        PHD Math. Sci.
Field of Specialization
Complex Analysis, Real Analysis, Partial differential equations.
Outline Activities
I am studying complex analysis of several complex variables.
It is important to understand the properties of holomorphic
functions on many kinds of domains in complex space.
In particular, the boundary behavior of these functions
can be represented in terms of the geometry of the
boundary of the respective domains.
I am interested in the class of pseudoconvex domains of finite
type. My viewpoint is from the singularity theory.
My students are also studying these thema from my viewpoints.
Research
Research Interests
• Complex analysis, Harmonic Analysis, Partial differential equations
keyword : holomorphic functions, asymptotic expansion, partial differential equation, complex geometry, Singularity theory
2000.10complex analysis.
Papers
 1 Joe Kamimoto, Newton polyhedra and order of contact on real hypersurfaces, J. Math. Soc. Japan, 2020.01. 2 Joe Kamimoto, Toshihiro Nose, Meromorphy of local zeta functions on some model cases, Journal of Functional Analysis, 278, 6, 1-25, 2020.04, It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as flat function, where and are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each case. Our results show new interesting phenomena in one of these cases. Actually, when , local zeta functions can be meromorphically extended to the half-plane and their poles on the half-plane are contained in the set.. 3 Joe Kamimoto, Toshihiro Nose, Nonpolar singularities of local zeta functions in some smooth case, Transactions of the American Mathematical Society, 10.1090/tran/7771, 372, 1, 661-676, 2019.01, It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this paper, the case of specific (nonreal analytic) smooth functions is precisely investigated. Indeed, asymptotic limits of the respective local zeta functions at some singularities in one direction are explicitly computed. Surprisingly, it follows from these behaviors that these local zeta functions have singularities different from poles.. 4 神本 丈, 野瀬敏洋, Asymptotic limit of oscillatory integrals with certain smooth phases, RIMS K\^oky\^uroku Bessatsu, 2017.09, 平坦な関数項を含む相関数について、振動積分の漸近挙動を正確に計算している．. 5 Joe Kamimoto, Toshihiro Nose, Newton polyhedra and weighted oscillatory integrals with smooth phases, Transactions of the American Mathematical Society, 10.1090/tran/6528, 368, 8, 5301-5361, 2016.01, In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals whose phases and weights are contained in a certain class of smooth functions, including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.. 6 Joe Kamimoto, Toshihiro Nose, Toric resolution of singularities in a certain class of C^{\infty} functions and asymptotic analysis of oscillatory integrals, J. Math. Soc. Univ. Tokyo , 23, 425-485, 2016.05, 実解析的という条件をはずした場合の滑らかな関数に関しては、その扱いが非常に困難になることはよく知られている。このような場合について、特異点解消という代数幾何の分野では、困難な結果を得た。これを応用して、単に滑らかな場合について、振動積分や局所ゼータ関数についての詳細な結果をえた。. 7 神本 丈, 野瀬敏洋, On meromorphic continuation of local zeta functions,, Proceedings of KSCV10.　F. Bracci et al. (eds.), Complex Analysis and Geometry, Springer , Proceedings in Mathematics and Statistics. , 144, 187-195, 2015.08, 局所ゼータ関数の解析接続に関して、最新の結果を報告している。. 8 Joe Kamimoto and Toshihiro Nose, On oscillatory integrals with C^{\infty} phases, Suriken Kokyuroku, Bessatsu, B40, 31-40, 2013.05, 相関数がなめらかな振動積分について、バルチェンコの結果を一般化した。. 9 Koji Cho, Joe Kamimoto, Toshihiro Nose, Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude, Journal of the Mathematical Society of Japan, 10.2969/jmsj/06520521, 65, 2, 521-562, 2013.08, The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that the amplitude has a zero at a critical point of the phase. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.. 10 Joe Kamimoto, Toshihiro Nose, Asymptotic analysis of weighted oscillatory integrals via Newton polyhedra, Proceedings of the 19th ICFIDCAA Hiroshima 2011, 3-12, 2013.06, 重み付き振動積分の漸近挙動をニュートン多面体の情報を用いて解析している。. 