## Seminar information archive

Seminar information archive ～12/09｜Today's seminar 12/10 | Future seminars 12/11～

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Beyond the Weinstein conjecture (ENGLISH)

**Dan Cristofaro-Gardiner**(University of California, Santa Cruz)Beyond the Weinstein conjecture (ENGLISH)

[ Abstract ]

The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the "standard" contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the "standard" contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Higher order families of lines and Fano manifolds covered by linear

spaces

(Japanese (writing in English))

**Taku Suzuki**(Utsunomiya)Higher order families of lines and Fano manifolds covered by linear

spaces

(Japanese (writing in English))

[ Abstract ]

In this talk, for an embedded Fano manifold $X$, we introduce higher

order families of lines and a new invariant $S_X$. They are line

versions of higher order minimal families of rational curves and the

invariant $N_X$ which were introduced in my previous talk on 4th

November 2016. In addition, $S_X$ is related to the dimension of

covering linear spaces. Our goal is to classify Fano manifolds $X$ which

have large $S_X$.

In this talk, for an embedded Fano manifold $X$, we introduce higher

order families of lines and a new invariant $S_X$. They are line

versions of higher order minimal families of rational curves and the

invariant $N_X$ which were introduced in my previous talk on 4th

November 2016. In addition, $S_X$ is related to the dimension of

covering linear spaces. Our goal is to classify Fano manifolds $X$ which

have large $S_X$.

#### Lectures

13:00-14:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Langlands-Rapoport for the Modular Curve

**Sander Mack-Crane**(University of California, Berkeley)Langlands-Rapoport for the Modular Curve

[ Abstract ]

We discuss a concrete version of the Langlands-Rapoport conjecture in the case of the modular curve, and use this case to illuminate some of the more abstract features of the Langlands-Rapoport conjecture for general (abelian type) Shimura varieties.

We discuss a concrete version of the Langlands-Rapoport conjecture in the case of the modular curve, and use this case to illuminate some of the more abstract features of the Langlands-Rapoport conjecture for general (abelian type) Shimura varieties.

#### Numerical Analysis Seminar

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Various aspects of numerical analysis (Japanese)

**Norikazu Saito**(University of Tokyo)Various aspects of numerical analysis (Japanese)

#### Seminar on Probability and Statistics

15:00-16:10 Room #052 (Graduate School of Math. Sci. Bldg.)

### 2018/05/07

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Proper holomorphic mappings and generalized pseudoellipsoids (JAPANESE)

**Atsushi Hayashimoto**(National Institute of Technology, Nagano College)Proper holomorphic mappings and generalized pseudoellipsoids (JAPANESE)

[ Abstract ]

We study the classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions.

Huang proved some classification theorems of proper holomorphic mappings between balls of different dimensions, which are called gap theorems. Our present theorems are their weakly pseudoconvex versions.

In the theorem, classified mapping is so-called a variables splitting mapping and each component is derived from a homogeneous proper polynomial mapping between balls.

The essential methods are the ''good'' decompositions of CR vector bundle and reduction the mapping under consideration to the mapping of balls. By this reduction, we can apply Huang's gap theorem.

We study the classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions.

Huang proved some classification theorems of proper holomorphic mappings between balls of different dimensions, which are called gap theorems. Our present theorems are their weakly pseudoconvex versions.

In the theorem, classified mapping is so-called a variables splitting mapping and each component is derived from a homogeneous proper polynomial mapping between balls.

The essential methods are the ''good'' decompositions of CR vector bundle and reduction the mapping under consideration to the mapping of balls. By this reduction, we can apply Huang's gap theorem.

#### Tokyo Probability Seminar

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

[ Reference URL ]

http://www.taksagawa.com

**Takahiro SAGAWA**(Faculty of Engineering, The University of Tokyo)(JAPANESE)

[ Reference URL ]

http://www.taksagawa.com

#### FMSP Lectures

15:00-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Introduction to the Langlands-Rapoport conjecture (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_SugWooShin.pdf

**Sug Woo Shin**(University of California, Berkeley)Introduction to the Langlands-Rapoport conjecture (ENGLISH)

[ Abstract ]

In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.

