We find 101,052 new extreme divisors on ${\bar{\mathit{M}}}_{0,7}$ (in 31 ${\mathit{S}}_7$-orbits) and millions of extreme nef curves over characteristic 0. Over characteristic 2, we identify two more ${\mathit{S}}_7$-orbits of extreme divisors and prove ${\overline{\text{Eff}}}^{\mathit{k}}({\bar{\mathit{M}}}_{0,\mathit{n}})$ is strictly larger over characteristic 2 than it is over characteristic 0 for all $1\le \mathit{k}\le \mathit{n}-6$. For each such k, we provide explicit cycles that are extreme in ${\text{Eff}}^{\mathit{k}}({\bar{\mathit{M}}}_{0,\mathit{n}})$ over characteristic 2 but external to ${\overline{\text{Eff}}}^{\mathit{k}}({\bar{\mathit{M}}}_{0,\mathit{n}})$ over characteristic 0. We apply our method of finding new extreme divisors to compute $\text{Eff}({\bar{\mathit{M}}}_{0,\mathcal{A}})$ for $\mathcal{A}=(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1)$, proving it is polyhedral over any field, and conjecture a description of $\text{Eff}({\text{Bl}}_{\mathit{e}}{\overline{\mathit{L}\mathit{M}}}_7)$.

We introduce new notions of log jet spaces. Mildly singular spaces are “smooth” in log geometry, so their log jet spaces behave like the jet spaces of smooth varieties. Myriad examples contrast log jet spaces with the usual jet spaces of schemes.

We then compute the log Kähler differentials of the log jet and arc spaces after de Fernex and Docampo. We obtain some of their applications to the structure of the log jet space. We conclude with comparison remarks with other log jet spaces.

The Chow ring of ${\mathcal{M}}_{\mathit{g}}$ is known to be generated by tautological classes for $\mathit{g}\le 9$. Meanwhile, the first example of a nontautological class on ${\mathcal{M}}_{\mathit{g}}$ is the fundamental class of the bielliptic locus in ${\mathcal{M}}_{12}$, due to van Zelm. It remains open if the Chow rings of ${\mathcal{M}}_{10}$ and ${\mathcal{M}}_{11}$ are generated by tautological classes. In these cases, a natural first place to look is at the bielliptic locus. In genus 10, it is already known that classes supported on the bielliptic locus are tautological. Here, we prove that all classes supported on the bielliptic locus are tautological in genus 11. By Looijenga’s vanishing theorem, this implies that they all vanish.

We show that the Bergman metric of the ball quotients ${\mathbb{B}}^2/\mathrm{\Gamma }$, where Γ is a finite and fixed point free group, is Kähler–Einstein if and only if Γ is trivial. As a consequence, we characterize the unit ball ${\mathbb{B}}^2$, among two-dimensional Stein spaces with isolated normal singularities, proving an algebraic version of Cheng’s conjecture for two-dimensional Stein spaces.

In this paper, we develop a technique for discovering (noneffective) irrational rays at the boundary of the Mori cone for linear systems on a general blowup of the plane and give examples of such irrational rays.

In this paper, we study the algebraic structure of mapping class group $Mod(\mathit{X})$ of 3-manifolds X that fiber as a circle bundle over a surface ${\mathit{S}}^1\to \mathit{X}\to {\mathit{S}}_{\mathit{g}}$. There is an exact sequence $1\to {\mathit{H}}^1({\mathit{S}}_{\mathit{g}})\to Mod(\mathit{X})\to Mod({\mathit{S}}_{\mathit{g}})\to 1$. We relate this to the Birman exact sequence and determine when this sequence splits.

Based on the work of Harada, Nowroozi, and Van Tuyl, who provided particular length two virtual resolutions for finite sets of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$, we prove that the vast majority of virtual resolutions of a pair for minimal elements of the multigraded regularity in this setting are of Hilbert–Burch type. We give explicit descriptions of these short virtual resolutions that depend only on the number of points. Moreover, despite initial evidence, we show that these virtual resolutions are not always short and give sufficient conditions for their length to be three.

