313 results
Search Results
2. Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur
- Author
-
Kevin Zumbrun and Benjamin Texier
- Subjects
Conservation law ,Kullback–Leibler divergence ,Standard molar entropy ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Regular polygon ,Min entropy ,Shock strength ,01 natural sciences ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Uniqueness ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS
- Published
- 2014
3. The geometry of diagonal groups
- Author
-
Peter J. Cameron, Cheryl E. Praeger, Csaba Schneider, R. A. Bailey, University of St Andrews. Pure Mathematics, University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra, and University of St Andrews. Statistics
- Subjects
Mathematics(all) ,South china ,Primitive permutation group ,General Mathematics ,Diagonal group ,T-NDAS ,Library science ,Group Theory (math.GR) ,O'Nan-Scott Theorem ,01 natural sciences ,Hospitality ,FOS: Mathematics ,NCAD ,Mathematics - Combinatorics ,QA Mathematics ,0101 mathematics ,Diagonal semilattice ,QA ,Cartesian lattice ,Mathematics ,business.industry ,20B05 ,Applied Mathematics ,010102 general mathematics ,Latin square ,Semilattice ,Latin cube ,010101 applied mathematics ,Hamming graph ,Research council ,Diagonal graph ,Combinatorics (math.CO) ,business ,Mathematics - Group Theory ,Partition - Abstract
Part of the work was done while the authors were visiting the South China University of Science and Technology (SUSTech), Shenzhen, in 2018, and we are grateful (in particular to Professor Cai Heng Li) for the hospitality that we received.The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no.EP/R014604/1), where further work on this paper was undertaken. In particular we acknowledge a Simons Fellowship (Cameron) and a Kirk Distinguished Visiting Fellowship (Praeger) during this programme. Schneider thanks the Centre for the Mathematics of Symmetry and Computation of The University of Western Australia and Australian Research Council Discovery Grant DP160102323 for hosting his visit in 2017 and acknowledges the support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.:421624/2018-3). Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our combinatorial axioms. However, for m≥3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with m
- Published
- 2022
4. An improvement on Furstenberg’s intersection problem
- Author
-
Han Yu
- Subjects
Combinatorics ,Intersection ,Applied Mathematics ,General Mathematics ,Bounded function ,010102 general mathematics ,Dimension (graph theory) ,Zero (complex analysis) ,0101 mathematics ,Invariant (mathematics) ,Dynamical system (definition) ,01 natural sciences ,Mathematics - Abstract
In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim A 2 + dim A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .
- Published
- 2021
5. Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks
- Author
-
Armando Martino, Stefano Francaviglia, Francaviglia, Stefano, and Martino, Armando
- Subjects
Outer space, conjugacy problem, automorphisms of free groups, graphs ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Group Theory (math.GR) ,Train track map ,Automorphism ,Lipschitz continuity ,01 natural sciences ,Convexity ,Free product ,Metric (mathematics) ,FOS: Mathematics ,20E06, 20E36, 20E08 ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Group Theory ,Mathematics - Abstract
This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems., 50 pages. Originally part of arXiv:1703.09945 . We decided to split that paper following the recommendations of a referee. Updated subsequent to acceptance by Transactions of the American Mathematical Society
- Published
- 2021
6. The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces
- Author
-
Paolo Mantero
- Subjects
Monomial ,Pure mathematics ,Mathematics::Commutative Algebra ,Betti number ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Algebraic geometry ,Star (graph theory) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Representation theory ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,Young tableau ,0101 mathematics ,Commutative algebra ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)
- Published
- 2020
7. Extremal growth of Betti numbers and trivial vanishing of (co)homology
- Author
-
Jonathan Montaño and Justin Lyle
- Subjects
Pure mathematics ,Conjecture ,Mathematics::Commutative Algebra ,Betti number ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Local ring ,Homology (mathematics) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,13D07, 13D02, 13C14, 13H10, 13D40 ,01 natural sciences ,Injective function ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ for all large $i$ implies $M$ has finite projective dimension or $N$ has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with a result of Gasharov and Peeva, provide sufficient conditions for $R$ to satisfy trivial vanishing; we provide sharpened conditions when $R$ is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke., to appear in Trans. Amer. Math. Soc
- Published
- 2020
8. Flow equivalence of G-SFTs
- Author
-
Toke Meier Carlsen, Søren Eilers, and Mike Boyle
- Subjects
Pure mathematics ,Finite group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,Dynamical Systems (math.DS) ,01 natural sciences ,Matrix (mathematics) ,Group action ,Flow (mathematics) ,FOS: Mathematics ,Equivariant map ,Mathematics - Dynamical Systems ,0101 mathematics ,Connection (algebraic framework) ,Equivalence (measure theory) ,Group ring ,Mathematics - Abstract
In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G$=\mathbb Z_2$, we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata., The paper has been augmented considerably and the second version is now 81 pages long. This version has been accepted for publication in Transactions of the American Mathematical Society
- Published
- 2020
9. Ultrametric properties for valuation spaces of normal surface singularities
- Author
-
Evelia R. García Barroso, Patrick Popescu-Pampu, Pedro Daniel González Pérez, and Matteo Ruggiero
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,Block (permutation group theory) ,14B05, 14J17, 32S25 ,Intersection number ,Function (mathematics) ,01 natural sciences ,Linear subspace ,Combinatorics ,Mathematics - Algebraic Geometry ,Tree (descriptive set theory) ,Singularity ,FOS: Mathematics ,0101 mathematics ,Normal surface ,Algebraic Geometry (math.AG) ,Ultrametric space ,Mathematics - Abstract
Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := \dfrac{(L \cdot A) \: (L \cdot B)}{A \cdot B}$, where $A \cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P{\l}oski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conversely, if $u_L$ is an ultrametric, then $X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $L$ to be an arbitrary semivaluation on $X$ and by defining $u_L$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $X$ is arborescent, and without any restriction on $X$ we exhibit special subspaces of the space of semivaluations in restriction to which $u_L$ is still an ultrametric., Comment: 50 pages, 14 figures. Final version
- Published
- 2019
10. The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation
- Author
-
Łukasz Kubat, Eric Jespers, Arne Van Antwerpen, Mathematics, Algebra, and Faculty of Sciences and Bioengineering Sciences
- Subjects
Monoid ,Semidirect product ,Yang–Baxter equation ,Applied Mathematics ,General Mathematics ,Prime ideal ,010102 general mathematics ,Subalgebra ,Semiprime ,Normal extension ,Mathematics - Rings and Algebras ,Jacobson radical ,01 natural sciences ,Algebra ,Rings and Algebras (math.RA) ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
For a finite involutive non-degenerate solution $(X,r)$ of the Yang--Baxter equation it is known that the structure monoid $M(X,r)$ is a monoid of I-type, and the structure algebra $K[M(X,r)]$ over a field $K$ share many properties with commutative polynomial algebras, in particular, it is a Noetherian PI-domain that has finite Gelfand--Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid $M(X,r)$ and the algebra $K[M(X,r)]$ is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of $M(X,r)$ as a regular submonoid in the semidirect product $A(X,r)\rtimes\mathrm{Sym}(X)$, where $A(X,r)$ is the structure monoid of the rack solution associated to $(X,r)$, we prove that $K[M(X,r)]$ is a module finite normal extension of a commutative affine subalgebra. In particular, $K[M(X,r)]$ is a Noetherian PI-algebra of finite Gelfand--Kirillov dimension bounded by $|X|$. We also characterize, in ring-theoretical terms of $K[M(X,r)]$, when $(X,r)$ is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of $M(X,r)$. These results allow us to control the prime spectrum of the algebra $K[M(X,r)]$ and to describe the Jacobson radical and prime radical of $K[M(X,r)]$. Finally, we give a matrix-type representation of the algebra $K[M(X,r)]/P$ for each prime ideal $P$ of $K[M(X,r)]$. As a consequence, we show that if $K[M(X,r)]$ is semiprime then there exist finitely many finitely generated abelian-by-finite groups, $G_1,\dotsc,G_m$, each being the group of quotients of a cancellative subsemigroup of $M(X,r)$ such that the algebra $K[M(X,r)]$ embeds into $\mathrm{M}_{v_1}(K[G_1])\times\dotsb\times \mathrm{M}_{v_m}(K[G_m])$., A subtle mistake in the proof of Theorem 4.4 has been corrected (will appear in a corrigendum et addendum, TAMS). In the latter paper we also strengthen some of the results by removing the "square free'' condition in Section 5 and in this paper we also prove new homological equivalences in Theorem 4.4
- Published
- 2019
11. Good coverings of Alexandrov spaces
- Author
-
Takao Yamaguchi and Ayato Mitsuishi
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Homotopy ,010102 general mathematics ,Stability (learning theory) ,Fibration ,Metric Geometry (math.MG) ,Type (model theory) ,Curvature ,Space (mathematics) ,Mathematics::Algebraic Topology ,01 natural sciences ,53C20, 53C23 ,Mathematics - Metric Geometry ,Mathematics::Category Theory ,Bounded function ,FOS: Mathematics ,Mathematics::Differential Geometry ,Isomorphism ,0101 mathematics ,Mathematics - Abstract
In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove the stability of the isomorphism classes of the nerves of good coverings in the non-collapsing case. In the proof, we need a version of Perelman's fibration theorem, which is also proved in this paper., Minor change basically on the proof of Theorem 1.2 in Section 5
- Published
- 2019
12. An 𝐿^{𝑝} theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions
- Author
-
Jennifer Chayes, Yufei Zhao, Henry Cohn, and Christian Borgs
- Subjects
Random graph ,Discrete mathematics ,Dense graph ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,01 natural sciences ,Power law ,Limit theory ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Equivalence (formal languages) ,Mathematics - Probability ,Mathematics - Abstract
We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper., Comment: 44 pages
- Published
- 2019
13. Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials
- Author
-
Daniel Erman, Steven V Sam, and Andrew Snowden
- Subjects
Pure mathematics ,General Mathematics ,media_common.quotation_subject ,MathematicsofComputing_GENERAL ,Hilbert's basis theorem ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematics - Algebraic Geometry ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Ideal (ring theory) ,0101 mathematics ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics ,media_common ,Conjecture ,Hilbert's syzygy theorem ,Mathematics::Commutative Algebra ,Degree (graph theory) ,Applied Mathematics ,010102 general mathematics ,13A02, 13D02 ,Mathematics - Commutative Algebra ,Infinity ,Bounded function ,symbols ,010307 mathematical physics - Abstract
Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin
- Published
- 2018
14. A curvature-free 𝐿𝑜𝑔(2𝑘-1) theorem
- Author
-
Florent Balacheff and Louis Merlin
- Subjects
Discrete mathematics ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,16. Peace & justice ,Curvature ,Mathematics::Geometric Topology ,01 natural sciences ,Volume entropy ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper presents a curvature-free version of the Log ( 2 k − 1 ) \text {Log}(2k-1) Theorem of Anderson, Canary, Culler, and Shalen [J. Differential Geometry 44 (1996), pp. 738–782]. It generalizes a result by Hou [J. Differential Geometry 57 (2001), no. 1, pp. 173–193] and its proof is rather straightforward once we know the work by Lim [Trans. Amer. Math. Soc. 360 (2008), no. 10, pp. 5089–5100] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the Collar Lemma in all dimensions.
