Updated on 2021/12/06

MONDEN Naoyuki

Organization
Faculty of Natural Science and Technology Associate Professor

### Degree

• 博士(理学) ( 大阪大学 )

• 位相幾何学

### Research Areas

• Natural Science / Geometry

### Professional Memberships

• 日本数学会

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### Committee Memberships

• N-KOOKセミナー   世話人

2020.1

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• 研究集会「ひねる代数～Hurwitz actionとその周辺～」   世話人

2019.10 - 2020.1

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• 研究集会「ハンドル体結び目とその周辺10・Hurwitz action 7」   世話人

2017.7 - 2017.10

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• 研究集会「Hurwitz action 6」   世話人

2016.7 - 2016.10

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• 研究集会「Hurwitz action 5」   世話人

2015.10 - 2016.1

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• 研究集会「Hurwitz action～HINERU～」   世話人

2014.10 - 2015.1

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• 研究集会「Hurwitz action～ひねる代数～」   世話人

2013.10 - 2014.1

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• 研究集会「Hurwitz actionとその周辺」   世話人

2012.10 - 2013.1

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• 研究集会「Hurwitz action」   世話人

2011.10 - 2012.1

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• 関西低次元トポロジー若手セミナー   世話人

2009.9 - 2013.3

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### Papers

• THE EXISTENCE OF AN INDECOMPOSABLE MINIMAL GENUS TWO LEFSCHETZ FIBRATION Reviewed

Anar Akhmedov, Naoyuki Monden

OSAKA JOURNAL OF MATHEMATICS   58 ( 1 )   29 - 36   2021.1

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：OSAKA JOURNAL OF MATHEMATICS

It was shown by Usher that any fiber sum of Lefschetz fibrations over S-2 is minimal, which was conjectured by Stipsicz. We prove that the converse does not hold by showing that there exists a genus-2 indecomposable minimal Lefschetz fibration (IMLF for short).

• Genus 2 Lefschetz fibrations with b(2)(+)=1 and c(1)(2)=1, 2 Reviewed

Anar Akhmedov, Naoyuki Monden

KYOTO JOURNAL OF MATHEMATICS   60 ( 4 )   1419 - 1451   2020.12

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：DUKE UNIV PRESS

In this article, we construct a family of genus 2 Lefschetz fibrations f(n) : X-theta n -> S-2 with e(X-theta n) = 11, b(2)(+) (X-theta n) = 1, and c(1)(2) (X-theta n) = 1 by applying a single lantern substitution to the twisted fiber sums of Matsumoto's genus 2 Lefschetz fibration over S-2. Moreover, we compute the fundamental group of X-theta n and show that it is isomorphic to the trivial group if n = -3 or -1, Z if n = -2, and Z(vertical bar n+2 vertical bar) for all integers n not equal -3, -2, -1. Also, we prove that our fibrations admit -2 section, that their total spaces are symplectically minimal, and that they have symplectic Kodaira dimension kappa = 2. In addition, using techniques developed over the past decade with other authors, we also construct the genus 2 Lefschetz fibrations over S-2 with c(1)(2) = 1, 2 and chi = 1 via the fiber sums of Matsumoto's and Xiao's genus 2 Lefschetz fibrations, and present some applications in constructing exotic smooth structures on small 4-manifolds with b(2)(+) = 1 and b(2)(+) = 3.

• Signatures of surface bundles and scl of a Dehn twist Reviewed

Naoyuki Monden

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES   100 ( 3 )   957 - 986   2019.12

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY

The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which depends on the signature. This provides a new upper bound on the minimal base genus for fixed signature and fiber genus. The second example gives a new asymptotic upper bound for this number in the case that fiber genus is odd. The third example has a small Euler characteristic. The last is a non-holomorphic example. The second aim is to improve upper bounds for stable commutator lengths of Dehn twists by giving factorizations of powers of Dehn twists as products of commutators. One of the factorizations is used to construct the second examples of surface bundles. As a corollary, we see that there is a gap between the stable commutator length of the Dehn twist along a non-separating curve in the mapping class group and that in the hyperelliptic mapping class group if the genus of the surface is greater than or equal to 8.

• NONHOLOMORPHIC LEFSCHETZ FIBRATIONS WITH (-1)-SECTIONS Reviewed

Noriyuki Hamada, Ryoma Kobayashi, Naoyuki Monden

PACIFIC JOURNAL OF MATHEMATICS   298 ( 2 )   375 - 398   2019.2

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PACIFIC JOURNAL MATHEMATICS

We construct two types of nonholomorphic Lefschetz fibrations over S-2 with (-1)-sections-hence, they are fiber sum indecomposable-by giving the corresponding positive relators. One type of the two does not satisfy the slope inequality (a necessary condition for a fibration to be holomorphic) and has a simply connected total space, and the other has a total space that cannot admit any complex structure in the first place. These give an alternative existence proof for nonholomorphic Lefschetz pencils without Donaldson's theorem.