We prove that among all constant-width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant-width body obtained by rotation of the Reuleaux triangle about an axis of symmetry.

Let $\mathit{d}\ge 2$ and V belong to a reverse Hölder class. Let u be a strong solution to the Schrödinger equation

$$-\mathrm{\Delta }\mathit{u}+\mathit{V}\mathit{u}=\mathit{f}\text{in}{\mathbb{R}}^{\mathit{d}}.$$

For an appropriate Musielak–Orlicz function φ, we show that

$$\Vert {\mathit{D}}^2\mathit{u}{\Vert }_{{\mathit{L}}^{\mathit{\phi }(·)}({\mathbb{R}}^{\mathit{d}})}+\Vert \mathit{V}\mathit{u}{\Vert }_{{\mathit{L}}^{\mathit{\phi }(·)}({\mathbb{R}}^{\mathit{d}})}\le \mathit{C}\Vert \mathit{f}{\Vert }_{{\mathit{L}}^{\mathit{\phi }(·)}({\mathbb{R}}^{\mathit{d}})}.$$

We use the moment method of Wood to study the distribution of random finite modules over a countable Dedekind domain with finite quotients, generated by taking cokernels of random $\mathit{n}\times \mathit{n}$ matrices with entries valued in the domain. Previously, Wood found that when the entries of a random $\mathit{n}\times \mathit{n}$ integral matrix are not too concentrated modulo a prime, the asymptotic distribution (as $\mathit{n}\to \infty$) of the cokernel matches the Cohen and Lenstra conjecture on the distribution of class groups of imaginary quadratic fields. We develop and prove a condition that produces a similar universality result for random matrices with entries valued in a countable Dedekind domain with finite quotients.

We show that the category of quasi-coherent Cartier crystals is equivalent to the category of unit Cartier modules on an F-finite Noetherian ring R and that these equivalent categories have finite global dimension by showing that every quasi-coherent Cartier crystal has a finite injective resolution. The length of the resolution is uniformly bounded by a bound only depending on R. Our result should be viewed as a generalization of the main result of Ma [Ma14] showing that the category of unit $\mathit{R}[\mathit{F}]$-modules over an F-finite regular ring R has finite global dimension $dim\mathit{R}+1$.

We study the Hodge conjecture for powers of K3 surfaces and show that if the Kuga–Satake correspondence is algebraic for a family of K3 surfaces of generic Picard number 16, then the Hodge conjecture holds for all powers of any K3 surface in that family.

We provide information on diffeotopy groups of exotic smoothings of punctured 4-manifolds, extending previous results on diffeotopy groups of exotic ${\mathbb{R}}^4$s. In particular, we prove that for a smoothable 4-manifold M and for a nonempty discrete set of points $\mathit{S}⊊\mathring{\mathit{M}}$, there are uncountably many distinct smoothings of $\mathit{M}\setminus \mathit{S}$ whose diffeotopy groups are uncountable.

We then prove that for a smoothable 4-manifold M and for a nonempty discrete set of points $\mathit{S}⊊\mathring{\mathit{M}}$, there exists a smoothing of $\mathit{M}\setminus \mathit{S}$ whose diffeotopy groups have similar properties as ${\mathcal{R}}_{\mathit{U}}$, Freedman and Taylor’s universal ${\mathbb{R}}^4$.

Moreover, we prove that if M is nonsmoothable, then both results still hold under the assumption that $|\mathit{S}|\ge 2$.

The goal of this paper is to accurately describe the maximal zero-free region of the independence polynomial for graphs of bounded degree, for large degree bounds. In the previous work with de Boer, Guerini, and Regts, it was demonstrated that this zero-free region coincides with the normality region of the related occupation ratios. These ratios form a discrete semigroup that is in a certain sense generated by finitely many rational maps. We will show that as the degree bound converges to infinity, the properly rescaled normality regions converge to a limit domain, which can be described as the maximal boundedness component of a semigroup generated by infinitely many exponential maps.

