## Seminar information archive

Seminar information archive ～04/25｜Today's seminar 04/26 | Future seminars 04/27～

### 2019/04/25

#### Applied Analysis

16:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

The porous medium equation on noncompact Riemannian manifolds with initial datum a measure

(English)

On sharp large deviations for the bridge of a general diffusion

(English)

**Matteo Muratori**(Polytechnic University of Milan) 16:00-17:00The porous medium equation on noncompact Riemannian manifolds with initial datum a measure

(English)

[ Abstract ]

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds. We show existence of solutions that take a finite Radon measure as initial datum, possibly sign-changing. We then prove uniqueness in the class of nonnegative solutions, upon assuming a quadratic lower bound on the Ricci curvature. Our result is "optimal" in the sense that any weak solution necessarily solves a Cauchy problem with initial datum a finite Radon measure. Moreover, as byproducts of the techniques we employ, we obtain some new results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions and related properties of potentials of positive Radon measures. Finally, we briefly discuss some work in progress regarding stability of the porous medium equation with respect to the Wasserstein distance, on Riemannian manifolds with Ricci curvature bounded below.

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds. We show existence of solutions that take a finite Radon measure as initial datum, possibly sign-changing. We then prove uniqueness in the class of nonnegative solutions, upon assuming a quadratic lower bound on the Ricci curvature. Our result is "optimal" in the sense that any weak solution necessarily solves a Cauchy problem with initial datum a finite Radon measure. Moreover, as byproducts of the techniques we employ, we obtain some new results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions and related properties of potentials of positive Radon measures. Finally, we briefly discuss some work in progress regarding stability of the porous medium equation with respect to the Wasserstein distance, on Riemannian manifolds with Ricci curvature bounded below.

**Maurizia Rossi**(University of Pisa) 17:00-18:00On sharp large deviations for the bridge of a general diffusion

(English)

[ Abstract ]

In this talk we provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided b enjoyes a simple condition that is always satisfied in dimension 1. On the other hand, we show that the drift can be influential if this assumption is not satisfied. This talk is based on a joint work with P. Baldi and L. Caramellino.

In this talk we provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided b enjoyes a simple condition that is always satisfied in dimension 1. On the other hand, we show that the drift can be influential if this assumption is not satisfied. This talk is based on a joint work with P. Baldi and L. Caramellino.

#### Information Mathematics Seminar

16:50-18:35 Room #128 (Graduate School of Math. Sci. Bldg.)

Birth and Development of Modern Cryptography (JAPANESE)

**Tatsuaki Okamoto**(NTT)Birth and Development of Modern Cryptography (JAPANESE)

[ Abstract ]

Cryptography Seminar

Cryptography Seminar

### 2019/04/24

#### Number Theory Seminar

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

P^1-localisation and a possible definition of arithmetic Kodaira-Spencer classes (English)

**Joseph Ayoub**(University of Zurich)P^1-localisation and a possible definition of arithmetic Kodaira-Spencer classes (English)

[ Abstract ]

A^1-localisation is a universal construction which produces "cohomology theories" for which the affine line A^1 is contractible. It plays a central role in the theory of motives à la Morel-Voevodsky. In this talk, I'll discuss the analogous construction where the affine line is replaced by the projective line P^1. This is the P^1-localisation which is arguably an unnatural construction since it produces "cohomology theories" for which the projective line P^1 is contractible. Nevertheless, I'll explain a few positive results and some computations around this construction which naturally lead to a definition of Kodaira-Spencer classes of arithmetic nature. (Unfortunately, it is yet unclear if these classes are really interesting and nontrivial.)

A^1-localisation is a universal construction which produces "cohomology theories" for which the affine line A^1 is contractible. It plays a central role in the theory of motives à la Morel-Voevodsky. In this talk, I'll discuss the analogous construction where the affine line is replaced by the projective line P^1. This is the P^1-localisation which is arguably an unnatural construction since it produces "cohomology theories" for which the projective line P^1 is contractible. Nevertheless, I'll explain a few positive results and some computations around this construction which naturally lead to a definition of Kodaira-Spencer classes of arithmetic nature. (Unfortunately, it is yet unclear if these classes are really interesting and nontrivial.)

#### Algebraic Geometry Seminar

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Varieties of dense globally F-split type with a non-invertible polarized

endomorphism

**Shou Yoshikawa**(Tokyo)Varieties of dense globally F-split type with a non-invertible polarized

endomorphism

[ Abstract ]

Broustet and Gongyo conjectured that if a normal projective variety X has a non-invertible polaried endomorphism, then X is of Calabi-Yau type. Furthermore, Schwede and Smith conjectured that a projective variety is of Calabi-Yau type if and only if of dense globally F-split type. Therefore it is a natural question to ask if a normal projective variety X has a non-invertible polaried endomorphism, then X is of dense globally F-split type. In this talk, I will introduce simple points and difficult points of the question. Furthermore I will give the affirmative answer of my question for 2-dimensional case.

