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Publisher: Cambridge University Press   (Total: 387 journals)

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 Forum of Mathematics, SigmaNumber of Followers: 1     Open Access journal ISSN (Online) 2050-5094 Published by Cambridge University Press  [387 journals]
• GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

• Authors: SPENCER BACKMAN; MATTHEW BAKER, CHI HO YUEN
Abstract: Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$ ). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.40
Issue No: Vol. 7 (2019)

• HIGH ORDER PARACONTROLLED CALCULUS

• Authors: ISMAËL BAILLEUL; FRÉDÉRIC BERNICOT
Abstract: We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations $\mathsf{E}$ and  $\mathsf{F}$ . We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701},\end{eqnarray}$$ on a 3-dimensional closed manifold, and the generalized KPZ equation $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701}+g(u)(\unicode[STIX]{x2202}u)^{2},\end{eqnarray}$$ driven by a $(1+1)$ -dimensional space/time white noise.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.44
Issue No: Vol. 7 (2019)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$A_{\infty+}(\mathbb{R})$ ++++++++++++ ++++++++ +++++DENSITY+AND+TRANSPORT+EQUATION+IN+ ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$\text{BMO}(\mathbb{R})$ ++++++++++++ ++++++++ ++++&rft.title=Forum+of+Mathematics,+Sigma&rft.issn=2050-5094&rft.date=2019&rft.volume=7&rft.aulast=JIANG&rft.aufirst=RENJIN&rft.au=RENJIN+JIANG&rft.au=KANGWEI+LI,+JIE+XIAO&rft_id=info:doi/10.1017/fms.2019.41">FLOW WITH $A_{\infty }(\mathbb{R})$ DENSITY AND TRANSPORT EQUATION IN
$\text{BMO}(\mathbb{R})$

• Authors: RENJIN JIANG; KANGWEI LI, JIE XIAO
Abstract: We show that, if $b\in L^{1}(0,T;L_{\operatorname{loc}}^{1}(\mathbb{R}))$ has a spatial derivative in the John–Nirenberg space $\operatorname{BMO}(\mathbb{R})$ , then it generates a unique flow $\unicode[STIX]{x1D719}(t,\cdot )$ which has an $A_{\infty }(\mathbb{R})$ density for each time $t\in [0,T]$ . Our condition on the map $b$ is not only optimal but also produces a sharp quantitative estimate for the density. As a killer application we achieve the well-posedness for a Cauchy problem of the transport equation in $\operatorname{BMO}(\mathbb{R})$ .
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.41
Issue No: Vol. 7 (2019)

• COVER TIME FOR THE FROG MODEL ON TREES

• Authors: CHRISTOPHER HOFFMAN; TOBIAS JOHNSON, MATTHEW JUNGE
Abstract: The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $\unicode[STIX]{x1D707}$ on the full $d$ -ary tree of height $n$ . If $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}(d^{2})$ , all of the vertices are visited in time $\unicode[STIX]{x1D6E9}(n\log n)$ with high probability. Conversely, if $\unicode[STIX]{x1D707}=O(d)$ the cover time is $\exp (\unicode[STIX]{x1D6E9}(\sqrt{n}))$ with high probability.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.37
Issue No: Vol. 7 (2019)

• BOREL DENSITY FOR APPROXIMATE LATTICES

• Authors: MICHAEL BJÖRKLUND; TOBIAS HARTNICK, THIERRY STULEMEIJER
Abstract: We extend classical density theorems of Borel and Dani–Shalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate lattices. Our proofs are based on the observation that Zariski closures of approximate subgroups are close to algebraic subgroups. Our main tools are stationary joinings between the hull dynamical systems of discrete approximate subgroups and their Zariski closures.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.39
Issue No: Vol. 7 (2019)

• ON TORUS ACTIONS OF HIGHER COMPLEXITY

• Authors: JÜRGEN HAUSEN; CHRISTOFF HISCHE, MILENA WROBEL
Abstract: We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of ‘general arrangement varieties’ and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.35
Issue No: Vol. 7 (2019)