- Published
- 2023
15. On symmetric linear diffusions
- Author
-
Liping Li and Jiangang Ying
- Subjects
Discrete mathematics ,Representation theorem ,Dirichlet form ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Disjoint sets ,01 natural sciences ,Dirichlet distribution ,010104 statistics & probability ,symbols.namesake ,Closure (mathematics) ,symbols ,Interval (graph theory) ,Countable set ,0101 mathematics ,Mathematics - Abstract
The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.
- Published
- 2018
16. On period relations for automorphic 𝐿-functions I
- Author
-
Fabian Januszewski
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Statistics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Period (music) ,Mathematics - Abstract
This paper is the first in a series of two dedicated to the study of period relations of the type L ( 1 2 + k , Π ) ∈ ( 2 π i ) d ⋅ k Ω ( − 1 ) k \bf Q ( Π ) , 1 2 + k critical , \begin{equation*} L\Big (\frac {1}{2}+k,\Pi \Big )\;\in \;(2\pi i)^{d\cdot k}\Omega _{(-1)^k}\textrm {\bf Q}(\Pi ),\quad \frac {1}{2}+k\;\text {critical}, \end{equation*} for certain automorphic representations Π \Pi of a reductive group G . G. In this paper we discuss the case G = G L ( n + 1 ) × G L ( n ) . G=\mathrm {GL}(n+1)\times \mathrm {GL}(n). The case G = G L ( 2 n ) G=\mathrm {GL}(2n) is discussed in part two. Our method is representation theoretic and relies on the author’s recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation Π \Pi under consideration. The new period relations we prove are in accordance with Deligne’s Conjecture on special values of L L -functions, and the author expects this method to apply to other cases as well.
- Published
- 2018
17. Functions of triples of noncommuting self-adjoint operators under perturbations of class $\boldsymbol {S}_p$
- Author
-
V. V. Peller
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,16. Peace & justice ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Mathematics - Spectral Theory ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,010307 mathematical physics ,Complex Variables (math.CV) ,0101 mathematics ,Spectral Theory (math.SP) ,Self-adjoint operator ,Mathematics - Abstract
In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten--von Neumann norm $\boldsymbol S_p$, $1\le p\le\infty$, for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$. In other words, we prove that for $p\in[1,\infty]$, there is no constant $K>0$ such that the inequality \begin{align*} \|f(A_1,B_1,C_1)&-f(A_2,B_2,C_2)\|_{\boldsymbol S_p}\\[.1cm] &\le K\|f\|_{B_{\infty,1}^1} \max\big\{\|A_1-A_2\|_{\boldsymbol S_p},\|B_1-B_2\|_{\boldsymbol S_p},\|C_1-C_2\|_{\boldsymbol S_p}\big\} \end{align*} holds for an arbitrary function $f$ in $B_{\infty,1}^1({\Bbb R}^3)$ and for arbitrary finite rank self-adjoint operators $A_1,\,B_1,\,C_1,\,A_2,\,B_2$ and $C_2$., 14 pages. arXiv admin note: substantial text overlap with arXiv:1606.08961
- Published
- 2018
18. Wave front sets of reductive Lie group representations II
- Author
-
Benjamin Harris
- Subjects
Wavefront ,Induced representation ,Applied Mathematics ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Wave front set ,Lie group ,(g,K)-module ,01 natural sciences ,Algebra ,Representation of a Lie group ,0103 physical sciences ,FOS: Mathematics ,Tempered representation ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups., Accepted to Transactions of the American Mathematical Society
- Published
- 2017
19. (Logarithmic) densities for automatic sequences along primes and squares
- Author
-
Boris Adamczewski, Clemens Müllner, Michael Drmota, Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Combinatoire, théorie des nombres (CTN), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and European Project: 648132,H2020,ERC-2014-CoG,ANT(2015)
- Subjects
FOS: Computer and information sciences ,Automatic sequence ,Mathematics - Number Theory ,Logarithm ,Formal Languages and Automata Theory (cs.FL) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Multiplicative function ,Computer Science - Formal Languages and Automata Theory ,Function (mathematics) ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Prime (order theory) ,Combinatorics ,Transfer (group theory) ,Aperiodic graph ,Bounded function ,Primary: 11B85, 11L20, 11N05, Secondary: 11A63, 11L03 ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - Abstract
In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lema\'nczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function., Comment: 35 pages. We added an Appendix concerning upper densities of subsequences of automatic sequences
- Published
- 2021
20. Constructing highly regular expanders from hyperbolic Coxeter groups
- Author
-
Jeroen Schillewaert, Alexander Lubotzky, François Thilmany, Marston Conder, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
- Subjects