We prove that away from the real axis this boundedness component avoids a neighborhood of the boundary of the limit cardioid, answering a recent question by Andreas Galanis. We also give an exact formula for the boundary of the boundedness component near the positive real boundary point.

Let ${\mathit{S}}_{\mathit{g}}$ be a closed orientable surface of genus $\mathit{g}>1$. Consider the minimal asymptotic translation length ${\mathit{L}}_{\mathcal{T}}(\mathit{k},\mathit{g})$ on the Teichmüller space of ${\mathit{S}}_{\mathit{g}}$, among pseudo-Anosov mapping classes of ${\mathit{S}}_{\mathit{g}}$ acting trivially on k-dimensional subspaces of ${\mathit{H}}_1({\mathit{S}}_{\mathit{g}})$, $0\le \mathit{k}\le 2\mathit{g}$. The asymptote of ${\mathit{L}}_{\mathcal{T}}(\mathit{k},\mathit{g})$ for extreme cases $\mathit{k}=0,2\mathit{g}$ have been shown by several authors. Jordan Ellenberg asked whether there is a lower bound for ${\mathit{L}}_{\mathcal{T}}(\mathit{k},\mathit{g})$ interpolating the known results on ${\mathit{L}}_{\mathcal{T}}(0,\mathit{g})$ and ${\mathit{L}}_{\mathcal{T}}(2\mathit{g},\mathit{g})$, which was affirmatively answered by Agol, Leininger, and Margalit.

In this paper, we study an analogue of Ellenberg’s question, replacing Teichmüller spaces with curve complexes. We provide lower and upper bound on the minimal asymptotic translation length ${\mathit{L}}_{\mathcal{C}}(\mathit{k},\mathit{g})$ on the curve complex, whose lower bound interpolates the known results on ${\mathit{L}}_{\mathcal{C}}(0,\mathit{g})$ and ${\mathit{L}}_{\mathcal{C}}(2\mathit{g},\mathit{g})$.

Finally, for each g, we construct a non-Torelli pseudo-Anosov ${\mathit{f}}_{\mathit{g}}\in Mod({\mathit{S}}_{\mathit{g}})$ which does not normally generate $Mod({\mathit{S}}_{\mathit{g}})$, so that the asymptotic translation length of ${\mathit{f}}_{\mathit{g}}$ on the curve complex decays faster than a constant multiple of $1/\mathit{g}$ as $\mathit{g}\to \infty$. From this, we provide a restriction on how small the asymptotic translation lengths on curve complexes should be if the similar phenomenon as in the work of Lanier and Margalit on Teichmüller spaces holds for curve complexes.

Let $\mathit{N}\ge 2$ and let $\mathrm{Out}({\mathit{F}}_{\mathit{N}})$ be the outer automorphism group of a non-abelian free group of rank N. Let ${\mathrm{IA}}_{\mathit{N}}(\mathbb{Z}/3\mathbb{Z})$ be the finite index subgroup of $\mathrm{Out}({\mathit{F}}_{\mathit{N}})$, which is the kernel of the natural action of $\mathrm{Out}({\mathit{F}}_{\mathit{N}})$ on ${\mathit{H}}_1({\mathit{F}}_{\mathit{N}},\mathbb{Z}/3\mathbb{Z})$. We show that ${\mathrm{IA}}_{\mathit{N}}(\mathbb{Z}/3\mathbb{Z})$ is an R-group, that is, for every $\mathit{\varphi },\mathit{\psi }\in {\mathrm{IA}}_{\mathit{N}}(\mathbb{Z}/3\mathbb{Z})$, if there exists $\mathit{k}\ge 1$ such that ${\mathit{\varphi }}^{\mathit{k}}={\mathit{\psi }}^{\mathit{k}}$, then $\mathit{\varphi }=\mathit{\psi }$. This answers a question of Handel and Mosher. We then use the fact that ${\mathrm{IA}}_{\mathit{N}}(\mathbb{Z}/3\mathbb{Z})$ is an R-group in order to prove that the normalizer in ${\mathrm{IA}}_{\mathit{N}}(\mathbb{Z}/3\mathbb{Z})$ of every abelian subgroup of ${\mathrm{IA}}_{\mathit{N}}(\mathbb{Z}/3\mathbb{Z})$ is equal to its centralizer. We finally give an alternative proof of a result, due to Feighn and Handel, that the centralizer of an element of $\mathrm{Out}({\mathit{F}}_{\mathit{N}})$, which has only finitely many periodic orbits of conjugacy classes of maximal cyclic subgroups of ${\mathit{F}}_{\mathit{N}}$, is virtually abelian.

