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European Mathematical Society Publishing House
2024-03-29 12:05:50
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=19&iss=12&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
19
2017
12
On a long range segregation model
Luis
Caffarelli
University of Texas at Austin, USA
Stefania
Patrizi
University of Texas at Austin, USA
Veronica
Quitalo
Purdue University, West Lafayette, USA
Segregation of populations, free boundary problems, long-range interactions
In this work we study the properties of segregation processes modeled by a family of equations $$L(u_i) (x) = u_i(x)\: F_i (u_1, \ldots, u_K)(x)\qquad i=1,\ldots, K$$ where $F_i (u_1, \ldots, u_K)(x)$ is a non-local factor that takes into consideration the values of the functions $u_j$ in a full neighborhood of $x.$ We consider as a model problem $$\Delta u_i^\epsilon (x) = \frac1{\epsilon^2} u_i^\epsilon (x)\sum_{i\neq j} H(u_j^\epsilon)(x)$$ where $\epsilon$ is a small parameter and $H(u_j^\epsilon)(x)$ is for instance $$H(u_j^\epsilon)(x)= \int_{\mathcal{B}_1 (x)} u_j^\epsilon (y)\, dy \:\: \mathrm {or} \:\: H(u_j^\epsilon)(x)= \mathrm {sup}_{y\in \mathcal{B}_1(x)} u_j^\epsilon (y).$$ Here $\mathcal{B}_1(x)$ is the unit ball centered at $x$ with respect to a smooth, uniformly convex norm $\rho$ of $\mathbb R^n$. Heuristically, this will force the populations to stay at $\rho$-distance 1, one from each other, as $\epsilon\to0$.
Partial differential equations
Integral equations
3575
3628
10.4171/JEMS/747
http://www.ems-ph.org/doi/10.4171/JEMS/747
11
20
2017
Triple Massey products and absolute Galois groups
Ido
Efrat
Ben Gurion University of the Negev, Beer-Sheva, Israel
Eliyahu
Matzri
Ben Gurion University of the Negev, Beer-Sheva, Israel
Triple Massey products, absolute Galois groups, Galois cohomology
Let $p$ be a prime number, $F$ a field containing a root of unity of order $p$, and $G_F$ the absolute Galois group. Extending results of Hopkins, Wickelgren, Mináč and Tân, we prove that the triple Massey product $H^1(G_F)^3\to H^2(G_F)$ contains 0 whenever it is non-empty. This gives a new restriction on the possible profinite group structure of $G_F$.
Field theory and polynomials
Associative rings and algebras
3629
3640
10.4171/JEMS/748
http://www.ems-ph.org/doi/10.4171/JEMS/748
11
20
2017
Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production
Youshan
Tao
Dong Hua University, Shanghai, China
Michael
Winkler
University of Paderborn, Germany
Chemotaxis, infinite-time blow-up, critical mass
We study the Neumann initial-boundary problem for the chemotaxis system \[ \left\{ \begin{array}{ll} u_t= \Delta u - \nabla \cdot (u\nabla v), & x\in \Omega, \, t>0, \\ 0=\Delta v - \mu(t)+w, & x\in \Omega, \, t>0, \\ \tau w_t + \delta w = u, & x\in \Omega, \, t>0, \end{array} \right. \qquad \qquad (\star) \] in the unit disk $\Omega:=B_1(0)\subset R^2$, where $\delta\ge 0$ and $\tau>0$ are given parameters and $\mu(t):=\frac{1}{|\Omega|} \int_\Omega w(x,t)dx$, $t>0$. It is shown that this problem exhibits a novel type of critical mass phenomenon with regard to the formation of singularities, which drastically differs from the well-known threshold property of the classical Keller-Segel system, as obtained upon formally taking $\tau\to 0$, in that it refers to blow-up in infinite time rather than in finite time. Specifically, it is first proved that for any sufficiently regular nonnegative initial data $u_0$ and $w_0$, ($\star$) possesses a unique global classical solution. In particular, this shows that in sharp contrast to classical Keller-Segel-type systems reflecting immediate signal secretion by the cells themselves, the indirect mechanism of signal production in ($\star$) entirely rules out any occurrence of blow-up in finite time. However, within the framework of radially symmetric solutions it is next proved that - whenever $\delta>0$ and $\int_\Omega u_08\pi\delta$, one can find initial data such that $\int_\Omega u_0=m$, and and the corresponding solution satisfies $$\|u(\cdot,t)\|_{L^\infty(\Omega)} \to \infty \qquad \mbox{as } t\to\infty.$$
Partial differential equations
Biology and other natural sciences
3641
3678
10.4171/JEMS/749
http://www.ems-ph.org/doi/10.4171/JEMS/749
11
20
2017
Mean quantum percolation
Charles
Bordenave
University of Toulouse, France
Arnab
Sen
University of Minnesota, Minneapolis, USA
Bálint
Virág
University of Toronto, Canada
Expected spectral measure, continuous spectra, sparse random graphs, supercritical percolation, unimodular tree, Erdős–Rényi graph
We study the spectrum of adjacency matrices of random graphs. We develop two techniques to lower bound the mass of the continuous part of the spectral measure or the density of states. As an application, we prove that the spectral measure of bond percolation in the two-dimensional lattice contains a non-trivial continuous part in the supercritical regime. The same result holds for the limiting spectral measure of a supercritical Erdős–Rényi graph and for the spectral measure of a unimodular random tree with at least two ends. We give examples of random graphs with purely continuous spectrum.