• YANG–BAXTER FIELD FOR SPIN HALL–LITTLEWOOD SYMMETRIC FUNCTIONS

• Authors: ALEXEY BUFETOV; LEONID PETROV
Abstract: Employing bijectivization of summation identities, we introduce local stochastic moves based on the Yang–Baxter equation for $U_{q}(\widehat{\mathfrak{sl}_{2}})$ . Combining these moves leads to a new object which we call the spin Hall–Littlewood Yang–Baxter field—a probability distribution on two-dimensional arrays of particle configurations on the discrete line. We identify joint distributions along down-right paths in the Yang–Baxter field with spin Hall–Littlewood processes, a generalization of Schur processes. We consider various degenerations of the Yang–Baxter field leading to new dynamic versions of the stochastic six-vertex model and of the Asymmetric Simple Exclusion Process.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.36
Issue No: Vol. 7 (2019)

• CUTTING A PART FROM MANY MEASURES

• Authors: PAVLE V. M. BLAGOJEVIĆ; NEVENA PALIĆ, PABLO SOBERÓN, GÜNTER M. ZIEGLER
Abstract: Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^{d}$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint.We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ ...
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.33
Issue No: Vol. 7 (2019)

• NORMAL FUNCTIONS FOR ALGEBRAICALLY TRIVIAL CYCLES ARE ALGEBRAIC FOR
ARITHMETIC REASONS

• Authors: JEFFREY D. ACHTER; SEBASTIAN CASALAINA-MARTIN, CHARLES VIAL
Abstract: For families of smooth complex projective varieties, we show that normal functions arising from algebraically trivial cycle classes are algebraic and defined over the field of definition of the family. In particular, the zero loci of those functions are algebraic and defined over such a field of definition. This proves a conjecture of Charles.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.34
Issue No: Vol. 7 (2019)

• CALDERÓN’S INVERSE PROBLEM WITH A FINITE NUMBER OF MEASUREMENTS

• Authors: GIOVANNI S. ALBERTI; MATTEO SANTACESARIA
Abstract: We prove that an $L^{\infty }$ potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace ${\mathcal{W}}$ . As a corollary, we obtain a similar result for Calderón’s inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces ${\mathcal{W}}$ , including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of $\dim {\mathcal{W}}$ .
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.31
Issue No: Vol. 7 (2019)

• EISENSTEIN–KRONECKER SERIES VIA THE POINCARÉ BUNDLE

• Authors: JOHANNES SPRANG
Abstract: A classical construction of Katz gives a purely algebraic construction of Eisenstein–Kronecker series using the Gauß–Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$ -adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein–Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz’ two-variable $p$ -adic Eisenstein measure through $p$ -adic theta functions of the Poincaré bundle.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.29
Issue No: Vol. 7 (2019)

• VALUE PATTERNS OF MULTIPLICATIVE FUNCTIONS AND RELATED SEQUENCES

• Authors: TERENCE TAO; JONI TERÄVÄINEN
Abstract: We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is ‘approximately multiplicative’ and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern $n+1\in A$ , $n+2\in A$ , $n+3\in A$ is positive as long as $A$ has density greater than $\frac{1}{3}$ . Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of $A$ having density exactly $\frac{1}{3}$ , below which one would need nontrivial information on the local distribution of $A$ in Bohr sets to proceed. We apply our results first to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors $P^{+}(n)$ ,
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.28
Issue No: Vol. 7 (2019)

• LOGARITHMIC DE RHAM COMPARISON FOR OPEN RIGID SPACES

• Authors: SHIZHANG LI; XUANYU PAN
Abstract: In this note, we prove the logarithmic $p$ -adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\unicode[STIX]{x1D70B},1)$ (in a certain sense) with respect to $\mathbb{F}_{p}$ -local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.27
Issue No: Vol. 7 (2019)

• THE EXPLICIT MORDELL CONJECTURE FOR FAMILIES OF CURVES

• Authors: SARA CHECCOLI; FRANCESCO VENEZIANO, EVELINA VIADA
Abstract: In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin–Demjanenko and the analysis of our explicit examples is carried to conclusion.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.20
Issue No: Vol. 7 (2019)