• On stable commutator lengths of Dehn twists along separating curves Reviewed

Naoyuki Monden, Kazuya Yoshihara

JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS   26 ( 10 )   2017.9

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD

We give new upper bounds on the stable commutator lengths of Dehn twists along separating curves in the mapping class group of a closed oriented surface. The estimates of these upper bounds are O(1/g), where g is the genus of the surface.

• Positive factorizations of mapping classes Reviewed

R. Inanc Baykur, Naoyuki Monden, Jeremy Van Horn-Morris

ALGEBRAIC AND GEOMETRIC TOPOLOGY   17 ( 3 )   1527 - 1555   2017

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：GEOMETRY & TOPOLOGY PUBLICATIONS

In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers g and n, we tell whether or not there exist genus-g Lefschetz pencils with n base points, and if there are, what the maximal Euler characteristic is whenever it is bounded above. We observe that only symplectic 4-manifolds of general type can attain arbitrarily large topology regardless of the genus and the number of base points of Lefschetz pencils on them.

• Constructing Lefschetz fibrations via daisy substitutions Reviewed

Anar Akhmedov, Naoyuki Monden

KYOTO JOURNAL OF MATHEMATICS   56 ( 3 )   501 - 529   2016.9

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：DUKE UNIV PRESS

We construct new families of nonhyperelliptic Lefschetz fibrations by applying the daisy substitutions to the families of words (c(1)c(2) . . c(2g-1)c(2g)c(2g+1)(2)c(2g) . c(2g-1) . . . c(2)c(1))(2) = 1, (c(1)c(2) . . . c(2g)c(2g+1))(2g+2) = 1, and (c(1)c(2) . . c(2g-1)c(2g))(2(2g+1)) = 1 in the mapping class group Gamma(g) of the closed orientable surface of genus g, and we study the sections of these Lefschetz fibrations. Furthermore, we show that the total spaces Of some of these Lefschetz fibrations are irreducible exotic 4-manifolds, and we compute their Seiberg Witten invariants. By applying the knot surgery to the family of Lefschetz fibrations obtained from the word (c(1)c(2) . . . c(2g)c(2g+1))(2g+2) = 1 via daisy substitutions, we also construct an infinite family of pairwise nondiffeomorphic irreducible symplectic and nonsymplectic 4-manifolds homeomorphic to (g(2) - g + 1)CP2 #(3g(2) - g (k - 3) + 2k + 3)(CP) over bar (2) for any g >= 3 and k= 2,...,g +1.

• LEFSCHETZ PENCILS AND FINITELY PRESENTED GROUPS Reviewed

Ryoma Kobayashi, Naoyuki Monden

PACIFIC JOURNAL OF MATHEMATICS   282 ( 2 )   359 - 388   2016.6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PACIFIC JOURNAL MATHEMATICS

From the works of Gompf and Donaldson, it is known that every finitely presented group can be realized as the fundamental group of the total space of a Lefschetz pencil. We give an alternative proof of this fact by providing the monodromy explicitly. In the proof, we give an alternative construction of the monodromy of Gurtas' fibration and a lift of that to the mapping class group of a surface with two boundary components.

• ON STABLE COMMUTATOR LENGTH IN HYPERELLIPTIC MAPPING CLASS GROUPS Reviewed

Danny Calegari, Naoyuki Monden, Masatoshi Sato

PACIFIC JOURNAL OF MATHEMATICS   272 ( 2 )   323 - 351   2014.12

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PACIFIC JOURNAL MATHEMATICS

We give a new upper bound on the stable commutator length of Dehn twists in hyperelliptic mapping class groups and determine the stable commutator length of some elements. We also calculate values and the defects of homogeneous quasimorphisms derived from omega-signatures and show that they are linearly independent in the mapping class groups of pointed 2-spheres when the number of points is small.

• LEFSCHETZ FIBRATIONS WITH SMALL SLOPE Reviewed

Naoyuki Monden

PACIFIC JOURNAL OF MATHEMATICS   267 ( 1 )   243 - 256   2014.1

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PACIFIC JOURNAL MATHEMATICS

We construct Lefschetz fibrations over S-2 which do not satisfy the slope inequality. This disproves a conjecture of Hain.

• ON ROOTS OF DEHN TWISTS Reviewed

Naoyuki Monden

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS   44 ( 3 )   987 - 1001   2014

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ROCKY MT MATH CONSORTIUM

Margalit and Schleimer [4] discovered a nontrivial root of the Dehn twist about a nonseparating curve on a closed oriented connected surface. We give a complete set of conjugacy invariants for such a root by using a classification theorem of Matsumoto and Montesinos [5, 6] for pseudo-periodic maps of negative twists. As an application, we determine the range of degree for roots of a Dehn twist.