Probability theory and stochastic processes
3679
3707
10.4171/JEMS/750
http://www.ems-ph.org/doi/10.4171/JEMS/750
11
20
2017
The wild McKay correspondence and $p$-adic measures
Takehiko
Yasuda
Osaka University, Japan
McKay correspondence, $p$-adic measures, wild quotient singularities, stringy invariants, mass formulas
We prove a version of the wild McKay correspondence by using $p$-adic measures. This result provides new proofs of mass formulas for extensions of a local field by Serre, Bhargava and Kedlaya.
Algebraic geometry
Number theory
3709
3743
10.4171/JEMS/751
http://www.ems-ph.org/doi/10.4171/JEMS/751
11
20
2017
Entropy and a convergence theorem for Gauss curvature flow in high dimension
Pengfei
Guan
McGill University, Montreal, Canada
Lei
Ni
University of California at San Diego, La Jolla, USA
Gauss curvature flow, entropy, support functions, regularity, convergence
We prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in $C^\{infty}$-topology to a smooth strictly convex soliton as $t$ goes to infinity is obtained as a consequence of these estimates together with an earlier result of Andrews. The estimates are established via the study of an entropy functional for convex bodies.
Partial differential equations
Differential geometry
Global analysis, analysis on manifolds
3735
3761
10.4171/JEMS/752
http://www.ems-ph.org/doi/10.4171/JEMS/752
11
20
2017
Integral transforms for coherent sheaves
David
Ben-Zvi
University of Texas at Austin, USA
David
Nadler
University of California at Berkeley, USA
Anatoly
Preygel
New York, USA
Fourier–Mukai transforms, coherent sheaves, derived categories, derived algebraic geometry
The theory of integral, or Fourier–Mukai, transforms between derived categories of sheaves is a well established tool in noncommutative algebraic geometry. General “kernel theorems” represent all reasonable linear functors between categories of perfect complexes (or their “large” version, quasi-coherent complexes) on schemes and stacks over some fixed base as integral kernels in the form of complexes (of the same nature) on the fiber product. However, for many applications in mirror symmetry and geometric representation theory one is interested instead in the bounded derived category of coherent sheaves (or its “large” version, ind-coherent sheaves), which differs from perfect complexes (and quasi-coherent complexes) once the underlying variety is singular. In this paper, we prove general kernel theorems for linear functors between derived categories of coherent sheaves over a base in terms of integral kernels on the fiber product. Namely, we identify coherent kernels with functors taking perfect complexes to coherent complexes (an analogue of the classical Schwartz kernel theorem), and kernels which are coherent relative to the source with functors taking all coherent complexes to coherent complexes. The proofs rely on key aspects of the “functional analysis” of derived categories, namely the distinction between small and large categories and its measurement using t-structures. These are used in particular to correct the failure of integral transforms on ind-coherent complexes to correspond to ind-coherent complexes on a fiber product. The results are applied in a companion paper to the representation theory of the affine Hecke category, identifying affine character sheaves with the spectral geometric Langlands category in genus one.
Algebraic geometry
Commutative rings and algebras
3763
3812
10.4171/JEMS/753
http://www.ems-ph.org/doi/10.4171/JEMS/753
11
20
2017
The motivic Steenrod algebra in positive characteristic
Marc
Hoyois
Massachusetts Institute of Technology, Cambridge, USA
Shane
Kelly
Tokyo Institute of Technology, Japan
Paul
Østvær
University of Oslo, Norway
The motivic Steenrod algebra and its dual
Let $S$ be an essentially smooth scheme over a field and $\ell\neq\mathrm {char}\: S$ a prime number. We show that the algebra of bistable operations in the mod $\ell$ motivic cohomology of smooth $S$-schemes is generated by the motivic Steenrod operations. This was previously proved by Voevodsky for $S$ a field of characteristic zero. We follow Voevodsky's proof but remove its dependence on characteristic zero by using etale cohomology instead of topological realization and by replacing resolution of singularities with a theorem of Gabber on alterations.
Algebraic geometry
$K$-theory
3813
3849
10.4171/JEMS/754
http://www.ems-ph.org/doi/10.4171/JEMS/754
11
20
2017
Face numbers of sequentially Cohen–Macaulay complexes and Betti numbers of componentwise linear ideals
Karim
Adiprasito
Hebrew University Jerusalem, Israel
Anders
Björner
Royal Institute of Technology, Stockholm, Sweden
Afshin
Goodarzi
Freie Universität Berlin, Germany
Simplicial complex, face numbers, Stanley–Reisner rings, sequential Cohen–Macaulayness, componentwise linear ideals
A numerical characterization is given of the $h$-triangles of sequentially Cohen–Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible in such a complex, generalizing the classical Macaulay–Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree ≤ $d$ and shifted pure $(d–1)$-dimensional simplicial complexes
Combinatorics
Commutative rings and algebras
3851
3865
10.4171/JEMS/755
http://www.ems-ph.org/doi/10.4171/JEMS/755
11
20
2017
Computing the Teichmüller polynomial
Erwan
Lanneau
Université Grenoble Alpes, Grenoble, France
Ferrán
Valdez
UNAM, Morelia, Mexico
Teichmüller polynomial, pseudo-Anosov homeomorphism, Thurston norm
The Teichmüller polynomial of a fibered 3-manifold, introduced in [McM00], plays a useful role in the construction of mapping classes having a small stretch factor. We provide a general algorithm for computing the Teichmüller polynomial given a pseudo-Anosov mapping class obtained as a loop in a train track automaton. As a byproduct, our algorithm allows us to derive all the relevant information on the topology of various fibers that belong to a fibered face.
Dynamical systems and ergodic theory
Functions of a complex variable
3867
3910
10.4171/JEMS/756
http://www.ems-ph.org/doi/10.4171/JEMS/756
11
20
2017