• THE CHERN–SCHWARTZ–MACPHERSON CLASS OF AN EMBEDDABLE SCHEME

• Authors: PAOLO ALUFFI
Abstract: The Chern–Schwartz–MacPherson class of a hypersurface in a nonsingular variety may be computed directly from the Segre class of the Jacobian subscheme of the hypersurface; this has been known for a number of years. We generalize this fact to arbitrary embeddable schemes: for every subscheme $X$ of a nonsingular variety  $V$ , we define an associated subscheme $\mathscr{Y}$ of a projective bundle $\mathscr{V}$ over $V$ and provide an explicit formula for the Chern–Schwartz–MacPherson class of $X$ in terms of the Segre class of  $\mathscr{Y}$ in  $\mathscr{V}$ . If $X$ is a local complete intersection, a version of the result yields a direct expression for the Milnor class of $X$ .For
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.25
Issue No: Vol. 7 (2019)

• FREE FINITE GROUP ACTIONS ON RATIONAL HOMOLOGY 3-SPHERES

• Authors: ALEJANDRO ADEM; IAN HAMBLETON
Abstract: We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.24
Issue No: Vol. 7 (2019)

• UNIFORM SPANNING FORESTS OF PLANAR GRAPHS

• Authors: TOM HUTCHCROFT; ASAF NACHMIAS
Abstract: We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that the same exponents hold universally over this entire class of graphs provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geometry.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.14
Issue No: Vol. 7 (2019)

• PARALLEL WEIGHT 2 POINTS ON HILBERT MODULAR EIGENVARIETIES AND THE PARITY
CONJECTURE

• Authors: CHRISTIAN JOHANSSON; JAMES NEWTON
Abstract: Let $F$ be a totally real field and let $p$ be an odd prime which is totally split in $F$ . We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over $F$ with weight varying only at a single place $v$ above $p$ . For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if $[F:\mathbb{Q}]$ is odd), by reducing to the case of parallel weight $2$ . As another consequence of our results on partial eigenvarieties, we show, still under the assumption that $p$ is totally split in $F$ , that the ‘full’ (dimension
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.23
Issue No: Vol. 7 (2019)

• THE POLYTABLOID BASIS EXPANDS POSITIVELY INTO THE WEB BASIS

Abstract: We show that the transition matrix from the polytabloid basis to the web basis of the irreducible $\mathfrak{S}_{2n}$ -representation of shape $(n,n)$ has nonnegative integer entries. This proves a conjecture of Russell and Tymoczko [Int. Math. Res. Not., 2019(5) (2019), 1479–1502].
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.22
Issue No: Vol. 7 (2019)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$L$ ++++++++++++ ++++++++ ++++-FUNCTIONS&rft.title=Forum+of+Mathematics,+Sigma&rft.issn=2050-5094&rft.date=2019&rft.volume=7&rft.aulast=SHANKAR&rft.aufirst=ARUL&rft.au=ARUL+SHANKAR&rft.au=ANDERS+SÖDERGREN,+NICOLAS+TEMPLIER&rft_id=info:doi/10.1017/fms.2019.18">SATO–TATE EQUIDISTRIBUTION OF CERTAIN FAMILIES OF ARTIN $L$ -FUNCTIONS

• Authors: ARUL SHANKAR; ANDERS SÖDERGREN, NICOLAS TEMPLIER
Abstract: We study various families of Artin $L$ -functions attached to geometric parametrizations of number fields. In each case we find the Sato–Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.18
Issue No: Vol. 7 (2019)

• ELASTIC GRAPHS

• Authors: DYLAN P. THURSTON
Abstract: An elastic graph is a graph with an elasticity associated to each edge. It may be viewed as a network made out of ideal rubber bands. If the rubber bands are stretched on a target space there is an elastic energy. We characterize when a homotopy class of maps from one elastic graph to another is loosening, that is, decreases this elastic energy for all possible targets. This fits into a more general framework of energies for maps between graphs.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.4
Issue No: Vol. 7 (2019)