• SECTIONS OF SURFACE BUNDLES AND LEFSCHETZ FIBRATIONS Reviewed

R. Inanc Baykur, Mustafa Korkmaz, Naoyuki Monden

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY   365 ( 11 )   5999 - 6016   2013.11

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC

We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the base genus h are positive, we prove that the adjunction bound 2h-2 is the only universal bound on the self-intersection number of a section of any such genus g bundle and fibration. As a side result, in the mapping class group of a surface with boundary, we calculate the precise value of the commutator lengths of all powers of a Dehn twist about a boundary component, concluding that the stable commutator length of such a Dehn twist is 1/2. We furthermore prove that there is no upper bound on the number of critical points of genus-g Lefschetz fibrations over surfaces with positive genera admitting sections of maximal self-intersection, for g >= 2.

• Degree of roots of disk twists on 3-dimensional handlebodies Reviewed

Susumu Hirose, Naoyuki Monden

GEOMETRIAE DEDICATA   164 ( 1 )   73 - 82   2013.6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER

Margalit and Schleimer (Geom Topol 13(3):1495-1497, 2009) discovered a nontrivial root of the Dehn twist about a nonseparating curve on a closed oriented connected surface. McCullough and Rajeevsarathy (Geom Dedicata 151(1):397-409, 2011) and Monden (Rocky Mt J Math, to appear) obtained the evaluation of the degrees of roots of Dehn twists. In this paper, we discuss existence and degrees of homeomorphisms whose power is equal to disk twist about a nonseparating disk in the mapping class group of the 3-dimensional handlebody.

• Generating the mapping class group by torsion elements of small order Reviewed

Naoyuki Monden

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY   154 ( 1 )   41 - 62   2013.1

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS

We show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated by 3 elements of order 3. Moreover, we show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated and by 4 elements of order 4.

• On Minimal Number of Singular Fibers in a Genus-2 Lefschetz Fibration Reviewed

Naoyuki Monden

TOKYO JOURNAL OF MATHEMATICS   35 ( 2 )   483 - 490   2012.12

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TOKYO JOURNAL MATHEMATICS EDITORIAL OFFICE ACAD CENTER

We show that the minimal number of singular fibers in a genus-2 Lefschetz fibration over a closed surface of genus h is equal to 5 if h >= 3, 5 or 6 if h = 2 and 6 or 7 if h = 1.

• On upper bounds on stable commutator lengths in mapping class groups Reviewed

Naoyuki Monden

TOPOLOGY AND ITS APPLICATIONS   159 ( 4 )   1085 - 1091   2012.3

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV

We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating curve on an oriented closed surface of genus 2 are not equal to each other. (C) 2011 Elsevier B.V. All rights reserved.

• Generating the Mapping Class Group of a Punctured Surface by Involutions Reviewed

Naoyuki Monden

TOKYO JOURNAL OF MATHEMATICS   34 ( 2 )   303 - 312   2011.12

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TOKYO JOURNAL MATHEMATICS EDITORIAL OFFICE ACAD CENTER

Let Sigma(g,b) denote a closed oriented surface of genus g with b punctures and let Mod(Sigma(g,b)) denote its mapping class group. Kassabov showed that Mod(Sigma(g,b)) is generated by 4 involutions if g > 7 or g = 7 and b is even, 5 involutions if g > 5 or g = 5 and b is even, and 6 involutions if g > 3 or g = 3 and b is even. We proved that Mod(Sigma(g,b)) is generated by 4 involutions if g = 7 and b is odd, and 5 involutions if g = 5 and b is odd.

• The mapping class group of a punctured surface is generated by three elements Reviewed

Naoyuki Monden

HIROSHIMA MATHEMATICAL JOURNAL   41 ( 1 )   1 - 9   2011.3

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HIROSHIMA UNIV, GRAD SCH SCI

Let Mod(Sigma(g,p)) be the mapping class group of a closed oriented surface Sigma(g,p) of genus g >= 1 with p punctures. Wajnryb proved that Mod(Sigma(g,0)) is generated by two elements. Korkmaz proved that one of these generators may be taken to be a Dehn twist. Korkmaz also proved the same result in the case of Mod(Sigma(g,1)). For p >= 2, we prove that Mod(Sigma(g,p)) is generated by three elements.