Vertex (graph theory) ,Diagram (category theory) ,General Mathematics ,Polytope ,Group Theory (math.GR) ,01 natural sciences ,GRAPHS ,Combinatorics ,20F55, 05C48 (Primary), 51F15, 22E40, 05C25 (Secondary) ,FOS: Mathematics ,SUBGROUPS ,Mathematics - Combinatorics ,0101 mathematics ,Quotient ,Mathematics ,Science & Technology ,Group (mathematics) ,Applied Mathematics ,010102 general mathematics ,Coxeter group ,Physical Sciences ,Expander graph ,Combinatorics (math.CO) ,Mathematics - Group Theory ,Regular polytope - Abstract
A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled., Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest Vinberg
- Published
- 2021
21. Hodge theory in combinatorics
- Author
-
Matthew Baker
- Subjects
Polynomial ,Sequence ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Algebraic geometry ,Chromatic polynomial ,01 natural sciences ,Matroid ,Unimodality ,Combinatorics ,Mathematics - Algebraic Geometry ,010104 statistics & probability ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Characteristic polynomial ,Mathematics - Abstract
George Birkhoff proved in 1912 that the number of proper colorings of a finite graph G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read's conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's result was subsequently refined and generalized by Huh and Katz, again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality. The breakthroughs of Huh and Huh-Katz left open the more general Rota-Welsh conjecture where graphs are generalized to (not necessarily representable) matroids and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and Huh-Katz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. But in 2015 Adiprasito, Huh, and Katz announced a proof of the Rota-Welsh conjecture based on a novel approach motivated by but not making use of any results from algebraic geometry. The authors first prove that the Rota-Welsh conjecture would follow from combinatorial analogues of the Hard Lefschetz Theorem and Hodge-Riemann relations in algebraic geometry. They then implement an elaborate inductive procedure to prove the combinatorial Hard Lefschetz Theorem and Hodge-Riemann relations using purely combinatorial arguments. We will survey these developments., Comment: 22 pages. This is an expository paper to accompany my lecture at the 2017 AMS Current Events Bulletin. v2: Numerous minor corrections
- Published
- 2017
22. Gromov hyperbolicity, the Kobayashi metric, and $\mathbb {C}$-convex sets
- Author
-
Andrew Zimmer
- Subjects
Pure mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Regular polygon ,Boundary (topology) ,Codimension ,01 natural sciences ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,Affine transformation ,Ball (mathematics) ,0101 mathematics ,Mathematics - Abstract
In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.
- Published
- 2017
23. Tame circle actions
- Author
-
Jordan Watts and Susan Tolman
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Kähler manifold ,Fixed point ,01 natural sciences ,Mathematics - Symplectic Geometry ,0103 physical sciences ,Symplectic category ,Slice theorem ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,53D20 (Primary) 53D05, 53B35 (Secondary) ,0101 mathematics ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic manifold ,Symplectic geometry ,Mathematics - Abstract
In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases., Comment: 25 pages
- Published
- 2017
24. Differentiability of the conjugacy in the Hartman-Grobman Theorem
- Author
-
Weinian Zhang, Kening Lu, and Wenmeng Zhang
- Subjects
Bump function ,0209 industrial biotechnology ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Invariant manifold ,02 engineering and technology ,01 natural sciences ,Hartman–Grobman theorem ,020901 industrial engineering & automation ,Conjugacy class ,Differentiable function ,0101 mathematics ,Mathematics - Abstract
The classical Hartman-Grobman Theorem states that a smooth diffeomorphism F ( x ) F(x) near its hyperbolic fixed point x ¯ \bar x is topological conjugate to its linear part D F ( x ¯ ) DF(\bar x) by a local homeomorphism Φ ( x ) \Phi (x) . In general, this local homeomorphism is not smooth, not even Lipschitz continuous no matter how smooth F ( x ) F(x) is. A question is: Is this local homeomorphism differentiable at the fixed point? In a 2003 paper by Guysinsky, Hasselblatt and Rayskin, it is shown that for a C ∞ C^\infty diffeomorphism F ( x ) F(x) , the local homeomorphism indeed is differentiable at the fixed point. In this paper, we prove for a C 1 C^1 diffeomorphism F ( x ) F(x) with D F ( x ) DF(x) being α \alpha -Hölder continuous at the fixed point that the local homeomorphism Φ ( x ) \Phi (x) is differentiable at the fixed point. Here, α > 0 \alpha >0 depends on the bands of the spectrum of F ′ ( x ¯ ) F’(\bar x) for a diffeomorphism in a Banach space. We also give a counterexample showing that the regularity condition on F ( x ) F(x) cannot be lowered to C 1 C^1 .