Ideals in infinite-dimensional polynomial rings that are invariant under the action of the monoid of increasing functions have been extensively studied recently. Of particular interest is the asymptotic behavior of truncations of such an ideal in finite-dimensional polynomial subrings. It has been conjectured that the Castelnuovo–Mumford regularity and projective dimension are eventual linear functions along such truncations. In the present paper we provide evidence for these conjectures. We show that for monomial ideals the projective dimension is eventually linear, whereas the regularity is asymptotically linear.

Let C be the rational normal curve of degree e in ${\mathbb{P}}^{\mathit{n}}$, and let $\mathit{X}\subset {\mathbb{P}}^{\mathit{n}}$ be a degree $\mathit{d}\ge 2$ hypersurface containing C. Coskun and Riedl show that the normal bundle ${\mathit{N}}_{\mathit{C}/\mathit{X}}$ is balanced for a general X. Larson studies the case of lines ($\mathit{e}=1$) and computes the dimension of the space of hypersurfaces for which ${\mathit{N}}_{\mathit{C}/\mathit{X}}$ has a given splitting type. In this paper, we work with any $\mathit{e}\ge 2$. We compute explicit examples of hypersurfaces for all possible splitting types, and for $\mathit{d}\ge 3$, we compute the dimension of the space of hypersurfaces for which ${\mathit{N}}_{\mathit{C}/\mathit{X}}$ has a given splitting type. For $\mathit{d}=2$, we give a lower bound on the maximum rank of quadrics X such that ${\mathit{N}}_{\mathit{C}/\mathit{X}}$ has a fixed splitting type.

This paper investigates exotic phenomena exhibited by links of disconnected surfaces with boundary properly embedded in the 4-ball. Our main results provide two different constructions of exotic pairs of surface links, which are Brunnian, meaning that all their proper sublinks are trivial. Furthermore, we modify these core constructions to vary the number of components in the exotic links, the genera of the components, and the number of components that must be removed before the surfaces become unlinked.

We study lattice points on hyperbolic circles centered at Heegner points of class number one. Our main result is that, on a density one subset of radii tending to infinity, the angles of such points equidistribute on the unit circle. To prove this, we establish a connection between lattice points and algebraic integers in the associated field having norm of a special form and satisfying a congruence condition. As a by-product of this, we obtain an explicit formulation of the classical hyperbolic circle problem as a shifted convolution sum for the function that counts the number of algebraic integers with given norm. Along the way, we also prove a lower bound for shifted B-numbers, which is done by sieve methods.

We prove that generic Hitchin representations are strongly dense: every pair of noncommuting elements in their image generate a Zariski-dense subgroup of ${\mathrm{SL}}_{\mathit{n}}(\mathbb{R})$. In the proof, we use a theorem of Rapinchuk, Benyash-Krivetz, and Chernousov to show that the set of Hitchin representations is Zariski-dense in the variety of representations of a surface group in ${\mathrm{SL}}_{\mathit{n}}(\mathbb{R})$.