[ Reference URL ]In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_SugWooShin.pdf

### 2018/05/02

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Classification of Rokhlin flows (English)

**Gabor Szabo**(Copenhagen Univ.)Classification of Rokhlin flows (English)

### 2018/04/24

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

BIRATIONAL BOUNDEDNESS OF RATIONALLY CONNECTED CALABI–YAU 3-FOLDS

(English)

**Wei-Chung Chen**(Tokyo)BIRATIONAL BOUNDEDNESS OF RATIONALLY CONNECTED CALABI–YAU 3-FOLDS

(English)

[ Abstract ]

Firstly, we show that rationally connected Calabi–Yau 3- folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ε-CY type form a birationally bounded family for ε > 0. Then we focus on ε-lc log Calabi–Yau pairs (X, B) such that coefficients of B are bounded from below away from zero. We show that such pairs are log bounded modulo flops. As a consequence, we show that rationally connected klt Calabi–Yau 3-folds with mld bounding away from 1 are bounded modulo flops.

Firstly, we show that rationally connected Calabi–Yau 3- folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ε-CY type form a birationally bounded family for ε > 0. Then we focus on ε-lc log Calabi–Yau pairs (X, B) such that coefficients of B are bounded from below away from zero. We show that such pairs are log bounded modulo flops. As a consequence, we show that rationally connected klt Calabi–Yau 3-folds with mld bounding away from 1 are bounded modulo flops.

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Singular Fibers of smooth maps and Cobordism groups (JAPANESE)

**Takahiro Yamamoto**(Tokyo Gakugei University)Singular Fibers of smooth maps and Cobordism groups (JAPANESE)

[ Abstract ]

Following the pioneering work of R.Thom, cobordism groups of smooth maps have been studied by some mathematicians. Especially, cobordism groups of Morse functions on closed manifolds was studied by O.Saeki and K.Ikegami, B.Kalmar. In this talk, we will introduce cobordism groups among Morse functions on compact manifolds with boundary and study the cobordism groups are trivial or not by using the theory of the universal complex of singular fibers of smooth maps.

Following the pioneering work of R.Thom, cobordism groups of smooth maps have been studied by some mathematicians. Especially, cobordism groups of Morse functions on closed manifolds was studied by O.Saeki and K.Ikegami, B.Kalmar. In this talk, we will introduce cobordism groups among Morse functions on compact manifolds with boundary and study the cobordism groups are trivial or not by using the theory of the universal complex of singular fibers of smooth maps.

### 2018/04/23

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Degeneration and bifurcation of quadratic endomorphisms of $\mathbb{P}^2$ towards a Hénon map (JAPANESE)

**Yûsuke Okuyama**(Kyoto Institute of Technology)Degeneration and bifurcation of quadratic endomorphisms of $\mathbb{P}^2$ towards a Hénon map (JAPANESE)

[ Abstract ]

The space of quadratic holomorphic endomorphisms of P^2 (over C) is

canonically identified with the complement of the zero locus of the

resultant form on P^{17}, and all H¥'enon maps, which are (the only)

interesting ones among all the quadratic polynomial automorphisms of C^2,

live in this zero locus.

We will talk about our joint work with Fabrizio Bianchi (Imperial College,

London) on the (algebraic) degeneration of quadratic endomorphisms of C^2

towards H¥'enon maps in terms of Berteloot-Bianchi-Dupont's

bifurcation/unstability theory of holomorphic families of endomorphisms of P^k,

which mostly generalizes Ma¥~n¥'e-Sad-Sullivan, Lyubich, and DeMarco's seminal

and similar theory on P^1.

Some preliminary knowledge on ergodic theory and pluripotential theory

would be desirable, but not be assumed.

The space of quadratic holomorphic endomorphisms of P^2 (over C) is

canonically identified with the complement of the zero locus of the

resultant form on P^{17}, and all H¥'enon maps, which are (the only)

interesting ones among all the quadratic polynomial automorphisms of C^2,

live in this zero locus.

We will talk about our joint work with Fabrizio Bianchi (Imperial College,

London) on the (algebraic) degeneration of quadratic endomorphisms of C^2

towards H¥'enon maps in terms of Berteloot-Bianchi-Dupont's

bifurcation/unstability theory of holomorphic families of endomorphisms of P^k,

which mostly generalizes Ma¥~n¥'e-Sad-Sullivan, Lyubich, and DeMarco's seminal

and similar theory on P^1.

Some preliminary knowledge on ergodic theory and pluripotential theory

would be desirable, but not be assumed.