- Published
- 2017
25. Isoperimetric properties of the mean curvature flow
- Author
-
Or Hershkovits
- Subjects
Pure mathematics ,Mean curvature flow ,Mean curvature ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,Upper and lower bounds ,Geometric measure theory ,0103 physical sciences ,Hausdorff measure ,010307 mathematical physics ,0101 mathematics ,Isoperimetric inequality ,Constant (mathematics) ,Mathematics - Abstract
In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self-evident. The first is a genuine, 5-line proof, for the isoperimetric inequality for k k -cycles in R n \mathbb {R}^n , with a constant differing from the optimal constant by a factor of only k \sqrt {k} , as opposed to a factor of k k k^k produced by all of the other soft methods. The second is a 3-line proof of a lower bound for extinction for arbitrary co-dimensional, singular, mean curvature flows starting from cycles, generalizing the main result of Giga and Yama-uchi (1993). We then turn to use the above-mentioned relation to prove a bound on the parabolic Hausdorff measure of the space-time track of high co-dimensional, singular, mean curvature flow starting from a cycle, in terms of the mass of that cycle. This bound is also reminiscent of a Michael-Simon isoperimetric inequality. To prove it, we are led to study the geometric measure theory of Euclidean rectifiable sets in parabolic space and prove a co-area formula in that setting. This formula, the proof of which occupies most of this paper, may be of independent interest.
- Published
- 2017
26. On embedding certain partial orders into the P-points under Rudin-Keisler and Tukey reducibility
- Author
-
Dilip Raghavan and Saharon Shelah
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Boolean algebra (structure) ,010102 general mathematics ,Ultrafilter ,Natural number ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,symbols ,Embedding ,Continuum (set theory) ,0101 mathematics ,Partially ordered set ,Continuum hypothesis ,Axiom ,Mathematics - Abstract
The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.
- Published
- 2017
27. On the constant scalar curvature Kähler metrics (II)—Existence results
- Author
-
Jingrui Cheng and Xiuxiong Chen
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Constant (mathematics) ,01 natural sciences ,Mathematics ,Scalar curvature ,Mathematical physics - Abstract
In this paper, we apply our previous estimates in Chen and Cheng [On the constant scalar curvature Kähler metrics (I): a priori estimates, Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness ofKK-energy in terms ofL1L^1geodesic distanced1d_1in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of theKK-energy in(E1,d1)(\mathcal {E}^1, d_1)are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilizedL1L^1geodesic ray where theKK-energy is non-increasing, which is a weak version of a conjecture by Donaldson. The continuity path proposed by Xiuxiong Chen [Ann. Math. Qué. 42 (2018), pp. 69–189] is instrumental in the above proofs.
- Published
- 2021
28. On the constant scalar curvature Kähler metrics (I)—A priori estimates
- Author
-
Jingrui Cheng and Xiuxiong Chen
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,A priori and a posteriori ,010307 mathematical physics ,0101 mathematics ,Constant (mathematics) ,01 natural sciences ,Scalar curvature ,Mathematics - Abstract
In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of aC0C^0bound for the Kähler potential.
- Published
- 2021
29. On pro-2 identities of 2×2 linear groups
- Author
-
Efim Zelmanov and David BenEzra
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics education ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let F ^ \hat {F} be a free pro- p p non-abelian group, and let Δ \Delta be a commutative Noetherian complete local ring with a maximal ideal I I such that c h a r ( Δ / I ) = p > 0 \mathrm {char}(\Delta /I)=p>0 . Zubkov [Sibirsk. Mat. Zh. 28 (1987), pp. 64–69] showed that when p ≠ 2 p\neq 2 , the pro- p p congruence subgroup \[ G L 2 1 ( Δ ) = ker ( G L 2 ( Δ ) ⟶ Δ → Δ / I G L 2 ( Δ / I ) ) GL_{2}^{1}(\Delta )=\ker (GL_{2}(\Delta )\overset {\Delta \to \Delta /I}{\longrightarrow }GL_{2}(\Delta /I)) \] admits a pro- p p identity, i.e., there exists an element 1 ≠ w ∈ F ^ 1\neq w\in \hat {F} that vanishes under any continuous homomorphism F ^ → G L 2 1 ( Δ ) \hat {F}\to GL_{2}^{1}(\Delta ) . In this paper we investigate the case p = 2 p=2 . The main result is that when c h a r ( Δ ) = 2 \mathrm {char}(\Delta )=2 , the pro- 2 2 group G L 2 1 ( Δ ) GL_{2}^{1}(\Delta ) admits a pro- 2 2 identity. This result was obtained by the use of trace identities that originate in PI-theory.
- Published
- 2021
30. Positivity of direct images of fiberwise Ricci-flat metrics on Calabi-Yau fibrations
- Author
-
Georg Schumacher, Young-Jun Choi, and Matthias Braun
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,Image (category theory) ,010102 general mathematics ,Fibered knot ,Kähler manifold ,01 natural sciences ,Triviality ,Base (group theory) ,Kähler–Einstein metric ,Differential Geometry (math.DG) ,FOS: Mathematics ,Calabi–Yau manifold ,Mathematics::Differential Geometry ,Complex Variables (math.CV) ,0101 mathematics ,Complex manifold ,32Q25, 32Q20, 32G05, 32W20 ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
Let $X$ be a K\"ahler manifold which is fibered over a complex manifold $Y$ such that every fiber is a Calabi-Yau manifold. Let $\omega$ be a fixed K\"ahler form on $X$. By Yau's theorem, there exists a unique Ricci-flat K\"ahler form $\rho\vert_{X_y}$ for each fiber, which is cohomologous to $\omega\vert_{X_y}$. This family of Ricci-flat K\"ahler forms $\rho\vert_{X_y}$ induces a smooth $(1,1)$-form $\rho$ on $X$ with a normalization condition. In this paper, we prove that the direct image of $\rho^{n+1}$ is positive on the base $Y$. We also discuss several byproducts, among them the local triviality of families of Calabi-Yau manifolds., Comment: 25 pages. Minor changes
- Published
- 2021
31. A lower bound for the double slice genus
- Author
-
Wenzhao Chen
- Subjects
Degree (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Geometric topology ,Geometric Topology (math.GT) ,Alexander polynomial ,Mathematics::Geometric Topology ,01 natural sciences ,Upper and lower bounds ,57K10, 57K31 ,Combinatorics ,Mathematics - Geometric Topology ,Mathematics::Algebraic Geometry ,Genus (mathematics) ,FOS: Mathematics ,0101 mathematics ,Ribbon knot ,Mathematics ,Knot (mathematics) ,Slice genus - Abstract
In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound in the topological category., 18 pages, 6 figures, to appear in Trans. Amer. Math. Soc
