Geometric and topological rigidity for compact submanifolds of odd dimension

We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn(n 5)is an odd-dimensional compact submanifold with parallel mean curvature in Sn+p,and if RicM(n-2-1n)(1+H2)and Hδn,whereδn is an explicit positive constant depending only on n,then M is a totally umbilical sphere.Here H is the mean curvature of M.Moreover,we prove that if Mn(n 5)is an odd-dimensional compact submanifold in the space form Fn+p(c)with c 0,and if RicM(n-2-εn)(c+H2),whereεn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere.     

Complex projective towers and their cohomological rigidity up to dimension six

A complex projective tower, or simply a ?P-tower, is an iterated complex projective fibration starting from a point. In this paper we classify all six-dimensional ?P-towers up to diffeomorphism, and as a consequence we show that all such manifolds are cohomologically rigid, i.e., they are completely determined up to diffeomorphism by their cohomology rings.     

K-theoretic invariants for Floer homology

This paper defines two K-theoretic invariants, Wh ₁ and Wh ₂, for individual and one-parameter families of Floer chain complexes. The chain complexes are generated by intersection points of two Lagrangian submanifolds of a symplectic manifold, and the boundary maps are determined by holomorphic curves connecting pairs of intersection points. The paper proves that Wh ₁ and Wh ₂ do not depend on the choice of almost complex structures and are invariant under Hamiltonian deformations. The proof of this invariance uses properties of holomorphic curves, parametric gluing theorems, and a stabilization process. Submitted: April 2001, Revised: December 2001, Final version: February 2002.     

A K-theoretic note on geometric quantization

We give a purely K-theoretic proof of a case of the “quantization commutes with reduction” result, conjectured by Guillemin and Sternberg and proved by Meinrenken and Vergne. We show that the quantization is simply a pushforward in K-theory, and use Lerman's symplectic cutting and the localization theorem in equivariant K-theory to prove that quantization commutes with reduction. The case where G=S ¹ and the action is free on the zero level set of the moment map is addressed. Received: 9 March 1999     

Neutral-fermionic presentation of the K-theoretic Q-function

Journal of Algebraic Combinatorics - We show a new neutral-fermionic presentation of Ikeda-Naruse’s K-theoretic Q-functions, which represent a Schubert class in the K-theory of coherent...     

Discrete groups of slow subgroup growth

It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least $n^{\frac{{\log n}}{{\log \log n}}} $ . In this paper we prove the existence of a finitely generated group whose subgroup growth is of type $n^{\frac{{\log n}}{{(\log \log n)^2 }}} $ . This is the slowest non-polynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth typen ^logn is also realized. The proofs involve analysis of the subgroup structure of finite alternating groups and finite simple groups in general. For example, we show there is an absolute constantc such that, ifT is any finite simple group, thenT has at mostn ^{c logn} subgroups of indexn.     

Instanton counting on blowup. II. K-theoretic partition function

We study Nekrasov's deformed partition function $Z(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta)$ of 5-dimensional supersymmetric Yang-Mills theory compactified on a circle. Mathematically it is the generating function of the characters of the coordinate rings of the moduli spaces of instantons on $\mathbb R^4$. We show that it satisfies a system of functional equations, called blowup equations, whose solution is unique. As applications, we prove (a) $F(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta) = \varepsilon_1\varepsilon_2 \log Z(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta)$ is regular at $\varepsilon_1 = \varepsilon_2 = 0$ (a part of Nekrasov's conjecture), and (b) the genus $1$ parts, which are first several Taylor coefficients of $F(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta)$, are written explicitly in terms of $\tau = d^2 F(0,0,\vec{a};\mathfrak q,\boldsymbol\beta)/da^2$ in rank $2$ case.     

Fractal dimension evolution and spatial replacement dynamics of urban growth

This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to interpret the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann’s equation. For the normalized data, Boltzmann’s equation is just equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is thus made in this article that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos mirror a process of complex replacement.     

Julia lines of entire functions of slow growth

We obtain sufficient conditions under which the Julia lines of entire functions of slow growth do not have finite exceptional values. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 829–834, June, 2006.     

Laplace transform of generalized functions of slow growth

We present certain special sections of the theory of functions of several variables and of the theory of generalized functions, which are the basic mathematical tools for certain directions in modern quantum field theory. In passing we study the properties of certain special algebras of analytic functions. A considerable part of the results in the article are due to the author.     

Resistance dimension,random walk dimension and fractal dimension

In this paper, we generalize a theorem due to Telcs concerning random walks on infinite graphs, which describes the relation of random walk dimension, fractal dimension and resistance dimension. Moreover, we obtain a reasonable upper bound and lower bound on the hitting time in terms of resistance for some nice graphs. In fact, the conditions given in this paper are weaker than those obtained by A. Telcs.Partly supported by National Natural Science Foundation and State Educational Committee of China.     

On K-theoretic invariants of semigroup C*-algebras attached to number fields

We show that semigroup C*-algebras attached to ax+b$ax+b$-semigroups over rings of integers determine number fields up to arithmetic equivalence, under the assumption that the number fields have the same number of roots of unity. For finite Galois extensions, this means that the semigroup C*-algebras are isomorphic if and only if the number fields are isomorphic.     

On elliptic equations and systems with critical growth in dimension two

We consider nonlinear elliptic equations of the form −Δu = g(u) in Ω, u = 0 on ∂Ω, and Hamiltonian-type systems of the form −Δu = g(v) in Ω, −Δv = f(u) in Ω, u = 0 and v = 0 on ∂Ω, where Ω is a bounded domain in ℝ² and f, g ∈ C(ℝ) are superlinear nonlinearities. In two dimensions the maximal growth (= critical growth) of f and g (such that the problem can be treated variationally) is of exponential type, given by Pohozaev-Trudinger-type inequalities. We discuss existence and nonexistence results related to the critical growth for the equation and the system. A natural framework for such equations and systems is given by Sobolev spaces, which provide in most cases an adequate answer concerning the maximal growth involved. However, we will see that for the system in dimension 2, the Sobolev embeddings are not sufficiently fine to capture the true maximal growths. We will show that working in Lorentz spaces gives better results. Dedicated to Professor S. Nikol’skii on the occasion of his 100th birthday     

Finitistic dimension and restricted injective dimension

We study the relations between finitistic dimensions and restricted injective dimensions. Let R be a ring and T a left R-module with A = End _R T. If _R T is selforthogonal, then we show that rid(T _A ) ? findim(A _A ) ? findim( _R T) + rid(T _A ). Moreover, if R is a left noetherian ring and T is a finitely generated left R-module with finite injective dimension, then rid(T _A ) ? findim(A _A ) ? fin.inj.dim( _R R)+rid(T _A ). Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.     

Behavior on rays of Dirichlet series of slow growth

The topic under consideration is the behavior on the rays arg z=, ¦¦ < /2, of an entire function F represented by a Dirichlet series, absolutely convergent in the whole plane, with exponents _n > 0 such that n= , n +, where q(r) is a proximate order and Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1603–1613, December, 1991.