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Nobuhiro Nakamura

Publications and source records attributed to Nobuhiro Nakamura.

16 recordsLinked to original sources
arXiv2025

Notions of simple type for Bauer--Furuta invariants

By extending the notion of simple type for the Seiberg--Witten invariant of a 4-manifold, we introduce notions of BF blowup simple type and BF homogeneous type for the Bauer--Furuta invariant and study their applications. Specifically, we show that the existence of an immersed 2-sphere with a certain condition guarantees BF blowup simple type. As an application, we determine the Bauer--Furuta invariant of a 4-manifold obtained by a logarithmic transformation along a torus in a fishtail neighborhood. We also give constraints on gluing decompositions of 4-manifolds by using BF homogeneous type. To prove these results, we also give gluing formulae and an immersed adjunction inequality for Bauer--Furuta invariants.

math.GT↗
arXiv2024

Extracting effective solutions hidden in large language models via generated comprehensive specialists: case studies in developing electronic devices

Recently, many studies have increasingly explored the use of large language models (LLMs) to generate research ideas and scientific hypotheses. However, real-world research and development often require solving complex, interdisciplinary challenges where solutions may not be readily found through existing knowledge related to the problem. Therefore, it is desirable to leverage the vast, comprehensive knowledge of LLMs to generate effective, breakthrough solutions by integrating various perspectives from other disciplines. Here, we propose SELLM (Solution Enumeration via comprehensive List and LLM), a framework leveraging LLMs and structured guidance using MECE (Mutually Exclusive, Collectively Exhaustive) principles, such as International Patent Classification (IPC) and the periodic table of elements. SELLM systematically constructs comprehensive expert agents from the list to generate cross-disciplinary and effective solutions. To evaluate SELLM's practicality, we applied it to two challenges: improving light extraction in organic light-emitting diode (OLED) lighting and developing electrodes for next-generation memory materials. The results demonstrate that SELLM significantly facilitates the generation of effective solutions compared to cases without specific customization or effort, showcasing the potential of SELLM to enable LLMs to generate effective solutions even for challenging problems.

cs.CL↗
arXiv2023

Upper bounds for virtual dimensions of Seiberg-Witten moduli spaces

Given a closed four-manifold with $b_1=0$ and a prime number $p$, we prove that for any mod $p^r$ basic class, the virtual dimension of the Seiberg-Witten moduli space is bounded above by $2r(p-1)-2$ under some conditions on $r$ and $b_2^+$. As an application, we obtain adjunction inequalities for embedded surfaces with negative self-intersection number.

math.GT↗
arXiv2021

A note on exotic families of 4-manifolds

We present a pair of smooth fiber bundles over the circle with a common $4$-dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber bundles; (3) they are not isomorphic to each other as smooth fiber bundles. In particular, we exhibit an example with non-simply-connected fiber.

math.GT↗
arXiv2021

The simple type conjecture for mod 2 Seiberg-Witten invariants

We prove that, under a simple condition on the cohomology ring, every closed 4-manifold has mod 2 Seiberg-Witten simple type. This result shows that there exists a large class of topological 4-manifolds such that all smooth structures have mod 2 simple type, and yet some have non-vanishing (mod 2) Seiberg-Witten invariants. As corollaries, we obtain adjunction inequalities and show that, under a mild topological condition, every geometrically simply connected closed 4-manifold has the vanishing mod 2 Seiberg-Witten invariant for at least one orientation.

math.GT↗
arXiv2020

Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds

We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of 4-manifolds $M$ for which the inclusion maps $\operatorname{Diff}(M) \hookrightarrow \operatorname{Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin 4-manifolds.

math.GT↗
arXiv2020

Real structures and the $\mathrm{Pin}^-(2)$-monopole equations

We investigate the $\mathrm{Pin}^-(2)$-monopole invariants of symplectic $4$-manifolds and Kähler surfaces with real structures. We prove the nonvanishing theorem for real symplectic $4$-manifolds which is an analogue of Taubes' nonvanishing theorem of the Seiberg-Witten invariants for symplectic $4$-manifolds. Furthermore, the Kobayashi-Hitchin type correspondence for real Kähler surfaces is given.

math.DG↗
arXiv2020

Constraints on families of smooth 4-manifolds from $\mathrm{Pin}^{-}(2)$-monopole

Using the Seiberg-Witten monopole equations, Baraglia recently proved that for most of simply-connected closed smooth $4$-manifolds $X$, the inclusions $\mathrm{Diff}(X) \hookrightarrow \mathrm{Homeo}(X)$ are not weak homotopy equivalences. In this paper, we generalize Baraglia's result using the $\mathrm{Pin}^{-}(2)$-monopole equations instead. We also give new examples of $4$-manifolds $X$ for which $π_{0}(\mathrm{Diff}(X)) \to π_{0}(\mathrm{Homeo}(X))$ are not surjections.

math.GT↗
arXiv2015

Yamabe Invariants and the $\mathrm{Pin}^-(2)$-monopole Equations

We compute the Yamabe invariants for a new infinite class of closed $4$-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations, the $\mathrm{Pin}^-(2)$-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the normalised Ricci flow with uniformly bounded scalar curvature.

math.DG↗
arXiv2015

$\mathrm{Pin}^-(2)$-monopole invariants

We introduce a diffeomorphism invariant of $4$-manifolds, the $\mathrm{Pin}^-(2)$-monopole invariant, defined by using the $\mathrm{Pin}^-(2)$-monopole equations. We compute the invariants of several $4$-manifolds, and prove gluing formulae. By using the invariants, we construct exotic smooth structures on the connected sum of an elliptic surface $E(n)$ with arbitrary number of the $4$-manifolds of the form of $S^2\timesΣ$ or $S^1\times Y$ where $Σ$ is a compact Riemann surface with positive genus and $Y$ is a closed $3$-manifold. As another application, we give an estimate of the genus of surfaces embedded in a $4$-manifold $X$ representing a class $α\in H_2(X;l)$, where $l$ is a local coefficient on $X$.

math.GT↗
arXiv2011

Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds

We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and give its applications to intersection forms with local coefficients of 4-manifolds. The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients. The second is a local coefficient version of Furuta's 10/8-inequality. As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

math.GT↗
arXiv2009

Smoothability of Z\times Z-actions on 4-manifolds

We construct a nonsmoothable Z\times Z-action on the connected sum of an Enriques surface and S^2\times S^2, such that each of generators is smoothable. We also construct a nonsmoothable self-homeomorphism on an Enriques surface.

math.GT↗
arXiv2008

Bauer-Furuta invariants under Z_2-actions

S.Bauer and M.Furuta defined a stable cohomotopy refinement of the Seiberg-Witten invariants. In this paper, we prove a vanishing theorem of Bauer-Furuta invariants for 4-manifolds with smooth Z/2-actions. As an application, we give a constraint on smooth Z/2-actions on homotopy K3#K3, and construct a nonsmoothable locally linear Z/2-action on K3#K3. We also construct a nonsmoothable locally linear Z/2-action on $K3$.

math.GT↗
arXiv2008

Nonsmoothable group actions on elliptic surfaces

Let G be a cyclic group of order 3, 5 or 7, and X=E(n) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is nonsmoothable with respect to infinitely many smooth structures on X. This extends the main result of our previous paper.

math.GT↗
arXiv2006

Pseudofree Z/3-actions on K3 surfaces

In this paper, we give a weak classification of locally linear pseudofree actions of the cyclic group of order 3 on a $K3$ surface, and prove the existence of such an action which can not be realized as a smooth action on the standard smooth $K3$ surface.

math.GT↗