#### Tokyo Probability Seminar

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Functional central limit theorems for non-symmetric random walks on nilpotent covering graphs (JAPANESE)

**Hiroshi KAWABI**(Faculty of Economics, Keio University)Functional central limit theorems for non-symmetric random walks on nilpotent covering graphs (JAPANESE)

### 2018/04/18

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

(Japanese)

**Doman Takata**(Univ. Tokyo)(Japanese)

#### Number Theory Seminar

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Fargues' conjecture in the GL_2-case (ENGLISH)

**Ildar Gaisin**(University of Tokyo)Fargues' conjecture in the GL_2-case (ENGLISH)

[ Abstract ]

Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of G-bundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the non-semi-stable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2-case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.

Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of G-bundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the non-semi-stable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2-case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.

#### Number Theory Seminar

17:10-18:10 Room #002 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Noriyuki Abe**(University of Tokyo)(JAPANESE)

### 2018/04/17

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Global existence of a strong solution to the Navier-Stokes equations in a curved thin domain (Japanese)

**Tatsu-Hiko Miura**(The University of Tokyo)Global existence of a strong solution to the Navier-Stokes equations in a curved thin domain (Japanese)

[ Abstract ]

In this talk, we consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a closed surface.

We establish the global-in-time existence of a strong solution for large data when the width of the thin domain is very small.

A key idea is to decompose a three-dimensional vector field into the average part which is almost two-dimensional and the residual part to which we can apply Poincaré type inequalities.

Such decomposition enables us to derive a good estimate for the inner product of the inertia term and the viscous term, which is essential for our arguments.

In this talk, we consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a closed surface.

We establish the global-in-time existence of a strong solution for large data when the width of the thin domain is very small.

A key idea is to decompose a three-dimensional vector field into the average part which is almost two-dimensional and the residual part to which we can apply Poincaré type inequalities.

Such decomposition enables us to derive a good estimate for the inner product of the inertia term and the viscous term, which is essential for our arguments.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

SNC log symplectic structures on Fano products (English/Japanese)

**Katsuhiko Okumura**(Waseda Univ. )SNC log symplectic structures on Fano products (English/Japanese)

[ Abstract ]

In 2014, Lima and Pereira gave a characterization of the even-dimensional projective space in terms of log symplectic Poisson structures. After that Pym gave an another more algebraic proof. In this talk, we will extend the result of Lima and Pereira to the case that the variety is a product of Fano varieties with the cyclic Picard group. This will be proved by extending Pym's proof. As a corollary, we will obtain a characterization of the projective space of all dimensions.

In 2014, Lima and Pereira gave a characterization of the even-dimensional projective space in terms of log symplectic Poisson structures. After that Pym gave an another more algebraic proof. In this talk, we will extend the result of Lima and Pereira to the case that the variety is a product of Fano varieties with the cyclic Picard group. This will be proved by extending Pym's proof. As a corollary, we will obtain a characterization of the projective space of all dimensions.

#### Numerical Analysis Seminar

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Introduction to Machine learning and its application to Medical diagnosis (Japanese)

**Yoshiki Sugitani**(Tohoku University)Introduction to Machine learning and its application to Medical diagnosis (Japanese)

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Tight contact structures on Seifert surface complements and knot invariants (ENGLISH)

**Tamás Kálmán**(Tokyo Institute of Technology)Tight contact structures on Seifert surface complements and knot invariants (ENGLISH)

[ Abstract ]

In joint work with Daniel Mathews, we examined complements of standard Seifert surfaces of special alternating links and used Honda's method to enumerate those tight contact structures on them whose dividing sets are isotopic to the link. The number turns out to be the leading coefficient of the Alexander polynomial. The proof is rather combinatorial in nature; for example, the Euler classes of the contact structures are identified with `hypertrees' in a certain hypergraph. Using earlier results with Hitoshi Murakami and Alexander Postnikov, this yields a connection between contact topology and the Homfly polynomial. We also found that the contact invariants of our structures form a basis for the sutured Floer homology of the manifold.

In joint work with Daniel Mathews, we examined complements of standard Seifert surfaces of special alternating links and used Honda's method to enumerate those tight contact structures on them whose dividing sets are isotopic to the link. The number turns out to be the leading coefficient of the Alexander polynomial. The proof is rather combinatorial in nature; for example, the Euler classes of the contact structures are identified with `hypertrees' in a certain hypergraph. Using earlier results with Hitoshi Murakami and Alexander Postnikov, this yields a connection between contact topology and the Homfly polynomial. We also found that the contact invariants of our structures form a basis for the sutured Floer homology of the manifold.