Broustet and Gongyo conjectured that if a normal projective variety X has a non-invertible polaried endomorphism, then X is of Calabi-Yau type. Furthermore, Schwede and Smith conjectured that a projective variety is of Calabi-Yau type if and only if of dense globally F-split type. Therefore it is a natural question to ask if a normal projective variety X has a non-invertible polaried endomorphism, then X is of dense globally F-split type. In this talk, I will introduce simple points and difficult points of the question. Furthermore I will give the affirmative answer of my question for 2-dimensional case.

### 2019/04/23

#### Tuesday Seminar on Topology

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

Higher Hochschild homology as a functor (ENGLISH)

**Christine Vespa**(Université de Strasbourg)Higher Hochschild homology as a functor (ENGLISH)

[ Abstract ]

Higher Hochschild homology generalizes classical Hochschild homology for rings. Recently, Turchin and Willwacher computed higher Hochschild homology of a finite wedge of circles with coefficients in the Loday functor associated to the ring of dual numbers over the rationals. In particular, they obtained linear representations of the groups Out(F_n) which do not factorize through GL(n,Z).

In this talk, I will begin by recalling what is Hochschild homology and higher Hochschild homology. Then I will explain how viewing higher Hochschild homology of a finite wedge of circles as a functor on the category of free groups provides a conceptual framework which allows powerful tools such as exponential functors and polynomial functors to be used. In particular, this allows the generalization of the results of Turchin and Willwacher; this gives rise to new linear representations of Out(F_n) which do not factorize through GL(n,Z).

(This is joint work with Geoffrey Powell.)

Higher Hochschild homology generalizes classical Hochschild homology for rings. Recently, Turchin and Willwacher computed higher Hochschild homology of a finite wedge of circles with coefficients in the Loday functor associated to the ring of dual numbers over the rationals. In particular, they obtained linear representations of the groups Out(F_n) which do not factorize through GL(n,Z).

In this talk, I will begin by recalling what is Hochschild homology and higher Hochschild homology. Then I will explain how viewing higher Hochschild homology of a finite wedge of circles as a functor on the category of free groups provides a conceptual framework which allows powerful tools such as exponential functors and polynomial functors to be used. In particular, this allows the generalization of the results of Turchin and Willwacher; this gives rise to new linear representations of Out(F_n) which do not factorize through GL(n,Z).

(This is joint work with Geoffrey Powell.)

### 2019/04/22

#### Seminar on Geometric Complex Analysis

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

Optimal destabilizer for a Fano manifold (Japanese)

**Tomoyuki Hisamoto**(Nayoya Univ.)Optimal destabilizer for a Fano manifold (Japanese)

[ Abstract ]

Around 2005, S. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence of the normalized Donaldson-Futaki invariants.

For a Fano manifold we construct a sequence of multiplier ideal sheaves from a new geometric flow and answer to Donaldson's question.

Around 2005, S. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence of the normalized Donaldson-Futaki invariants.

For a Fano manifold we construct a sequence of multiplier ideal sheaves from a new geometric flow and answer to Donaldson's question.

#### Numerical Analysis Seminar

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Superconvergence of the HDG method (Japanese)

**Issei Oikawa**(Hitotsubashi University )Superconvergence of the HDG method (Japanese)

#### Discrete mathematical modelling seminar

17:15-18:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Geometry of the Kahan-Hirota-Kimura discretization

**Yuri Suris**(Technische Universität Berlin)Geometry of the Kahan-Hirota-Kimura discretization

[ Abstract ]

We will report on some novel results concerning the bilinear discretization of quadratic vector fields.

We will report on some novel results concerning the bilinear discretization of quadratic vector fields.