- Published
- 2021
32. What are Lyapunov exponents, and why are they interesting?
- Author
-
Amie Wilkinson
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Spectral theory ,General Mathematics ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,02 engineering and technology ,Lyapunov exponent ,Barycentric subdivision ,Computer Science::Computational Geometry ,Equilateral triangle ,Translation (geometry) ,01 natural sciences ,Midpoint ,Mathematics - Spectral Theory ,Mathematics - Geometric Topology ,symbols.namesake ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Ergodic theory ,Mathematics - Dynamical Systems ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematical Physics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,Mathematical Physics (math-ph) ,37C40, 37D25, 37H15, 34D08, 37C60, 47B36, 32G15 ,Computer Science::Graphics ,symbols ,020201 artificial intelligence & image processing ,Schrödinger's cat - Abstract
This expository paper, based on a Current Events Bulletin talk at the January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and discusses the role they play in three areas: smooth ergodic theory, Teichm\"uller theory, and the spectral theory of one-frequency Schr\"odinger operators. The inspiration for this paper is the work of 2014 Fields Medalist Artur Avila, and his work in these areas is given special attention., Comment: 27 pages. To appear in the Bulletin of the AMS
- Published
- 2016
33. Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type
- Author
-
Wenxian Shen and Zhongwei Shen
- Subjects
Work (thermodynamics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Monotonic function ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Exponential growth ,Uniqueness ,0101 mathematics ,Exponential decay ,Mathematics - Abstract
The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, it is shown that the wave profile of the unique generalized traveling wave is of the same recurrence as the media. In particular, if the media is time almost periodic, then so is the wave profile of the unique generalized traveling wave.
- Published
- 2016
34. Derangements in subspace actions of finite classical groups
- Author
-
Robert M. Guralnick and Jason Fulman
- Subjects
Classical group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Codimension ,01 natural sciences ,Combinatorics ,Group of Lie type ,010201 computation theory & mathematics ,Simple group ,Bounded function ,Classification of finite simple groups ,CA-group ,0101 mathematics ,Subspace topology ,Mathematics - Abstract
This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.
- Published
- 2016
35. Classification of tile digit sets as product-forms
- Author
-
Chun-Kit Lai, Ka-Sing Lau, and Hui Rao
- Subjects
Polynomial (hyperelastic model) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,01 natural sciences ,Prime (order theory) ,010101 applied mathematics ,Combinatorics ,Matrix (mathematics) ,Tree (descriptive set theory) ,Integer ,Product (mathematics) ,0101 mathematics ,Cyclotomic polynomial ,Mathematics - Abstract
Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In our previous paper, we classified such tile digit sets ${\mathcal D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into product of cyclotomic polynomials. In this paper, we first show that a tile digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal D}\oplus{\mathcal L} = {\Bbb Z}^s$ for some discrete set ${\mathcal L}$). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on ${\Bbb R}^1$ together with our previous results to characterize explicitly all tile digit sets ${\mathcal D}\subset {\Bbb Z}$ with $A = p^{\alpha}q$ ($p, q$ distinct primes) as {\it modulo product-form} of some order, an advance of the previously known results for $A = p^\alpha$ and $pq$.
- Published
- 2016
36. On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces
- Author
-
Bruno de Mendonça Braga and Ilijas Farah
- Subjects
Pure mathematics ,Property (philosophy) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Rigidity (psychology) ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Metric space ,Coarse space ,FOS: Mathematics ,Property a ,0101 mathematics ,Algebra over a field ,Operator Algebras (math.OA) ,Coarse structure ,Mathematics - Abstract
Given a coarse space $(X,\mathcal{E})$, one can define a $\mathrm{C}^*$-algebra $\mathrm{C}^*_u(X)$ called the uniform Roe algebra of $(X,\mathcal{E})$. It has been proved by J. \v{S}pakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.
- Published
- 2020
37. Scattering for the 𝐿² supercritical point NLS
- Author
-
Riccardo Adami, Reika Fukuizumi, and Justin Holmer
- Subjects
Scattering ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Schrödinger equation ,Nonlinear point interaction ,01 natural sciences ,symbols.namesake ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,symbols ,NLS ,Point (geometry) ,010307 mathematical physics ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematical physics ,Mathematics - Abstract
We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. “Point” means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ± ∞ t\to \pm \infty in the L 2 L^2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.
