Iosif Polterovich
We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D. Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in math.DG/9905073.
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We embark on a tour that takes us through four closely related topics: the dual concepts of independence and spanning in finite abelian groups and the analogous dual concepts of designs and distance sets on spheres. We review some of the main known results in each area, mention several open questions, and discuss some connections among these four interesting topics.
On the combinatorics of Murai spheres and its applications
本文研究了Murai球体的组合结构，分类了其在不同维度下的类型，并探讨了其在凸多面体和色数计算中的应用。
We prove, in a purely combinatorial way, the spectral curve topological recursion for the problem of enumeration of bi-colored maps, which are dual objects to dessins d'enfant. Furthermore, we give a proof of the quantum spectral curve equation for this problem. Then we consider the generalized case of 4-colored maps and outline the idea of the proof of the corresponding spectral curve topological recursion.
Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycleson the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, andsatisfy remarkable algebraic properties such as the existence and uniqueness of a prime factorization.Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in directcorrespondence with…
The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein* over Z_2. We also introduce a Boolean cell complex which is an analogue for twisted involutions of the Coxeter complex. Several classical Coxeter complex properties are shared by our complex. When the group is finite, it is a shellable sphere, shelling orders being given by the linear extensions of the weak order on twisted involutions. Furthermore, the h-polynomial of the comple…
The combinatorics of Farey words and their traces
本文引入与3-流形和orbifold几何拓扑相关的三元Farey多项式族，通过抽象组合观点研究其递归定义和性质，为PSL(2,C)中秩二子群的分类问题提供实用方法。
The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the open string partition function of $\mathbb{C}^3$. We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and repres…
We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple proofs for a lot of results about $R$-diagonal elements. Our investigations do not assume the trace property for the considered linear functionals.
Trace monoids and heaps of pieces appear in various contexts in combinatorics. They also constitute a model used in computer science to describe the executions of asynchronous systems. The design of a natural probabilistic layer on top of the model has been a long standing challenge. The difficulty comes from the presence of commuting pieces and from the absence of a global clock. In this paper, we introduce and study the class of Bernoulli probability measures that we claim to be the simplest adequate probability measures on infinite traces. For this, we strongly rely on the theory of trace c…
On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties
本文从扇与多面体出发构造组合截断函数，证明其积分的收敛性与多项式性，推广了Arthur迹公式、Ehrhart多项式及Langlands引理，并给出环面簇上的几何解释。
Residue Formulae for the Trace on Affine Hecke Algebras
受Catalan组合学启发，利用Szenes-Vergne理论从Opdam生成函数导出仿射Hecke代数标准迹的留数公式，进而得到大腔中平移元素迹的乘积公式。
Embeddings of edge-colored dual graphs of balanced 3- and 4-manifolds
本文研究了平衡亏格在3-4维流形对偶图嵌入中的应用，建立了平衡亏格的下界和同胚判定条件，为图论和组合学提供了新视角。
Partial Resolutions and Noncrossing Combinatorics
本文研究了有限可约群的Hecke代数中心元素，通过部分Springer解析和非交叉集合构造，扩展了相关不变量的计算方法。
Discrete trace formulas and holomorphic functional calculus for the adjacency matrix of regular graphs
本文通过全纯函数演算研究正则图邻接矩阵，推导出离散迹公式，并为行走计数、Ihara-Bass公式等提供了新证明。
In this paper we define generalised spheres in buildings using the simplicial structure and Weyl distance in the building, and we derive an explicit formula for the cardinality of these spheres. We prove a generalised notion of distance regularity in buildings, and develop a combinatorial formula for the cardinalities of intersections of generalised spheres. Motivated by the classical study of algebras associated to distance regular graphs we investigate the algebras and modules of Hecke operators arising from our generalised distance regularity, and prove isomorphisms between these algebras a…
The Trace Method for Cotangent Sums
本文利用迹方法研究余切整数幂之和，通过将余切值实现为自伴矩阵的特征值并展开生成函数，得出显式求值，并提出 Riemann zeta 函数在偶数处的新近似。
On a trace formula for counting Eulerian cycles
该研究将计数欧拉回路的经典问题与同调谱图理论结合，提出显式迹公式，得到以亏格为参数的固定参数可处理算法，并开发对称性归约减少计算量。
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