In this paper, we analyze the fundamental group ${\mathit{\pi }}_1(\mathrm{\Sigma }\mathit{X},{\bar{\mathit{x}}}_0)$ of the reduced suspension $\mathrm{\Sigma }\mathit{X}$ where $(\mathit{X},{\mathit{x}}_0)$ is an arbitrary based Hausdorff space. We show that ${\mathit{\pi }}_1(\mathrm{\Sigma }\mathit{X},{\bar{\mathit{x}}}_0)$ is canonically isomorphic to a direct limit ${\underrightarrow{lim}}_{\mathit{A}\in \mathcal{P}}{\mathit{\pi }}_1(\mathrm{\Sigma }\mathit{A},{\bar{\mathit{x}}}_0)$ where each group ${\mathit{\pi }}_1(\mathrm{\Sigma }\mathit{A},{\bar{\mathit{x}}}_0)$ is isomorphic to a finitely generated free group or the infinite earring group. A direct consequence of this characterization is that ${\mathit{\pi }}_1(\mathrm{\Sigma }\mathit{X},{\bar{\mathit{x}}}_0)$ is locally free for any Hausdorff space X. Additionally, we show that $\mathrm{\Sigma }\mathit{X}$ is simply connected if and only if X is sequentially 0-connected at ${\mathit{x}}_0$.

The cancellation problem asks whether $\mathit{A}[{\mathit{X}}_1,{\mathit{X}}_2,\dots ,{\mathit{X}}_{\mathit{n}}]\cong \mathit{B}[{\mathit{Y}}_1,{\mathit{Y}}_2,\dots ,{\mathit{Y}}_{\mathit{n}}]$ implies $\mathit{A}\cong \mathit{B}$. Hamann introduced the class of steadfast rings as the rings for which a version of the cancellation problem considered by Abhyankar, Eakin, and Heinzer holds. By work of Asanuma, Hamann, and Swan steadfastness can be characterized in terms of p-seminormality, which is a variant of normality introduced by Swan. We prove that p-seminormality and steadfastness deform for reduced Noetherian local rings. We also prove that p-seminormality and steadfastness are stable under adjoining formal power series variables for reduced (not necessarily Noetherian) rings. Our methods also give new proofs of the facts that normality and weak normality deform, which are of independent interest.

We conjecture a structure formula for the $\mathrm{SU}(\mathit{r})$ Vafa–Witten partition function for surfaces with holomorphic 2-form. The conjecture is based on S-duality and a structure formula for the vertical contribution previously derived by the third-named author using Gholampour–Thomas’s theory of virtual degeneracy loci.

For ranks $\mathit{r}=2,3$, conjectural expressions for the partition function in terms of the theta functions of ${\mathit{A}}_{\mathit{r}-1}$, ${\mathit{A}}_{\mathit{r}-1}^{\vee }$ and Seiberg–Witten invariants were known. We conjecture new expressions for $\mathit{r}=4,5$ in terms of the theta functions of ${\mathit{A}}_{\mathit{r}-1}$, ${\mathit{A}}_{\mathit{r}-1}^{\vee }$, Seiberg–Witten invariants, and continued fractions studied by Ramanujan. The vertical part of our conjectures is proved for low virtual dimensions by calculations on nested Hilbert schemes.

The horizontal part of our conjectures gives predictions for virtual Euler characteristics of Gieseker–Maruyama moduli spaces of stable sheaves. In this case, our formulae are sums of universal functions with coefficients in Galois extensions of $\mathbb{Q}$. The universal functions, corresponding to different quantum vacua, are permuted under the action of the Galois group.

For $\mathit{r}=6,7$, we also find relations with Hauptmoduln of ${\mathrm{\Gamma }}_0(\mathit{r})$. We present K-theoretic refinements for $\mathit{r}=2,3,4$ involving weak Jacobi forms.

For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla and carried out by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erdős holds. We also characterize rational valued periodic functions for which the associated Dirichlet series vanishes at two different positive integers under some additional conditions.

We prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. These are concerned with the question of Ingham, who asked for optimal and explicit order estimates for the error term $\mathrm{\Delta }(\mathit{x}):=\mathit{\psi }(\mathit{x})-\mathit{x}$, given any zero-free region $\mathcal{D}(\mathit{\eta }):=\{\mathit{s}=\mathit{\sigma }+\mathit{i}\mathit{t}\in \mathbb{C}:\mathit{\sigma }:=\mathrm{\Re }\mathit{s}\ge 1-\mathit{\eta }(\mathit{t})\}$. In the classical case essentially sharp results are due to some 40 years old work of Pintz.