### MISC

• Naoyuki Monden

2021.3

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Let $\Sigma_{g,p}$ be a oriented connected surface of genus $g$ with $p$
punctures. We denote by $\mathcal{M}_{g,p}$ and $\mathcal{M}_{g,p}^\pm$ the
mapping class group and the extended mapping class group of $\Sigma_{g,p}$,
respectively. In this paper, we show that $\mathcal{M}_{g,p}$ and
$\mathcal{M}_{g,p}^\pm$ are generated by two element for $g\geq 3$ and $p\geq 0$.

• R. Inanc Baykur, Kenta Hayano, Naoyuki Monden

2019.3

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We study a symplectic surgery operation we call unchaining, which effectively
reduces the second Betti number and the symplectic Kodaira dimension at the
same time. Using unchaining, we give novel constructions of symplectic
Calabi-Yau surfaces from complex surfaces of general type, as well as from
rational and ruled surfaces via the natural inverse of this operation.
Combining the unchaining surgery with others, which all correspond to certain
monodromy substitutions for Lefschetz pencils, we provide further applications,
such as a complete resolution of a conjecture of Stipsicz on the existence of
exceptional sections in Lefschetz fibrations, new constructions of exotic
symplectic 4-manifolds, and inequivalent pencils of the same genera and the
same number of base points on families of symplectic 4-manifolds. Meanwhile, we
give a handy criterion for determining from the monodromy of a pencil whether
its total space is spin or not.

• On the geography of Lefschetz fibrations Reviewed

Naoyuki Monden

II   297 - 325   2015.2

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• LEFSCHETZ PENCILS AND FINITELY PRESENTED GROUPS (The Topology and the Algebraic Structures of Transformation Groups)

小林 竜馬, 門田 直之

数理解析研究所講究録   ( 1922 )   84 - 89   2014.10

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Language：Japanese   Publisher：京都大学数理解析研究所

• 様々な非正則Lefschetz fibrationとそれらの構成について

門田直之

第60回 トポロジーシンポジウム 講演集   31 - 40   2013.8

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• Generating the full mapping class group by involutions (Geometry of transformation groups and related topics)

Monden Naoyuki

RIMS Kokyuroku   ( 1612 )   135 - 145   2008.9

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Language：English   Publisher：Kyoto University

### Presentations

• 写像類群の最小の生成系について Invited

門田直之

早稲田双曲幾何幾何学的群論セミナー  2021.11.26

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Presentation type：Oral presentation (invited, special)

• 点付き写像類群の最小の生成系について Invited

門田直之

東京女子大学トポロジーセミナー  2021.10.9

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Presentation type：Oral presentation (invited, special)

• On minimal generating sets for the mapping class group of a punctured surface

門田直之

日本数学会2021年度秋季総合分科会  2021.9.14

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Presentation type：Oral presentation (general)

• 点付き写像類群の最小の生成系について

門田直之

研究集会「拡大KOOKセミナー2021」  2021.9.1

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Presentation type：Oral presentation (general)

• 点付き写像類群の最小の生成系について Invited

門田直之

研究集会「リーマン面に関連する位相幾何学」  2021.8.16

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Presentation type：Oral presentation (invited, special)

• 極小シンプレクティック4次元多様体の地誌学と球面上のLefschetz fibration Invited

門田直之

4-Dimensional Topology and Gauge Theory  2020.1.28

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Presentation type：Oral presentation (invited, special)

• ファイバー和分解不可能で極小な種数2のLefschetz fibrationの存在 Invited

門田直之

研究集会「リーマン面に関連する位相幾何学」  2019.9.9

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Presentation type：Oral presentation (invited, special)

• ファイバー和分解不可能で極小な Lefschetz fibration について Invited

門田直之

研究集会「特異点論とトポロジー」  2019.7.31

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Presentation type：Oral presentation (invited, special)

• A strategy to reduce the number of Johnson's generators Invited

Naoyuki Monden

Lefschetz Pencils and Low dimensional Topology,  2019.6.1

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Language：English   Presentation type：Oral presentation (invited, special)

• ファイバー和分解不可能で極小な種数2のLefschetz fibrationの存在 Invited

門田直之

トポロジー金曜セミナー  2019.2.4

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Presentation type：Oral presentation (invited, special)

• ファイバー和分解不可能で極小な種数2のLefschetz fibrationの存在 Invited

門田直之

４次元トポロジーセミナー  2018.12.7

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Presentation type：Oral presentation (invited, special)

• 極小シンプレクティック4次元多様体の地誌学と球面上のLefschetz fibration Invited

門田直之

東京理科大学 特異点・トポロジーセミナー  2018.7.12

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Presentation type：Oral presentation (invited, special)

• 写像類群における交換子長・安定交換子長について Invited

門田直之

談話会  2018.5.11

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Presentation type：Oral presentation (invited, special)

• Signatures of surface bundles over surfaces Invited

Naoyuki Monden

Low Dimensional Topology and Gauge Theory  2017.8.7

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Language：English   Presentation type：Oral presentation (invited, special)