### 2019/04/19

#### Operator Algebra Seminars

13:00-14:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Coarse decomposition of II$_1$ factors (English)

**Sorin Popa**(UCLA/Kyoto University)Coarse decomposition of II$_1$ factors (English)

### 2019/04/17

#### Number Theory Seminar

17:00-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On the Shafarevich conjecture for minimal surfaces of Kodaira dimension 0 (Japanese)

**Teppei Takamatsu**(University of Tokyo)On the Shafarevich conjecture for minimal surfaces of Kodaira dimension 0 (Japanese)

#### Operator Algebra Seminars

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

(日本語)

**Takashi Satomi**(Univ. Tokyo)(日本語)

### 2019/04/16

#### Tuesday Seminar on Topology

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

Thurston’s bounded image theorem (ENGLISH)

**Ken’ichi Ohshika**(Gakushuin University)Thurston’s bounded image theorem (ENGLISH)

[ Abstract ]

The bounded image theorem by Thurston constitutes an important step in the proof of his unifomisation theorem for Haken manifolds. Thurston’s original argument was never published and has been unknown up to now. It has turned out a weaker form of this theorem is enough for the proof, and books by Kappovich and by Otal use this weaker version. In this talk, I will show how to prove Thurston’s original version making use of more recent technology. This is joint work with Cyril Lecuire.

The bounded image theorem by Thurston constitutes an important step in the proof of his unifomisation theorem for Haken manifolds. Thurston’s original argument was never published and has been unknown up to now. It has turned out a weaker form of this theorem is enough for the proof, and books by Kappovich and by Otal use this weaker version. In this talk, I will show how to prove Thurston’s original version making use of more recent technology. This is joint work with Cyril Lecuire.

### 2019/04/15

#### Seminar on Geometric Complex Analysis

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

(Japanese)

**Takeo Ohsawa**(Nagoya Univ.)(Japanese)

### 2019/04/10

#### Number Theory Seminar

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

The geometry of the affine Springer fibers and Arthur's weighted orbital integrals (English)

**Zongbin Chen**(Yau Mathematical Sciences Center, Tsinghua University)The geometry of the affine Springer fibers and Arthur's weighted orbital integrals (English)

[ Abstract ]

The affine Springer fibers are geometric objects conceived for the study of orbital integrals. They have complicated geometric structures. We will explain our work on the geometry of affine Springer fibers, with emphasize on the construction of a fundamental domain, and show how the study of the affine Springer fibers can be reduced to that of its fundamental domain. As an application, we will explain how to calculate Arthur's weighted orbital integrals via counting points on the fundamental domain.

The affine Springer fibers are geometric objects conceived for the study of orbital integrals. They have complicated geometric structures. We will explain our work on the geometry of affine Springer fibers, with emphasize on the construction of a fundamental domain, and show how the study of the affine Springer fibers can be reduced to that of its fundamental domain. As an application, we will explain how to calculate Arthur's weighted orbital integrals via counting points on the fundamental domain.

### 2019/04/09

#### Tuesday Seminar of Analysis

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

The Poisson equation on Riemannian manifolds (English)

**Fabio Punzo**(Politecnico di Milano)The Poisson equation on Riemannian manifolds (English)

[ Abstract ]

The talk is concerned with the existence of solutions to the Poisson equation on complete non-compact Riemannian manifolds. In particular, the interplay between the Ricci curvature and the behaviour at infinity of the source function will be discussed. This is a joint work with G. Catino and D.D. Monticelli.

The talk is concerned with the existence of solutions to the Poisson equation on complete non-compact Riemannian manifolds. In particular, the interplay between the Ricci curvature and the behaviour at infinity of the source function will be discussed. This is a joint work with G. Catino and D.D. Monticelli.

#### Tuesday Seminar on Topology

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

Coulomb branches of 3d SUSY gauge theories (JAPANESE)

**Hiraku Nakajima**(Kavli IPMU, The University of Tokyo)Coulomb branches of 3d SUSY gauge theories (JAPANESE)

[ Abstract ]

I will give an introduction to a mathematical definition of Coulomb branches of 3-dimensional SUSY gauge theories, given by my joint work with Braverman and Finkelberg. I will emphasize on the role of hypothetical 3d TQFT associated with gauge theories.

I will give an introduction to a mathematical definition of Coulomb branches of 3-dimensional SUSY gauge theories, given by my joint work with Braverman and Finkelberg. I will emphasize on the role of hypothetical 3d TQFT associated with gauge theories.

### 2019/04/08

#### Numerical Analysis Seminar

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Crack growth model of viscoelastic material with the phase field approach (Japanese)

**Takeshi Takaishi**(Musashino University)Crack growth model of viscoelastic material with the phase field approach (Japanese)

### 2019/04/02

#### Tuesday Seminar on Topology

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

A topological interpretation of symplectic fillings of a normal surface singularity (ENGLISH)

**Jongil Park**(Seoul National University)A topological interpretation of symplectic fillings of a normal surface singularity (ENGLISH)

[ Abstract ]

One of active research areas in symplectic 4-manifolds is to classify symplectic fillings of certain 3-manifolds equipped with a contact structure.

Among them, people have long studied symplectic fillings of the link of a normal surface singularity. Note that the link of a normal surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure.

