isotopy invariants are defined for each multi-index $I=i_1i_2...i_m~(i_jin {1,...,n})$. Here $m$ is called the length. Let $r(I)$ denote the maximum number of times that any index appears in $I$. It is known that Milnor invariants with $r=1$, i.e., Milnor invariants for all multi-indices $I$ with $r(I)=1$, are link-homotopy invariant. N. Habegger and X. S. Lin showed that two string links are link-homotopic if and only if their Milnor invariants with $r=1$ coincide. This gives us that a link in $S^3$ is link-homotopic to a trivial link if and only if all Milnor invariants of the link with $r=1$ vanish. Although Milnor invariants with $r=2$ are not link-homotopy invariants, T. Fleming and the author showed that Milnor invariants with $rleq 2$ are self $Delta $-equivalence invariants. In this paper, we give a self $Delta $-equivalence classification of the set of $n$-component links in $S^3$ whose Milnor invariants with length $leq 2n-1$ and $rleq 2$ vanish. As a corollary, we have that a link is self $Delta $-equivalent to a trivial link if and only if all Milnor invariants of the link with $rleq 2$ vanish. This is a geometric characterization for links whose Milnor invariants with $rleq 2$ vanish. The chief ingredient in our proof is Habiro's clasper theory. We also give an alternate proof of a link-homotopy classification of string links by using clasper theory. [ABSTRACT FROM AUTHOR]

Let $G$ be a finite group, $S in mathit {Syl}_p(G)$, and $mathcal S$ be the set subgroups containing $S$. For $M in mathcal S$ and $V = Omega _1Z(O_p(M))$, the paper discusses the action of $M$ on $V$. Apart from other results, it is shown that for groups of parabolic characteristic $p$ either $S$ is contained in a unique maximal $p$-local subgroup, or there exists a maximal $p$-local subgroup in $M in mathcal S$ such that $V$ is a nearly quadratic 2F-module for $M$. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Murre's conjectures on the structure of rational Chow groups and exhibit explicit Chow--Künneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle, explicit Chow--Künneth projectors are obtained which satisfy Murre's conjectures, and the motivic Hard Lefschetz theorem is verified. [ABSTRACT FROM AUTHOR]

In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type. [ABSTRACT FROM AUTHOR]

We consider subgroups of $C^{\infty}$-diffeomorphisms of the circle $\mathbb S^1$ which act transitively on $4$-tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of $\mathbb S^1$. A stronger result concerning $C^{\infty}$-approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category. [ABSTRACT FROM AUTHOR]

This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically closed field $k$ of positive characteristic $p$. We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case $\mathfrak{g}= \operatorname{Lie}(\mathcal{G})$ is the Lie algebra of a smooth group $\mathcal{G}$. For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension ${\rm rk}(\mathfrak{g})$ is shown to be irreducible of dimension $\dim_k\mathfrak{g}-{\rm rk}(\mathfrak{g})$, with Premet's regular Cartan subalgebras belonging to the regular locus. [ABSTRACT FROM AUTHOR]

In this paper we construct an example of a complete minimal disk which is properly immersed in a ball of $\mathbb{R}^3$. [ABSTRACT FROM AUTHOR]

Let E be a CM number field and F its maximal real subfield. We prove a level-raising result for regular algebraic conjugate self-dual automorphic representations of GLn(AE). This generalizes previously known results of Thorne [Forum Math. Sigma 2 (2014)] by removing certain hypotheses occurring in that work. In particular, the level-raising prime p is allowed to be unramified as opposed to inert in F, the field E/F is not assumed to be everywhere unramified, and the field E is allowed to be a general CM field. [ABSTRACT FROM AUTHOR]

A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope $G(A)$ of a finite dimensional superalgebra $A$. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: $A$ is a minimal superalgebra if and only if the ideal of identities of $G(A)$ is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties $\mathcal{V}$ such that $\exp({\mathcal{V}})=d\ge 2$ and $\exp(\mathcal{U})