- Published
- 2020
38. Tame topology of arithmetic quotients and algebraicity of Hodge loci
- Author
-
Benjamin Bakker, Bruno Klingler, and Jacob Tsimerman
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics - Logic ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Logic (math.LO) ,Algebraic Geometry (math.AG) ,Quotient ,Topology (chemistry) ,Mathematics - Abstract
In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. $2)$ We prove that the period map associated to any pure polarized variation of integral Hodge structures $\mathbb{V}$ on a smooth complex quasi-projective variety $S$ is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. $3)$ As a corollary of $2)$ and of Peterzil-Starchenko's o-minimal Chow theorem we recover that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$, a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable $SL_2$-orbit theorem of Cattani-Kaplan-Schmid., 23 pages, final version. arXiv admin note: substantial text overlap with arXiv:1803.09384
- Published
- 2020
39. Sharp uncertainty principles on general Finsler manifolds
- Author
-
Wei Zhao, Libing Huang, and Alexandru Kristály
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Curvature ,Computer Science::Digital Libraries ,01 natural sciences ,Mathematics - Analysis of PDEs ,Rigidity (electromagnetism) ,FOS: Mathematics ,Computer Science::Mathematical Software ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,Finsler manifold ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fully characterizes the nature of the Finsler manifold in terms of three non-Riemannian quantities, namely, its reversibility and the vanishing of the flag curvature and $S$-curvature induced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. The optimality of our results are supported by Randers-type Finslerian examples originating from the Zermelo navigation problem., 29 pages; some references have been added; to appear in Trans. Amer. Math. Soc
- Published
- 2020
40. Betti tables of monomial ideals fixed by permutations of the variables
- Author
-
Satoshi Murai
- Subjects
Monomial ,Mathematics::Combinatorics ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,Polynomial ring ,010102 general mathematics ,Dimension (graph theory) ,Field (mathematics) ,Monomial ideal ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Table (information) ,01 natural sciences ,Combinatorics ,Integer ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of $I_n$ for all large intergers $n$ from the $\mathbb Z^m$-graded Betti table of $I_m$ for some integer $m$. Our main result shows that the projective dimension and the regularity of $I_n$ eventually become linear functions on $n$, confirming a special case of conjectures posed by Le, Nagel, Nguyen and R\"omer., Comment: 20 pages
- Published
- 2020
41. On singular vortex patches, II: Long-time dynamics
- Author
-
Tarek M. Elgindi and In-Jee Jeong
- Subjects
Cusp (singularity) ,Work (thermodynamics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dynamics (mechanics) ,01 natural sciences ,Exponential function ,Vortex ,Mathematics - Analysis of PDEs ,Classical mechanics ,Time dynamics ,FOS: Mathematics ,0101 mathematics ,Spiral ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In a companion paper, we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the companion work. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time., Comment: 17 pages, 6 figures
- Published
- 2020
42. An explicit Pólya-Vinogradov inequality via Partial Gaussian sums
- Author
-
Matteo Bordignon and Bryce Kerr
- Subjects
Pure mathematics ,Inequality ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Gaussian ,010102 general mathematics ,Vinogradov ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,symbols ,0101 mathematics ,Mathematics ,media_common - Abstract
In this paper we obtain a new fully explicit constant for the Pólya-Vinogradov inequality for squarefree modulus. Given a primitive character χ \chi to squarefree modulus q q , we prove the following upper bound: | ∑ 1 ⩽ n ⩽ N χ ( n ) | ⩽ c q log q , \begin{align*} \left | \sum _{1 \leqslant n\leqslant N} \chi (n) \right |\leqslant c \sqrt {q} \log q, \end{align*} where c = 1 / ( 2 π 2 ) + o ( 1 ) c=1/(2\pi ^2)+o(1) for even characters and c = 1 / ( 4 π ) + o ( 1 ) c=1/(4\pi )+o(1) for odd characters, with an explicit o ( 1 ) o(1) term. This improves a result of Frolenkov and Soundararajan for large q q . We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with multiplicative coefficients. This allows a power saving on the minor arcs rather than a factor of log q \log {q} as in previous approaches and is an important factor for fully explicit bounds.
- Published
- 2020
43. Entrance laws at the origin of self-similar Markov processes in high dimensions
- Author
-
Ting Yang, Bati Sengul, Andreas E. Kyprianou, and Victor Rivero
- Subjects
Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Process (computing) ,Markov process ,01 natural sciences ,symbols.namesake ,Law ,Convergence (routing) ,FOS: Mathematics ,symbols ,0101 mathematics ,Mathematics - Probability ,Mathematics - Abstract
In this paper we consider the problem of finding entrance laws at the origin for self-similar Markov processes in R d \mathbb {R}^d , killed upon hitting the origin. Under suitable assumptions, we show the existence of an entrance law and the convergence to this law when the process is started close to the origin. We obtain an explicit description of the process started from the origin as the time reversal of the original self-similar Markov process conditioned to hit the origin.