Here we consider a given system of Beurling primes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$, the corresponding integer counting function $\mathcal{N}(\mathit{x})$, and the respective error term ${\mathrm{\Delta }}_{\mathcal{P}}(\mathit{x}):={\mathit{\psi }}_{\mathcal{P}}(\mathit{x})-\mathit{x}$ in the PNT of Beurling, where ${\mathit{\psi }}_{\mathcal{P}}(\mathit{x})$ is the Beurling analogue of $\mathit{\psi }(\mathit{x})$. First we prove that if the Beurling zeta function ${\mathit{\zeta }}_{\mathcal{P}}$ does not vanish in $\mathcal{D}(\mathit{\eta })$, then the extension of Pintz’ result holds: $|{\mathrm{\Delta }}_{\mathcal{P}}(\mathit{x})|\le \mathit{x}exp((1+\mathit{\epsilon }){\mathit{\omega }}_{\mathit{\eta }}(log\mathit{x}))(\mathit{x}>{\mathit{x}}_0(\mathit{\epsilon }))$, where ${\mathit{\omega }}_{\mathit{\eta }}(\mathit{y}):=\mathcal{L}(\mathit{\eta })(\mathit{y})$ is the naturally occurring conjugate function—essentially the Legendre transform—of $\mathit{\eta }(\mathit{t})$, introduced into the field by Ingham. In the second part we prove a converse: if ${\mathit{\zeta }}_{\mathcal{P}}$ has an infinitude of zeroes in the given domain, then analogously to the classical case, $|{\mathrm{\Delta }}_{\mathcal{P}}(\mathit{x})|\ge \mathit{x}exp((1-\mathit{\epsilon }){\mathit{\omega }}_{\mathit{\eta }}(log\mathit{x}))$ holds “infinitely often”. This also shows that both main results are sharp apart from the arbitrarily small $\mathit{\epsilon }>0$.

The classical results of Pintz used many facts about the Riemann zeta function. Recently we worked out a number of analogous results—including some construction of quasi-optimal integration paths, a Riemann–von Mangoldt-type formula, a Carlson-type density theorem, and a Turán-type local density theorem—for the Beurling context, too. These, together with Turán’s power sum theory, all play some indispensable role in deriving the main results of the paper.

Enriched curves have been studied over algebraically closed fields by Mainò [10] and recently over general base schemes in [1]. In this paper, we study enriched curves from a logarithmic viewpoint: we give a succinct definition of the stack of rich log curves as an open substack of the stack of log curves and define an enriched curve to be a curve with a minimal rich log structure on it. This logarithmic viewpoint turns out to be a natural language for enriched structures, leading to a simple modular compactification. This modular compactification is a smooth log blowup of the stack of log curves, answering affirmatively two questions from [1]. We also generalize the concept of rich curves to r-rich curves and show similar results. We include a chapter phrasing some of the key definitions solely in the language of real tropical geometry.

Let Z be an affine algebraic variety and X be a smooth flexible variety. We develop some criteria under which Z admits a closed embedding into X. In particular, we show that if $dim\mathit{X}\ge max(2dim\mathit{Z}+1,dim\mathit{T}\mathit{Z})$ and X is isomorphic (as an algebraic variety) to a special linear group or to a symplectic group, then Z admits a closed embedding into X.

In this paper, we study the cohomology of vector bundles on ${\mathbb{P}}^{\mathit{n}}$ defined as kernels or cokernels of general maps ${\mathit{V}}_1\to {\mathit{V}}_2$, where ${\mathit{V}}_{\mathit{i}}$ are direct sums of line bundles or certain exceptional bundles. This problem is deeply connected with fundamental results on the ranks of multiplication maps between spaces of homogeneous polynomials. Our main result establishes an asymptotic cohomology vanishing theorem. We also characterize the stability of general Steiner bundles on ${\mathbb{P}}^{\mathit{n}}$ and give a criterion for such bundles to be ample.