In this talk, I’d like to investigate a topological surgery description for minimal symplectic fillings of the link of quotient surface singularities and weighted homogeneous surface singularities with a canonical contact structure. Explicitly, I’ll show that every minimal symplectic filling of the link of quotient surface singularities and weighted homogeneous surface singularities satisfying certain conditions can be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding surface singularity. This is joint work with Hakho Choi.

One of active research areas in symplectic 4-manifolds is to classify symplectic fillings of certain 3-manifolds equipped with a contact structure.

Among them, people have long studied symplectic fillings of the link of a normal surface singularity. Note that the link of a normal surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure.

In this talk, I’d like to investigate a topological surgery description for minimal symplectic fillings of the link of quotient surface singularities and weighted homogeneous surface singularities with a canonical contact structure. Explicitly, I’ll show that every minimal symplectic filling of the link of quotient surface singularities and weighted homogeneous surface singularities satisfying certain conditions can be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding surface singularity. This is joint work with Hakho Choi.

### 2019/03/27

#### Tuesday Seminar on Topology

17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

On a moduli space interpretation of the Turaev cobracket (ENGLISH)

**Florian Naef**(Université de Genève)On a moduli space interpretation of the Turaev cobracket (ENGLISH)

[ Abstract ]

Given an oriented surface, Goldman defines a Lie bracket on the vector space spanned by free homotopy classes of loops in terms of intersections. This Lie bracket is the universal version of the Atiyah-Bott Poisson structure on the moduli space of flat connections. Using self-intersections Turaev defines a Lie cobracket on loops. We give a possible interpretation of this structure on moduli spaces of flat connections in the form of a natural BV operator on the moduli space of flat connection with values in a super Lie algebra equipped with an odd pairing. This is joint work with A. Alekseev, J. Pulmann and P. Ševera.

Given an oriented surface, Goldman defines a Lie bracket on the vector space spanned by free homotopy classes of loops in terms of intersections. This Lie bracket is the universal version of the Atiyah-Bott Poisson structure on the moduli space of flat connections. Using self-intersections Turaev defines a Lie cobracket on loops. We give a possible interpretation of this structure on moduli spaces of flat connections in the form of a natural BV operator on the moduli space of flat connection with values in a super Lie algebra equipped with an odd pairing. This is joint work with A. Alekseev, J. Pulmann and P. Ševera.

### 2019/03/26

#### Tuesday Seminar on Topology

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

Cube capacities (ENGLISH)

**Michael Hutchings**(University of California, Berkeley)Cube capacities (ENGLISH)

[ Abstract ]

We define a new series of symplectic capacities using equivariant symplectic homology. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but can be computed in many more examples. In particular, we use these capacities to find many examples of symplectic embeddings of cubes where the cube is as large as possible. This is joint work with Jean Gutt.

We define a new series of symplectic capacities using equivariant symplectic homology. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but can be computed in many more examples. In particular, we use these capacities to find many examples of symplectic embeddings of cubes where the cube is as large as possible. This is joint work with Jean Gutt.

### 2019/03/22

#### Colloquium

13:00-17:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Mathematical structures of quantum mechanics and classical mechanics (日本語)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~shu/

Algebraic cyles, Periods and Motives (日本語)

[ Reference URL ]

http://gauss.ms.u-tokyo.ac.jp/index-j.html

Research on groups of homeomorphisms (日本語)

https://www.ms.u-tokyo.ac.jp/~tsuboi/

**Shu NAKAMURA**(The University of Tokyo) 13：00-14：00Mathematical structures of quantum mechanics and classical mechanics (日本語)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~shu/

**Tomohide TERASOMA**(The University of Tokyo) 14:30-15:30Algebraic cyles, Periods and Motives (日本語)

[ Reference URL ]

http://gauss.ms.u-tokyo.ac.jp/index-j.html

**Takashi TSUBOI**(The University of Tokyo) 16:00-17:00Research on groups of homeomorphisms (日本語)

[ Abstract ]

The homeomorphisms of a topological space form a group. The group seems to be too wild to be considered. In some cases it becomes a countable group but it is usually uncountable group. I have studied groups of homeomorphisms of topological spaces or groups of diffeomorphisms of manifolds which are related to invariants of foliations. I found several relationship between dynamical properties of group actions and homology of groups. There are many unsolved problems on the group of

homeomorphisms. I also intend to investigate more on the shape of groups. I would like to talk on such topics around groups of homeomorphisms.