- Published
- 2020
44. The distribution of sandpile groups of random regular graphs
- Author
-
András Mészáros
- Subjects
Distribution (number theory) ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,05C80, 15B52, 60B20 ,Directed graph ,01 natural sciences ,law.invention ,Combinatorics ,Invertible matrix ,law ,Random regular graph ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Limit (mathematics) ,Adjacency matrix ,0101 mathematics ,Heuristics ,Mathematics - Probability ,Mathematics - Abstract
We study the distribution of the sandpile group of random d-regular graphs. For the directed model, we prove that it follows the Cohen-Lenstra heuristics, that is, the limiting probability that the $p$-Sylow subgroup of the sandpile group is a given $p$-group $P$, is proportional to $|Aut(P)|^{-1}$. For finitely many primes, these events get independent in the limit. Similar results hold for undirected random regular graphs, where for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne. This answers an open question of Frieze and Vu whether the adjacency matrix of a random regular graph is invertible with high probability. Note that for directed graphs this was recently proved by Huang. It also gives an alternate proof of a theorem of Backhausz and Szegedy., Comment: We improved the presentation of the paper. More details are given in several proofs
- Published
- 2020
45. A flow method for the dual Orlicz–Minkowski problem
- Author
-
Jian Lu and YanNan Liu
- Subjects
Generalization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Flow method ,Monge–Ampère equation ,Support function ,01 natural sciences ,Dual (category theory) ,symbols.namesake ,Flow (mathematics) ,Gaussian curvature ,symbols ,0101 mathematics ,Minkowski problem ,Mathematics - Abstract
In this paper the dual Orlicz–Minkowski problem, a generalization of the L p L_p dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a new existence result of solutions to this problem for smooth measures.
- Published
- 2020
46. Random matrices with prescribed eigenvalues and expectation values for random quantum states
- Author
-
Elizabeth Meckes and Mark W. Meckes
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Observable ,01 natural sciences ,Hermitian matrix ,Distribution (mathematics) ,Quantum state ,Joint probability distribution ,Probability distribution ,0101 mathematics ,Random matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Given a collection λ _ = { λ 1 \underline {\lambda }=\{\lambda _1 , … \dots , λ n } \lambda _n\} of real numbers, there is a canonical probability distribution on the set of real symmetric or complex Hermitian matrices with eigenvalues λ 1 , … , λ n \lambda _1, \ldots , \lambda _n . In this paper, we study various features of random matrices with this distribution. Our main results show that under mild conditions, when n n is large, linear functionals of the entries of such random matrices have approximately Gaussian joint distributions. The results take the form of upper bounds on distances between multivariate distributions, which allows us also to consider the case when the number of linear functionals grows with n n . In the context of quantum mechanics, these results can be viewed as describing the joint probability distribution of the expectation values of a family of observables on a quantum system in a random mixed state. Other applications are given to spectral distributions of submatrices, the classical invariant ensembles, and to a probabilistic counterpart of the Schur–Horn theorem, relating eigenvalues and diagonal entries of Hermitian matrices.
- Published
- 2020
47. The family of perfect ideals of codimension 3, of type 2 with 5 generators
- Author
-
Witold Kraśkiewicz, Jerzy Weyman, Ela Celikbas, and Jai Laxmi
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Complete intersection ,Codimension ,Type (model theory) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,16. Peace & justice ,01 natural sciences ,13C40, 13D02, 13H10 ,0103 physical sciences ,FOS: Mathematics ,Key (cryptography) ,Multiplication ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
In this paper we define an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the Tor algebra. This family is likely to play a key role in classifying perfect ideals with five generators of type two., 11 pages
- Published
- 2020
48. Hermitian curvature flow on unimodular Lie groups and static invariant metrics
- Author
-
Luigi Vezzoni, Mattia Pujia, and Ramiro A. Lafuente
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Lie group ,01 natural sciences ,Hermitian matrix ,Nilpotent ,Unimodular matrix ,Differential Geometry (math.DG) ,FOS: Mathematics ,Hermitian manifold ,Mathematics::Differential Geometry ,0101 mathematics ,Abelian group ,Invariant (mathematics) ,Quotient ,Mathematics - Abstract
We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. The solution $g_t$ always exist for all positive times, and $(1 + t)^{-1}g_t$ converges as $t\to \infty$ in Cheeger-Gromov sense to a non-flat left-invariant soliton $(\bar G, \bar g)$. Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-K\"ahler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result in \cite{EFV} for the pluriclosed flow. In the last part of the paper we study HCF on Lie groups with abelian complex structures., Comment: 25 pages. Revised version. To appear in TAMS
- Published
- 2020
49. Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space
- Author
-
Ronggang Shi
- Subjects
Pointwise ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Lie group ,Dynamical Systems (math.DS) ,16. Peace & justice ,01 natural sciences ,Combinatorics ,Mathematics::Group Theory ,Subgroup ,Homogeneous space ,FOS: Mathematics ,Ergodic theory ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics ,Haar measure - Abstract
Let $\Gamma$ be a lattice of a semisimple Lie group $L$. Suppose that one parameter Ad-diagonalizable subgroup $\{g_t\}$ of $L$ acts ergodically on $L/\Gamma$ with respect to the probability Haar measure $\mu$. For certain proper subgroup $U$ of the unstable horospherical subgroup of $\{g_t\}$ we show that given $x\in L/\Gamma$ for almost every $u\in U$ the trajectory $\{g_tux: 0\le t\le T\}$ is uniformly distributed with respect to $\mu$ as $T\to \infty$., Comment: The detailed proof is modified a lot to make the paper easy to read
- Published
- 2020
50. Composition series for GKZ-systems
- Author
-
Jiangxue Fang
- Subjects
Pure mathematics ,14F10, 32S60 ,Composition series ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,FOS: Mathematics ,0101 mathematics ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics - Abstract
In this paper, we find a composition series of GKZ-systems with semisimple successive quotients. We also study the composition series of the corresponding perverse sheaves and compare these two composition series under the Riemann-Hilbert correspondence., 32 pages, appear to Transactions of AMS
- Published
- 2020
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.