• 曲面上の曲面束の符号数について Invited

門田直之

4次元トポロジーセミナー  2017.6.9

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Presentation type：Oral presentation (invited, special)

• 4次元シンプレクティック多様体の地誌学について

門田直之

琉球結び目セミナー  2016.12.17

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Presentation type：Oral presentation (general)

• (-1)-切断を持つ非正則Lefschetz fibration について Invited

門田直之

トポロジー金曜セミナー  2016.10.7

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Presentation type：Oral presentation (invited, special)

• 曲面上の曲面束の符号数について

門田直之

研究集会「拡大KOOKセミナー2016」  2016.8.24

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Presentation type：Oral presentation (general)

• 写像類群と低次元トポロジー Invited

門田直之

トポロジー新人セミナー  2016.8.9

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Presentation type：Oral presentation (invited, special)

• On stable commutator lengths of Dehn twists Invited

Naoyuki Monden

Differential Geometry and Symplectic Topology Seminar  2016.4.19

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Language：English   Presentation type：Oral presentation (invited, special)

• Stable commutator length of Dehn twists and the signatures of surface bundles Invited

Naoyuki Monden

Special Session on Low Dimensional and Symplectic Topology  2016.4.16

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Language：English   Presentation type：Oral presentation (invited, special)

• Positive factorizations of mapping classes Invited

門田直之

微分トポロジー16  2016.3.20

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Presentation type：Oral presentation (invited, special)

• Upper bounds on stable commutator lengths of Dehn twists} Invited

Naoyuki Monden

The 11-th East Asian School of Knots and Related Topics  2016.1.26

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Language：English   Presentation type：Oral presentation (general)

• Genus-2 Lefschetz fibrations with $b^+_2=1$ and $c^2_1=1,2$ Invited

門田直之

Hurwitz action  2016.1.10

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Presentation type：Oral presentation (invited, special)

• 4次元トポロジーとLefschetz fibrationの地誌学 Invited

門田直之

東京理科大学談話会  2015.12.18

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Presentation type：Oral presentation (invited, special)

• 写像類群における安定交換子長 Invited

門田直之

北海道大学幾何学コロキウム  2015.12.11

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Presentation type：Oral presentation (invited, special)

• 写像類群における安定交換子長 Invited

門田直之

N-KOOKセミナー  2015.10.31

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Presentation type：Oral presentation (invited, special)

• 4次元トポロジーとLefschetz fibrationの地誌学 Invited

門田直之

談話会  2015.8.8

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Presentation type：Oral presentation (invited, special)

• Positive factorizations of mapping classes Invited

門田直之

東工大トポロジーセミナー  2015.6.3

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Presentation type：Oral presentation (invited, special)

• Positive factorizations of mapping classes

R. Inanc Baykur, 門田直之, Jeremy Van Horn-Morris

日本数学会2015年度年会トポロジー分科会一般講演  2015.3.23

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Presentation type：Oral presentation (general)

• Introduction to Hurwitz action and positive factorizations of mapping classes Invited

門田直之

Hurwitz action  2015.1.10

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Presentation type：Oral presentation (invited, special)

• Twisted substitutions and fundamental groups of Lefschetz fibraitons Invited

Naoyuki Monden

Workshop on Topology and Invariants of 4-Manifolds  2014.8.26

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Language：English   Presentation type：Oral presentation (invited, special)

• レフシェッツ・ペンシルと有限表示群 Invited

門田直之

研究集会「変換群の位相幾何と代数構造」  2014.5.28

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Presentation type：Oral presentation (invited, special)

• Lefschetz pencils and finitely presented groups Invited

門田直之

N-KOOKセミナー  2014.5.17

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Presentation type：Oral presentation (invited, special)

• Non-holomorphic Lefschetz fibrations with (-1)-sections Invited

Naoyuki Monden

Friday Seminar on Knot Theory  2014.4.25

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Language：English   Presentation type：Oral presentation (invited, special)

• 代表的なレフシェッツ束空間の有理ブローダウン Invited

門田直之

研究集会「写像類群における関係とレフシェッツ束空間」  2014.3.19

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Presentation type：Oral presentation (invited, special)

• The geography problem of Lefschetz fibrations Invited

門田直之

トポロジー火曜セミナー  2013.10.1

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Presentation type：Oral presentation (invited, special)

• 様々な非正則Lefschetz fibrationとそれらの構成について Invited

第60回トポロジーシンポジウム  2013.8.6

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Presentation type：Oral presentation (invited, special)

• Dehn twistの安定交換子長について Invited

門田直之

東京女子大学トポロジーセミナー  2013.4.13

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Presentation type：Oral presentation (invited, special)

