Authors:ADAM SIMON LEVINE; TYE LIDMAN Abstract: We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to $S^{2}$ but do not admit a spine (that is, a piecewise linear embedding of $S^{2}$ that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer $d$ invariants. PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2019.11 Issue No:Vol. 7 (2019)

Authors:IAN LE Abstract: Let $S$ be a surface, $G$ a simply connected classical group, and $G^{\prime }$ the associated adjoint form of the group. We show that the moduli spaces of framed local systems ${\mathcal{X}}_{G^{\prime },S}$ and ${\mathcal{A}}_{G,S}$ , which were constructed by Fock and Goncharov [‘Moduli spaces of local systems and higher Teichmuller theory’, Publ. Math. Inst. Hautes Études Sci.103 (2006), 1–212], have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in that paper, and also allows one to quantize higher Teichmuller space, which was previously only possible when $G$ was of type $A$ . PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2019.5 Issue No:Vol. 7 (2019)

Authors:TIMOTHY C. BURNESS; DONNA M. TESTERMAN Abstract: Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic $p>0$ and let $X=\text{PSL}_{2}(p)$ be a subgroup of $G$ containing a regular unipotent element $x$ of $G$ . By a theorem of Testerman, $x$ is contained in a connected subgroup of $G$ of type $A_{1}$ . In this paper we prove that with two exceptions, $X$ itself is contained in such a subgroup (the exceptions arise when $(G,p)=(E_{6},13)$ or PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2019.12 Issue No:Vol. 7 (2019)

Authors:LÁSZLÓ ERDŐS; TORBEN KRÜGER, DOMINIK SCHRÖDER Abstract: We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion. PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2019.2 Issue No:Vol. 7 (2019)

Authors:LINQUAN MA; THOMAS POLSTRA, KARL SCHWEDE, KEVIN TUCKER Abstract: We study $F$ -signature under proper birational morphisms $\unicode[STIX]{x1D70B}:Y\rightarrow X$ , showing that $F$ -signature strictly increases for small morphisms or if $K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$ . In certain cases, we can even show that the $F$ -signature of $Y$ is at least twice as that of $X$ . We also provide examples of $F$ -signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses. PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2019.6 Issue No:Vol. 7 (2019)

Authors:CARLOS AMÉNDOLA; PETER FRIZ, BERND STURMFELS Abstract: The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures. PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2019.3 Issue No:Vol. 7 (2019)

Authors:CHRISTIAN REIHER; MATHIAS SCHACHT Abstract: We study forcing pairs for quasirandom graphs. Chung, Graham, and Wilson initiated the study of families ${\mathcal{F}}$ of graphs with the property that if a large graph $G$ has approximately homomorphism density $p^{e(F)}$ for some fixed $p\in (0,1]$ for every $F\in {\mathcal{F}}$ , then $G$ is quasirandom with density $p$ . Such families ${\mathcal{F}}$ are said to be forcing. Several forcing families were found over the last three decades and characterizing all bipartite graphs $F$ such that $(K_{2},F)$ is a forcing pair is a well-known open problem in the area of quasirandom graphs, which is closely related to Sidorenko’s conjecture. In fact, most of the known forcing families involve bipartite graphs only.We consider forcing pairs containing the triangle PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2019.7 Issue No:Vol. 7 (2019)

Authors:A. ASOK; J. FASEL, M. J. HOPKINS Abstract: Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$ , it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general. PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2018.16 Issue No:Vol. 7 (2019)

Authors:JIM BRYAN; MARTIJN KOOL Abstract: We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.)24 (2) (2018), 1527–1548, arXiv:math/1603.05271], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka. PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2019.1 Issue No:Vol. 7 (2019)

Authors:STEVEN V SAM; ANDREW SNOWDEN Abstract: Twisted commutative algebras (tca’s) have played an important role in the nascent field of representation stability. Let $A_{d}$ be the tca freely generated by $d$ indeterminates of degree 1. In a previous paper, we determined the structure of the category of $A_{1}$ -modules (which is equivalent to the category of $\mathbf{FI}$ -modules). In this paper, we establish analogous results for the category of $A_{d}$ -modules, for any $d$ . Modules over $A_{d}$ are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and we hope the results of this paper will eventually lead to improvements on those works. Our results also have implications in asymptotic commutative algebra. PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2018.27 Issue No:Vol. 7 (2019)

Authors:ANIRBAN BASAK; ELLIOT PAQUETTE, OFER ZEITOUNI Abstract: We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_{N}$ be a deterministic $N\times N$ matrix, and let $G_{N}$ be a complex Ginibre matrix. We consider the matrix ${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$ , where $\unicode[STIX]{x1D6FE}>1/2$ . With $L_{N}$ the empirical measure of eigenvalues of ${\mathcal{M}}_{N}$ , we provide a general deterministic equivalence theorem that ties $L_{N}$ to the singular values of $z-M_{N}$ , with $z\in \mathbb{C}$ . We then compute the limit of $L_{N}$ when PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2018.29 Issue No:Vol. 7 (2019)

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++++&rft.title=Forum+of+Mathematics,+Sigma&rft.issn=2050-5094&rft.date=2019&rft.volume=7&rft.aulast=BOXER&rft.aufirst=GEORGE&rft.au=GEORGE+BOXER&rft.au=FRANK+CALEGARI,+MATTHEW+EMERTON,+BRANDON+LEVIN,+KEERTHI+MADAPUSI+PERA,+STEFAN+PATRIKIS&rft_id=info:doi/10.1017/fms.2018.24">COMPATIBLE SYSTEMS OF GALOIS REPRESENTATIONS ASSOCIATED TO THE EXCEPTIONAL GROUP $E_{6}$

Authors:GEORGE BOXER; FRANK CALEGARI, MATTHEW EMERTON, BRANDON LEVIN, KEERTHI MADAPUSI PERA, STEFAN PATRIKIS Abstract: We construct, over any CM field, compatible systems of $l$ -adic Galois representations that appear in the cohomology of algebraic varieties and have (for all $l$ ) algebraic monodromy groups equal to the exceptional group of type $E_{6}$ . PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2018.24 Issue No:Vol. 7 (2019)

Authors:SŁAWOMIR SOLECKI Abstract: First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by the existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg–Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers’ Ramsey theorem. PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2018.28 Issue No:Vol. 7 (2019)

Authors:MATTHEW JENSSEN; FELIX JOOS, WILL PERKINS Abstract: We prove a lower bound on the entropy of sphere packings of $\mathbb{R}^{d}$ of density $\unicode[STIX]{x1D6E9}(d\cdot 2^{-d})$ . The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the $\unicode[STIX]{x1D6FA}(d\cdot 2^{-d})$ lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least $(1+o_{d}(1))\log (2/\sqrt{3})d\cdot 2^{-d}$ when the ratio of the fugacity parameter to the volume covered by a single sphere is at least $3^{-d/2}$ . Such a bound on the sphere packing density was first achieved by Rogers, with subsequent improvements to the leading constant by Davenport and Rogers, Ball, Vance, and Venkatesh. PubDate: 2019-01-01T00:00:00.000Z DOI: 10.1017/fms.2018.25 Issue No:Vol. 7 (2019)