• CODIMENSION TWO CYCLES IN IWASAWA THEORY AND ELLIPTIC CURVES WITH
SUPERSINGULAR REDUCTION

• Authors: ANTONIO LEI; BHARATHWAJ PALVANNAN
Abstract: A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s $2$ -variable $p$ -adic $L$ -functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a $2$ -variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field $K$ (where an odd prime $p$ splits) of an elliptic curve $E$ , defined over  $\mathbb{Q}$ , with good supersingular reduction at $p$ . On the analytic side, we consider eight pairs of $2$ -variable ...
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.17
Issue No: Vol. 7 (2019)

• SINGULARITY OF RANDOM SYMMETRIC MATRICES—A COMBINATORIAL APPROACH TO
IMPROVED BOUNDS

• Authors: ASAF FERBER; VISHESH JAIN
Abstract: Let $M_{n}$ denote a random symmetric $n\times n$ matrix whose upper-diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_{n}$ is singular with probability at most $(2+o(1))^{-n}$ . On the other hand, the best known upper bound on the singularity probability of $M_{n}$ , due to Vershynin (2011), is $2^{-n^{c}}$ , for some unspecified small constant $c>0$ . This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.21
Issue No: Vol. 7 (2019)

• GENERIC UNLABELED GLOBAL RIGIDITY

• Authors: STEVEN J. GORTLER; LOUIS THERAN, DYLAN P. THURSTON
Abstract: Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^{d}$ for some $n$ and some $d\geqslant 2$ . Each pair of points has a Euclidean distance in the configuration. Given some graph $G$ on $n$ vertices, we measure the point-pair distances corresponding to the edges of $G$ . In this paper, we study the question of when a generic $\mathbf{p}$ in $d$ dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair distances together with knowledge of $d$ and
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.16
Issue No: Vol. 7 (2019)

• DEFORMATION CONDITIONS FOR PSEUDOREPRESENTATIONS

• Authors: PRESTON WAKE; CARL WANG-ERICKSON
Abstract: Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna’s theory to pseudorepresentations.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.19
Issue No: Vol. 7 (2019)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$U(n,1)$ ++++++++++++ ++++++++ ++++&rft.title=Forum+of+Mathematics,+Sigma&rft.issn=2050-5094&rft.date=2019&rft.volume=7&rft.aulast=MARSHALL&rft.aufirst=SIMON&rft.au=SIMON+MARSHALL&rft.au=SUG+WOO+SHIN&rft_id=info:doi/10.1017/fms.2019.13">ENDOSCOPY AND COHOMOLOGY IN A TOWER OF CONGRUENCE MANIFOLDS FOR $U(n,1)$

• Authors: SIMON MARSHALL; SUG WOO SHIN
Abstract: By assuming the endoscopic classification of automorphic representations on inner forms of unitary groups, which is currently work in progress by Kaletha, Minguez, Shin, and White, we bound the growth of cohomology in congruence towers of locally symmetric spaces associated to $U(n,1)$ . In the case of lattices arising from Hermitian forms, we expect that the growth exponents we obtain are sharp in all degrees.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.13
Issue No: Vol. 7 (2019)

• A FILTRATION ON EQUIVARIANT BOREL–MOORE HOMOLOGY

• Authors: ARAM BINGHAM; MAHIR BILEN CAN, YILDIRAY OZAN
Abstract: Let $G/H$ be a homogeneous variety and let $X$ be a $G$ -equivariant embedding of $G/H$ such that the number of $G$ -orbits in $X$ is finite. We show that the equivariant Borel–Moore homology of $X$ has a filtration with associated graded module the direct sum of the equivariant Borel–Moore homologies of the $G$ -orbits. If $T$ is a maximal torus of $G$ such that each $G$ -orbit has a
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.15
Issue No: Vol. 7 (2019)

• A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

• Authors: DANIEL M. KANE; ROBERT C. RHOADES
Abstract: Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of $n$ without $k$ consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without $k$ consecutive parts. Andrews showed that when $k=2$ , the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For $k>2$ , the asymptotic properties of the generating functions have proved more difficult to obtain. Using $q$ -series identities and the $k=2$ case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case $k=3$ was given by Zagier.This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.8
Issue No: Vol. 7 (2019)