• Lefschetz fibrations with small slope

門田直之

日本数学会2013年度年会トポロジー分科会一般講演  2013.3.21

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Presentation type：Oral presentation (general)

• On stable commutator length of a Dehn twist Invited

Naoyuki Monden

Friday Seminar on Knot Theory  2013.2.1

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Language：English   Presentation type：Oral presentation (invited, special)

• Dehn twistの安定交換子長について

門田直之

結び目の数学V  2012.12.15

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Presentation type：Oral presentation (general)

• 4-manifolds and stable commutator length in mapping class groups Invited

門田直之

微分トポロジーセミナー  2012.11.27

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Presentation type：Oral presentation (invited, special)

• Lefschetz fibrations with small slope Invited

門田直之

リーマン面に関連する位相幾何学2012  2012.9.4

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Presentation type：Oral presentation (invited, special)

• Stable commutator length in mapping class groups, (I), (II) Invited

門田直之

鳥羽微分トポロジー2012  2012.8.7

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Presentation type：Oral presentation (invited, special)

• Lefschetz fibrations with small slope Invited

Naoyuki Monden

The Conference on Group Actions and Applications in Geometry, Topology and Analysis  2012.7.21

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Language：English   Presentation type：Oral presentation (invited, special)

• Lefschetz fibrationのgeography問題について Invited

門田直之

トポロジー・幾何セミナー  2012.7.10

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Presentation type：Oral presentation (invited, special)

• 曲面上の曲面束, Lefschetz束のsection Invited

N-KOOKセミナー  2012.6.23

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Presentation type：Oral presentation (invited, special)

• Lefschetz fibrationのgeography問題について} Invited

門田直之

談話会  2012.5.16

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Presentation type：Oral presentation (invited, special)

• On stable commutator length in hyperelliptic mapping class groups

Danny Calegari, 佐藤正寿, 門田直之

日本数学会2012年度年会トポロジー分科会一般講演  2012.3.28

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Presentation type：Oral presentation (general)

• Sections of surface bundles and Lefschetz fibrations

R. Inanc Baykur, Mustafa Korkmaz, 門田直之

日本数学会2012年度年会トポロジー分科会一般講演  2012.3.27

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Presentation type：Oral presentation (general)

• Lefschetzファイバー空間から見るトポロジーと代数幾何の違い Invited

門田直之

Hurwitz action  2012.1.29

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Presentation type：Oral presentation (invited, special)

• Sections of surface bundles and Lefschetz fibrations Invited

Naoyuki Monden

Friday Seminar on Knot Theory  2011.11.25

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Language：English   Presentation type：Oral presentation (invited, special)

• On stable commutator length in hyperelliptic mapping class groups

門田直之

4次元トポロジー研究集会  2011.11.7

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Presentation type：Oral presentation (general)

• Lefschetz fibrations over a torus admitting a section of square 0

門田直之

日本数学会2011年度秋季総合分科会トポロジー分科会一般講演  2011.9.28

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Presentation type：Oral presentation (general)

• On stable commutator length of a Dehn twist Invited

門田直之

リーマン面に関連する位相幾何学2011  2011.9.5

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Presentation type：Oral presentation (invited, special)

• Lefschetz fibrationのsectionとsingular fiberの本数 Invited

門田直之

Workshop on Lefschetz fibrations, broken Lefschetz fibrations and related topics  2011.3.29

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Presentation type：Oral presentation (invited, special)

• Lefschetz fibrations over a torus admitting a section of square 0

門田直之

日本数学会2011年度年会トポロジー分科会一般講演  2011.3.21

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Presentation type：Oral presentation (general)

• Lefschetz fibrationのsingular fiberの本数について Invited

門田直之

Spring Workshop 2011 on Low-Dimensional Topology and its Ramifications  2011.3.1

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Presentation type：Oral presentation (invited, special)

• On the minimal number of singular fibers in a genus-2 Lefschetz fibration Invited

Naoyuki Monden

The Seventh East Asian School of Knots and Related Topics  2011.1.12

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Language：English   Presentation type：Oral presentation (general)

• Genus-2 Lefschetz fibrationのsingular fiberの最小本数について

門田直之

結び目の数学II  2010.12.23

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Presentation type：Oral presentation (general)

• Genus-2 Lefschetz fibrationのsingular fiberの本数について Invited

門田直之

トポロジーの現在と未来  2010.12.22

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Presentation type：Oral presentation (invited, special)

• Lefschetz fibrations over a torus admitting a section of square 0

門田直之

4次元トポロジー研究集会  2010.11.16

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Presentation type：Oral presentation (general)

• On upper bounds of stable commutator length in mapping class groups

Naoyuki Monden

International Conference Japan-Mexico on Topology and its Applications  2010.9.27