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $\mathit{n}\le \mathit{N}$ and integer parts of polynomials evaluated at n. The error terms in our formulas are of various strengths depending on the Diophantine properties of the leading coefficients of these polynomials.

This article provides a deeper study of the Riesz transform commutators associated with the Neumann Laplacian operator ${\mathrm{\Delta }}_{\mathit{N}}$ on ${\mathbb{R}}^{\mathit{n}}$. Along the line of singular value estimates for Riesz transform commutators established by Janson and Wolff and Rochberg and Semmes, we establish a full range of Schatten-p class characterization for these commutators.

We show that the splinter property ascends along regular residue field extensions and along arbitrary regular maps in equal characteristic. We also study the splinter property of non-Noetherian rings, especially those related to ultrapowers, to the extent necessary for our main results.

We prove a generalized version of Evans and Griffith’s improved new intersection theorem: Let I be an ideal in a local ring R. If a finite free R-complex, concentrated in nonnegative degrees, has I-torsion homology in positive degrees, and the homology in degree 0 has an I-torsion minimal generator, then the length of the complex is at least $dim\mathit{R}-dim\mathit{R}/\mathit{I}$. This improves the bound $\mathrm{ht}\mathit{I}$ obtained by Avramov, Iyengar, and Neeman in 2018.

Given a metric space X, an analyst’s traveling salesman theorem for X gives a quantitative relationship between the length of a shortest curve containing any subset $\mathit{E}\subseteq \mathit{X}$ and a multi-scale sum measuring the “flatness” of E. The first such theorem was proven by Jones for $\mathit{X}={\mathbb{R}}^2$ and extended to $\mathit{X}={\mathbb{R}}^{\mathit{n}}$ by Okikiolu, whereas an analogous theorem was proven for Hilbert space, $\mathit{X}=\mathit{H}$, by Schul. Bishop has since shown that if one considers Jordan arcs, then the quantitative relationship given by Jones’ and Okikioulu’s results can be sharpened. This paper gives a full proof of Schul’s original necessary half of the traveling salesman theorem in Hilbert space and provides a sharpening of the theorem’s quantitative relationship when restricted to Jordan arcs analogous to Bishop’s aforementioned sharpening in ${\mathbb{R}}^{\mathit{n}}$.

A smooth projective variety Y is said to satisfy Bott vanishing if ${\mathrm{\Omega }}_{\mathit{Y}}^{\mathit{j}}\otimes \mathit{L}$ has no higher cohomology for every j and every ample line bundle L. Few examples are known to satisfy this property. Among them are toric varieties, as well as the quintic del Pezzo surface, recently shown by Totaro. Here we present a new class of varieties satisfying Bott vanishing, namely stable GIT quotients of ${({\mathbb{P}}^1)}^{\mathit{n}}$ by the action of ${\mathit{PGL}}_2$ over an algebraically closed field of characteristic zero. For this, we use the work done by Halpern-Leistner on the derived category of a GIT quotient and his version of the quantization theorem. We also see that, using similar techniques, we can recover Bott vanishing for the toric case.

In this article we obtain large deviation estimates for zeros of random holomorphic sections on punctured Riemann surfaces. These estimates are then employed to yield estimates for the respective hole probabilities. A particular case of relevance that is covered by our setting is that of cusp forms on arithmetic surfaces. Most of the results we obtain also allow for reasonably general probability distributions on holomorphic sections, which underlines the universal character of these estimates. Finally, we also extend our results to the case of certain higher dimensional complete Hermitian manifolds, which are not necessarily assumed to be compact.

We show that a conjecture of Kotschick about one-forms without zeros on compact Kähler manifolds follows in the case of simple Albanese torus from a conjecture of Bobadilla and Kollár about homologically trivial fibrations. As an application, we prove Kotschick’s conjecture for compact Kähler manifolds X with ${\mathit{b}}_1(\mathit{X})\ge 2dim\mathit{X}-2$, whose Albanese torus is simple.