• GOLDFELD’S CONJECTURE AND CONGRUENCES BETWEEN HEEGNER POINTS

• Authors: DANIEL KRIZ; CHAO LI
Abstract: Given an elliptic curve $E$ over $\mathbb{Q}$ , a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever $E$ has a rational 3-isogeny. We also prove the analogous result for the sextic twists of $j$ -invariant 0 curves. For a more general elliptic curve $E$ , we show that the number of quadratic twists of $E$ up to twisting discriminant $X$ of analytic rank 0 (respectively 1) is $\gg X/\log ^{5/6}X$ , improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between $p$ -adic logarithms of Heegner points and apply it in the special cases $p=3$ and
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.9
Issue No: Vol. 7 (2019)

• ENLARGED MIXED SHIMURA VARIETIES, BI-ALGEBRAIC SYSTEM AND SOME AX TYPE
TRANSCENDENTAL RESULTS

• Authors: ZIYANG GAO
Abstract: We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We study their bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its relation with the logarithmic Ax theorem and the Ax-Lindemann theorem which we shall prove. All these bi-algebraic and transcendental results extend their counterparts for mixed Shimura varieties. In the end we briefly discuss the André–Oort and Zilber–Pink type problems for enlarged mixed Shimura varieties.
PubDate: 2019-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.10
Issue No: Vol. 7 (2019)

• SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS

• 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)

• CLUSTER STRUCTURES ON HIGHER TEICHMULLER SPACES FOR CLASSICAL GROUPS

• 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)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$A_{1}$ ++++++++++++ ++++++++ ++++-TYPE+SUBGROUPS+CONTAINING+REGULAR+UNIPOTENT+ELEMENTS&rft.title=Forum+of+Mathematics,+Sigma&rft.issn=2050-5094&rft.date=2019&rft.volume=7&rft.aulast=BURNESS&rft.aufirst=TIMOTHY&rft.au=TIMOTHY+C.+BURNESS&rft.au=DONNA+M.+TESTERMAN&rft_id=info:doi/10.1017/fms.2019.12">$A_{1}$ -TYPE SUBGROUPS CONTAINING REGULAR UNIPOTENT ELEMENTS

• 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)

• RANDOM MATRICES WITH SLOW CORRELATION DECAY

• 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)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$F$ ++++++++++++ ++++++++ ++++-SIGNATURE+UNDER+BIRATIONAL+MORPHISMS&rft.title=Forum+of+Mathematics,+Sigma&rft.issn=2050-5094&rft.date=2019&rft.volume=7&rft.aulast=MA&rft.aufirst=LINQUAN&rft.au=LINQUAN+MA&rft.au=THOMAS+POLSTRA,+KARL+SCHWEDE,+KEVIN+TUCKER&rft_id=info:doi/10.1017/fms.2019.6">$F$ -SIGNATURE UNDER BIRATIONAL MORPHISMS

• 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)

• VARIETIES OF SIGNATURE TENSORS

• 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)

• FORCING QUASIRANDOMNESS WITH TRIANGLES

• 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)

• OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

• 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)

• DONALDSON–THOMAS INVARIANTS OF LOCAL ELLIPTIC SURFACES VIA THE
TOPOLOGICAL VERTEX

• 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)

• GL-EQUIVARIANT MODULES OVER POLYNOMIAL RINGS IN INFINITELY MANY VARIABLES.
II

• 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)

• REGULARIZATION OF NON-NORMAL MATRICES BY GAUSSIAN NOISE—THE BANDED
TOEPLITZ AND TWISTED TOEPLITZ CASES

• 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)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$E_{6}$ ++++++++++++ ++++++++ ++++&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)

• MONOID ACTIONS AND ULTRAFILTER METHODS IN RAMSEY THEORY

• 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)

• ON THE HARD SPHERE MODEL AND SPHERE PACKINGS IN HIGH DIMENSIONS

• 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)

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