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Language：English   Presentation type：Oral presentation (general)

• Disk twistのrootの次数について Invited

門田直之

リーマン面に関連する位相幾何学2010  2010.9.7

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Presentation type：Oral presentation (invited, special)

• On roots of Dehn twists Invited

門田直之

トポロジー火曜セミナー  2010.5.18

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Presentation type：Oral presentation (invited, special)

• Generating set of the mapping class group Invited

Naoyuki Monden

Friday Seminar on Knot Theory  2010.4.16

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Language：English   Presentation type：Oral presentation (invited, special)

• Disk twistのrootの次数について

門田直之

日本数学会2010年度年会トポロジー分科会一般講演  2010.3.24

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Presentation type：Oral presentation (general)

• On roots of Dehn twists Invited

Naoyuki Monden

2010.3.9

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Language：English   Presentation type：Oral presentation (invited, special)

• On roots of Dehn twists Invited

門田直之

Spring Workshop 2010 on Low-Dimensional Topology and its Ramifications  2010.3.5

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Presentation type：Oral presentation (invited, special)

• On roots of Dehn twists Invited

Naoyuki Monden

The Sixth East Asian School of Knots and Related Topics  2010.1.26

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Language：English   Presentation type：Oral presentation (general)

• Seminar on work of Hamenstadt 第7回 Invited

門田直之

低次元トポロジーセミナー  2009.11.7

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Presentation type：Oral presentation (invited, special)

• リーマン面の退化族と写像類群 Invited

門田直之

久留米セミナー  2009.11.7

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Presentation type：Oral presentation (invited, special)

• Dehn twistのrootについて Invited

門田直之

関西低次元トポロジー若手セミナー  2009.9.28

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Presentation type：Oral presentation (invited, special)

• Dehn twistのrootについて

門田直之

日本数学会2009年度秋季総合分科会トポロジー分科会一般講演  2009.9.25

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Presentation type：Oral presentation (general)

• Seminar on work of Hamenstadt 第3回 Invited

門田直之

低次元トポロジーセミナー  2009.6.2

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Presentation type：Oral presentation (invited, special)

• 小さな位数の元による写像類群の生成について

門田直之

日本数学会2009年度年会トポロジー分科会一般講演  2009.3.26

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Presentation type：Oral presentation (general)

• 向き付け可能な曲面の写像類群について Invited

門田直之

数学院生談話会  2009.2.12

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Presentation type：Oral presentation (invited, special)

• Generating the mapping class group of a punctured surface by involutions Invited

門田直之

Workshop on Geometry and Topology of Mapping class groups  2008.11.12

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Presentation type：Oral presentation (invited, special)

• Generating the mapping class group of a punctured surface by involutions

門田直之

日本数学会2008年度秋季総合分科会トポロジー分科会一般講演  2008.9.24

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Presentation type：Oral presentation (general)

• Generating the mapping class group of a punctured surface by 3 elements Invited

門田直之

阪大トポロジーセミナー  2008.7.29

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Presentation type：Oral presentation (invited, special)

• Generating the mapping class group of a punctured surface by involutions Invited

門田直之

変換群の幾何とその周辺  2008.5.22

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Presentation type：Oral presentation (invited, special)

• 位数2の元による点付き曲面の写像類群の生成について

門田直之

Winter Workshop 2008 on Low-Dimensional Topology and its Ramifications  2008.2.14

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Presentation type：Oral presentation (general)

### Research Projects

• 写像類群を用いたシンプレクティック４次元多様体の研究

Grant number：20K03613  2020.04 - 2025.03

日本学術振興会  科学研究費助成事業 基盤研究(C)  基盤研究(C)

門田 直之

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Grant amount：\4550000 （ Direct expense: \3500000 、 Indirect expense：\1050000 ）

• 写像類群による4次元トポロジーの地誌学と手術の研究

Grant number：16K17601  2016.04 - 2021.03

日本学術振興会  科学研究費助成事業 若手研究(B)  若手研究(B)

門田 直之

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Grant amount：\2600000 （ Direct expense: \2000000 、 Indirect expense：\600000 ）