A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the algebraic structure. In this paper we introduce a more general perspective on total symmetry and formulate a notion of “irreducibility” for totally symmetric sets in the general linear group. We classify irreducible totally symmetric sets, as well as those of maximal cardinality.

We study the Mori Dream Space (MDS) property for blowups of weighted projective planes at a general point and, more generally, blowups of toric surfaces defined by a rational plane triangle. The birational geometry of these varieties is largely governed by the existence of a negative curve in them, different from the exceptional curve of the blowup.

We consider a parameter space of all rational triangles, and within this space, we study how the negative curves and the MDS property vary. The main theme is that almost all known results about the MDS property can be explained by the variation of negative curves in the parameter space. We attempt to classify all negative curves. In particular, we construct two new infinite families of them, one of which is the first known infinite family of special negative curves.

For a compact oriented smooth n-manifold M and a codimension-1 homology class $\mathit{\varphi }\in H_{\mathit{n}-1}(\mathit{M},\partial \mathit{M})$, we investigate a simplicial complex ${\mathcal{S}}^{\mathrm{†}}(\mathit{M},\mathit{\varphi })$ relating the properly embedded hypersurfaces in M representing ϕ. Its definition is akin to that of other classical complexes, such as the curve complex of a surface or the Kakimizu complex of a knot, with the difference that hypersurfaces are not taken up to isotopy.

We prove that ${\mathcal{S}}^{\mathrm{†}}(\mathit{M},\mathit{\varphi })$ is connected and simply connected in every dimension n. We also show the connectedness of a similar complex ${\mathcal{T}}^{\mathrm{†}}(\mathit{M},\mathit{\varphi })$ adapted to the three-dimensional case, where only Thurston norm-realizing surfaces are considered. The connectedness results are transported to the complexes $\mathcal{S}(\mathit{M},\mathit{\varphi })$ and $\mathcal{T}(\mathit{M},\mathit{\varphi })$ where hypersurfaces are taken up to isotopy, and for $\mathit{n}=2$, the simple connectedness result carries over as well. We also briefly discuss extensions to a context studied by Turaev, where regular graphs in 2-complexes are used to represent one-dimensional cohomology classes.

We finish with two applications: we give an alternative proof of the fact that all Seifert surfaces for a fixed knot in a rational homology sphere are tube-equivalent, and we use the connectedness of ${\mathcal{T}}^{\mathrm{†}}(\mathit{M},\mathit{\varphi })$ to define a new ${\ell }^2$-invariant of two-dimensional homology classes in irreducible and boundary-irreducible oriented compact connected 3-manifolds with empty or toroidal boundary.

Let $(\mathit{C},{\mathit{p}}_1,\dots ,{\mathit{p}}_{\mathit{n}})$ be a fixed general pointed curve, and let $(\mathit{X},{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}})$ be a smooth hypersurface of degree e and dimension r with n general points. We consider the problem of enumerating maps $\mathit{f}:\mathit{C}\to \mathit{X}$ of degree d (as measured in the ambient projective space) such that $\mathit{f}({\mathit{p}}_{\mathit{i}})={\mathit{x}}_{\mathit{i}}$. When e is small compared to r and d is large compared to g, e, and r, these numbers have been computed first by passing to a virtual count in Gromov–Witten theory obtained by Buch–Pandharipande and then by showing (in the work of the author with Pandharipande) that the virtual counts are enumerative via an analysis of boundary contributions in the moduli space of stable maps. In this note, we give a simpler computation via projective geometry.

We define a Rasmussen s-invariant over the coefficient ring $\mathbb{Z}$ and show how it is related to the s-invariants defined over a field. A lower bound for the slice genus of a knot arising from it is obtained, and we give examples of knots for which this lower bound is better than all lower bounds coming from the s-invariants over fields. We also compare it to the Lipshitz–Sarkar refinement related to the first Steenrod square.