11 Koji Cho, Joe Kamimoto, Toshihiro Nose, Asymptotics of the Bergman function for semipositive holomorphic line bundles, Kyushu Journal of Mathematics, 10.2206/kyushujm.65.349, 65, 2, 349-382, 2011.11, In this paper, an asymptotic expansion of the Bergman function at a degenerate point is given for high powers of semipositive holomorphic line bundles on compact K̈ahler manifolds, whose Hermitian metrics have some kind of quasihomogeneous properties. In the sense of pointwise asymptotics, this expansion is a generalization of the expansion of Tian- Zelditch-Catlin-Lu in the positive line bundle case.. 12 趙　康治、神本　丈、野瀬敏洋, On the Bergman fuction for semipositive holomorphic line bundles, 数理解析研究所講究録, １６１３、ｐｐ１－５, 2008.09. 13 Bo Yong Chen, Joe Kamimoto, Takeo Ohsawa, Behavior of the Bergman kernel at infinity, Mathematische Zeitschrift, 10.1007/s00209-004-0676-6, 248, 4, 695-708, 2004.12, We give a precise decay rate of the Bergman kernel and metric at infinity on model domains, characterized in terms of certain convex polyhedron.. 14 Joe Kamimoto, Newton polyhedra and the Bergman kernel, Mathematische Zeitschrift, 10.1007/s00209-003-0554-7, 246, 3, 405-440, 2004.03, The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domainΩin ℂn+1, the Bergman kernel B(z) of Ωtakes the form near a boundary point p: B(Z) = Φ(w, ρ)/ρ2+2/dF (log(1/ρ))mF-1, where (w, ρ) is some polar coordinates on a nontangential cone Λ with apex at ρ and ρ means the distance from the boundary. Here Φ admits some asymptotic expansion with respect to the variables ρ1/m and log(1/ρ) as ρ → 0 on Λ The values of dF- > 0, mF ∈ ℤ + and m ∈ ℕ are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of Φ as ρ → 0 on Λ is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel.. 15 Joe Kamimoto, Non-analytic Bergman and Szegö kernels for weakly pseudoconvex tube domains in ℂ2, Mathematische Zeitschrift, 10.1007/PL00004843, 236, 3, 585-603, 2001.01, For any weakly pseudoconvex tube domain in ℂ2 with real analytic boundary, there exist points on the boundary off the diagonal where the Bergman kernel and the Szegö kernel fail to be real analytic.. 16 Joe Kamimoto, Haseo Ki, Young One Kim, On the multiplicities of the zeros of laguerre-pólya functions, Proceedings of the American Mathematical Society, 128, 1, 189-194, 2000.12, We show that all the zeros of the Fourier transforms of the functions exp(-x2m), m = 1,2,⋯, are real and simple. Then, using this result, we show that there are infinitely many polynomials p(x1,⋯, xn) such that for each (m1,⋯, mn) ∈ (ℕ \ {0})n the translates of the function p(x1,⋯, xn)exp (-∑j=1nxj2mj) generate L1(ℝn). Finally, we discuss the problem of finding the minimum number of monomials pα(x1,⋯, xn), α ∈ A, which have the property that the translates of the functions pα(x1,⋯, xn)exp(-∑j=1nxj2mj), α ∈ A, generate L1ℝn), for a given (m1,⋯,mn) ∈ (ℕ\{0})n.. 17 Joe Kamimoto, The Bergman kernel on weakly pseudoconvex tube domains in C2, Proceedings of the Japan Academy Series A: Mathematical Sciences, 10.3792/pjaa.75.12, 75, 2, 12-15, 1999.01. 18 Joe Kamimoto, On an integral of hardy and littlewood, Kyushu Journal of Mathematics, 10.2206/kyushujm.52.249, 52, 1, 249-263, 1998.01.
Presentations
 1 神本　丈, Asymptotic analysis of oscillatory integrals with degenerate phases, 偏微分方程式姫路研究集会, 2020.03. 2 神本　丈, Meromorphy of local zeta functions in smooth model cases, 研究集会「超局所解析と漸近解析」, 2019.11. 3 神本 丈, On analytic continuation of local zeta functions, 研究集会「New development of microlocal analysis and singular perturbation theory」, 2016.10, 局所ゼータ関数の解析接続に関して、現在までの研究および最新の研究の成果について、発表した。. 4 Newton　polyhedra and oscillatory integrals. 5 Newton polyhedra and asymptotic analysis of the Bergman kernel. 6 Joe NMN Kamimoto, On oscillatory integrals with smooth phases, Geometric Complex Analysis Tokyo 2012'', 2012.07, スムーズな相関数をもつ振動積分の漸近挙動をニュートン多面体の幾何学的な情報から導き出す様子を示した。. 7 Newton polyhedra and oscillatory integrals. 8 Newton polyhedra and oscillatory integrals. 9 Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude. 10 Newton polyhedra and asymptotic analysis of oscillatory integrals.
Educational
Educational Activities
My usual teaching activity is the following kinds of
lectures: 1. The fundamental lectures for
first or second undergraduate scientific students,
2. the lectures of special analysis for engineering
students,
3. the lectures of complex analysis for mathematical
students.
Moreover I have seminars with about 10 students. 