本研究の目的は, 4次元シンプレクティック多様体やLefschetz fibration全体の振舞いを調べることである. シンプレクティック構造は, 偶数次元の多様体に対し定義され, 複素構造に似て非常にきれいな構造である. Lefschetz fibrationとは“有限個のある型の特異ファイバーをもつ曲面上の曲面束”のようなものである.
任意の有限表示群に対し, その群を基本群にもつような4次元シンプレクティック多様体やLefschetz fibrationはこれまでにいくつか構成されていた. 一方, Noether lineの下の領域に任意の有限表示群を基本群にもつ4次元シンプレクティック多様体は構成されていたが, Lefschetz fibrationについてはいまだ見つかっていなかった. これまでに, 申請者は, 単連結で, 全空間が極小であり, Noether lineの下の領域にある多様体について, Lefschetz fibartionの構造が入る例を構成した.
上述の背景のもとで, 申請者は, 申請者の結果の一般化として, 任意の有限表示群を基本群にもち, 全空間が極小であり, Noether lineの下の領域にある多様体について, Lefschetz fibrationの構造が入る例を構成した. 特に, Noether lineの下の領域にある多様体は複素多様体ではないため, 得られたLefschetz fibrationは非正則であることがわかる. Lefschetz fibrationと写像類群の文字列である条件を満たすものが対応する. 申請者は, 与えたLefschetz fibrationにおいても対応する写像類群の文字列を具体的に与えている.

• Grant number：25800043  2013.04 - 2017.03

Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)  Grant-in-Aid for Young Scientists (B)

MONDEN NAOYUKI, AKHMEDOV ANAR, R. INANC REFIK Baykur, HAMADA NORIYUKI, VAN-HORN MORRIS JEREMY, KOBAYASHI RYOMA, YOSHIHARA KAZUYA

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Grant amount：\2600000 （ Direct expense: \2000000 、 Indirect expense：\600000 ）

We studied Lefschetz fibrations and Lefschetz pencils, which play an important role in 4-dimensional topology, using mapping class groups. In particular, we solved a problem on the Euler characteristics of Lefschetz pencils and constructed explicit examples that their existence are known but there are no example or that they have new properties. Moreover, as an application of the techniques we obtained in the study, we gave a result on stable commutator lengths in mapping class groups.

### Class subject in charge

• Topology （2021academic year） Late  - 木3,木4

• Excercises in Basic Geometry Aa （2021academic year） 1st semester  - 木5,木6

• Excercises in Basic Geometry Ab （2021academic year） Second semester  - 木5,木6

• Excercises in Basic Geometry A （2021academic year） 1st and 2nd semester  - 木5,木6

• Seminar in Geometry （2021academic year） Year-round  - その他

• Calculus I （2021academic year） 1st and 2nd semester  - 水1,水2

• Calculus I （2021academic year） 1st and 2nd semester  - 水1～2

• Calculus Ia （2021academic year） 1st semester  - 水1,水2

• Calculus Ib （2021academic year） Second semester  - 水1,水2

• Information Mathematics （2021academic year） 1st and 2nd semester  - 木3,木4

• Information Mathematics a （2021academic year） 1st semester  - 木3,木4

• Information Mathematics b （2021academic year） Second semester  - 木3,木4

• Introduction to Natural Science 1(Mathematics) （2021academic year） 1st semester  - 火3～4

• Introduction to Natural Science 2(Mathematics) （2021academic year） Second semester  - 火3～4

• Topology （2020academic year） Late  - 木3,木4

• Seminar in Geometry （2020academic year） Year-round  - その他

• Calculus I （2020academic year） 1st and 2nd semester  - 水1,水2

• Calculus Ia （2020academic year） 1st semester  - 水1,水2

• Calculus Ib （2020academic year） Second semester  - 水1,水2

• Calculus II （2020academic year） 3rd and 4th semester  - 火5,火6

• Calculus II （2020academic year） 3rd and 4th semester  - 水1,水2

• Calculus IIa （2020academic year） Third semester  - 火5,火6

• Calculus IIa （2020academic year） Third semester  - 水1,水2

• Calculus IIb （2020academic year） Fourth semester  - 火5,火6

• Calculus IIb （2020academic year） Fourth semester  - 水1,水2

• Calculus IIa （2020academic year） Third semester  - 火5,火6

• Calculus IIa （2020academic year） Third semester  - 水1,水2

• Calculus IIb （2020academic year） Fourth semester  - 火5,火6

• Calculus IIb （2020academic year） Fourth semester  - 水1,水2

• Calculus Ia （2020academic year） 1st semester  - 水1,水2

• Calculus Ib （2020academic year） Second semester  - 水1,水2

• Information Mathematics （2020academic year） 1st and 2nd semester  - 木3,木4

• Information Mathematics a （2020academic year） 1st semester  - 木3,木4

• Information Mathematics b （2020academic year） Second semester  - 木3,木4

• Glance at Mathematical Science B （2020academic year） Second semester  - 火1,火2

• Advanced Lecture on Mathematical Sciences C （2020academic year） Prophase  - その他

### Social Activities

• トポロジーの目で見た地球の形

Role(s)：Lecturer

大学院教育改革支援プログラム「数物から社会に発信・発進する人材の育成」・数物カフェ第1弾：阪大大学院生による数学・物理の最前線への招待1  2009.2.19

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