### 2018/04/16

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Metrics on a closed surface which maximize the first eigenvalue of the Laplacian (JAPANESE)

**Shin Nayatani**(Nagoya University)Metrics on a closed surface which maximize the first eigenvalue of the Laplacian (JAPANESE)

[ Abstract ]

In this talk, I will focus on recent progress on metrics that maximize the first eigenvalue of the Laplacian (under area normalization) on a closed surface. First, I introduce Hersch-Yang-Yau's inequality (1970, 1980), which was the starting point of the above problem. This is an inequality indicating that the first eigenvalue (precisely, the product of it with the area) is bounded from above by a constant depending only on the genus of the surface. Then I will outline the recent progress on the existence problem for maximizing metrics together with the relation with minimal surfaces in the sphere. Finally, I will discuss Jacobson-Levitin-Nadirashvili-Nigam-Polterovich's conjecture, which explicitly predicts maximizing metrics in the case of genus two, and the affirmative resolution of it (joint work with Toshihiro Shoda).

In this talk, I will focus on recent progress on metrics that maximize the first eigenvalue of the Laplacian (under area normalization) on a closed surface. First, I introduce Hersch-Yang-Yau's inequality (1970, 1980), which was the starting point of the above problem. This is an inequality indicating that the first eigenvalue (precisely, the product of it with the area) is bounded from above by a constant depending only on the genus of the surface. Then I will outline the recent progress on the existence problem for maximizing metrics together with the relation with minimal surfaces in the sphere. Finally, I will discuss Jacobson-Levitin-Nadirashvili-Nigam-Polterovich's conjecture, which explicitly predicts maximizing metrics in the case of genus two, and the affirmative resolution of it (joint work with Toshihiro Shoda).

### 2018/04/11

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Non-abelian cohomology and Diophantine geometry (ENGLISH)

**Minhyong Kim**(University of Oxford)Non-abelian cohomology and Diophantine geometry (ENGLISH)

[ Abstract ]

This lecture will review the construction of moduli schemes of torsors for sheaves of pro-unipotent groups and their applications to the resolution of Diophantine problems.

This lecture will review the construction of moduli schemes of torsors for sheaves of pro-unipotent groups and their applications to the resolution of Diophantine problems.

### 2018/04/10

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the Morse-Novikov number for 2-knots (JAPANESE)

**Hisaaki Endo**(Tokyo Institute of Technology)On the Morse-Novikov number for 2-knots (JAPANESE)

[ Abstract ]

Pajitnov, Rudolph and Weber defined the Morse-Novikov number for classical links and studied their undamental properties in 2001. This invariant has been investigated in relation to (twisted) Alexander polynomials and the (twisted) Novikov homology. In this talk, we define the Morse-Novikov number for 2-knots and show its several properties. In particular, we describe its relations to motion pictures and spin constructions for 2-knots. This talk is based on joint works with Andrei Pajitnov (Nantes University).

Pajitnov, Rudolph and Weber defined the Morse-Novikov number for classical links and studied their undamental properties in 2001. This invariant has been investigated in relation to (twisted) Alexander polynomials and the (twisted) Novikov homology. In this talk, we define the Morse-Novikov number for 2-knots and show its several properties. In particular, we describe its relations to motion pictures and spin constructions for 2-knots. This talk is based on joint works with Andrei Pajitnov (Nantes University).

### 2018/04/09

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Adjoint forms on algebraic varieties (English)

**Luca Rizzi**(Udine)Adjoint forms on algebraic varieties (English)

[ Abstract ]

The so called adjoint theory was introduced by A. Collino and G.P. Pirola in the case of smooth algebraic curves and then generalized by G.P. Pirola and F. Zucconi in the case of smooth algebraic varieties of arbitrary dimension.

The main idea of this theory is to study particular differential forms, called adjoint forms, on an algebraic variety to obtain information on the infinitesimal deformations of the variety itself.

The natural context for the application of this theory is given by Torelli-type problems, in particular infinitesimal Torelli problems.

The so called adjoint theory was introduced by A. Collino and G.P. Pirola in the case of smooth algebraic curves and then generalized by G.P. Pirola and F. Zucconi in the case of smooth algebraic varieties of arbitrary dimension.

The main idea of this theory is to study particular differential forms, called adjoint forms, on an algebraic variety to obtain information on the infinitesimal deformations of the variety itself.

The natural context for the application of this theory is given by Torelli-type problems, in particular infinitesimal Torelli problems.

#### Algebraic Geometry Seminar

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Commuting nilpotents, punctual Hilbert schemes and jet bundles (ENGLISH)

**David Hyeon**(Seoul National University)Commuting nilpotents, punctual Hilbert schemes and jet bundles (ENGLISH)

[ Abstract ]

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via two different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via two different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

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