[ Reference URL ]The homeomorphisms of a topological space form a group. The group seems to be too wild to be considered. In some cases it becomes a countable group but it is usually uncountable group. I have studied groups of homeomorphisms of topological spaces or groups of diffeomorphisms of manifolds which are related to invariants of foliations. I found several relationship between dynamical properties of group actions and homology of groups. There are many unsolved problems on the group of

homeomorphisms. I also intend to investigate more on the shape of groups. I would like to talk on such topics around groups of homeomorphisms.

https://www.ms.u-tokyo.ac.jp/~tsuboi/

### 2019/03/13

#### Seminar on Probability and Statistics

14:00-15:10 Room #052 (Graduate School of Math. Sci. Bldg.)

On parameter estimation of hidden Markov processes

**Yury A. Kutoyants**(Laboratoire Manceau de Mathématiques, Le Mans University)On parameter estimation of hidden Markov processes

[ Abstract ]

We present a survey of several results devoted to parameter estimation of partially observed models. The hidden processes are Ornstein-Uhlenbeck process and Telegraph process. We describe the asymptotic behavior of the MLE and BE of the unknown parameters of hidden processes and special attention is paid to a new class of estimators called Multi-step MLE-processes, which have the same asymptotic properties as the MLE but can be calculated much easier than MLE.

The corresponding articles are

1.Kutoyants Yu. A., " On the multi-step MLE-process for ergodic

diffusion", (arXiv 1504.01869) Stochastic Processes and their

Applications, 2017, 127, 2243-2261.

2.Khasminskii, R. Z. and Kutoyants, Yu. A. "On parameter estimation of

hidden telegraph process". (arXiv:1509.02704 ) Bernoulli, 2018, 24, 3,

2064-2090.

3.Kutoyants, Yu. A. "On parameter estimation of hidden

Ornstein-Uhlenbeck process", Journal of Multivariate Analysis. 2019,

169, 1, 248-263.

4.Kutoyants, Yu. A. "On parameter estimation of hidden ergodic

Ornstein-Uhlenbeck process", 2019, submitted (arXiv:1902.08500)

We present a survey of several results devoted to parameter estimation of partially observed models. The hidden processes are Ornstein-Uhlenbeck process and Telegraph process. We describe the asymptotic behavior of the MLE and BE of the unknown parameters of hidden processes and special attention is paid to a new class of estimators called Multi-step MLE-processes, which have the same asymptotic properties as the MLE but can be calculated much easier than MLE.

The corresponding articles are

1.Kutoyants Yu. A., " On the multi-step MLE-process for ergodic

diffusion", (arXiv 1504.01869) Stochastic Processes and their

Applications, 2017, 127, 2243-2261.

2.Khasminskii, R. Z. and Kutoyants, Yu. A. "On parameter estimation of

hidden telegraph process". (arXiv:1509.02704 ) Bernoulli, 2018, 24, 3,

2064-2090.

3.Kutoyants, Yu. A. "On parameter estimation of hidden

Ornstein-Uhlenbeck process", Journal of Multivariate Analysis. 2019,

169, 1, 248-263.

4.Kutoyants, Yu. A. "On parameter estimation of hidden ergodic

Ornstein-Uhlenbeck process", 2019, submitted (arXiv:1902.08500)

### 2019/03/05

#### Tuesday Seminar of Analysis

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

The structure of minimal surfaces near polyhedral cones (English)

**Nicholas Edelen**(Massachusetts Institute of Technology)The structure of minimal surfaces near polyhedral cones (English)

[ Abstract ]

We prove a regularity theorem for minimal varifolds which resemble a cone $C_0$ over an equiangular geodesic net. For varifold classes admitting a ``no-hole'' condition on the singular set, we additionally establish regularity near the cone $C_0 \times R^m$. Our result implies the following generalization of Taylor's structure theorem for soap bubbles: given an $n$-dimensional soap bubble $M$ in $R^{n+1}$, then away from an $(n-3)$-dimensional set, $M$ is locally $C^{1,\alpha}$ equivalent to $R^n$, a union of three half-$n$-planes meeting at $120$ degrees, or an $(n-2)$-line of tetrahedral junctions. This is joint work with Maria Colombo and Luca Spolaor.

We prove a regularity theorem for minimal varifolds which resemble a cone $C_0$ over an equiangular geodesic net. For varifold classes admitting a ``no-hole'' condition on the singular set, we additionally establish regularity near the cone $C_0 \times R^m$. Our result implies the following generalization of Taylor's structure theorem for soap bubbles: given an $n$-dimensional soap bubble $M$ in $R^{n+1}$, then away from an $(n-3)$-dimensional set, $M$ is locally $C^{1,\alpha}$ equivalent to $R^n$, a union of three half-$n$-planes meeting at $120$ degrees, or an $(n-2)$-line of tetrahedral junctions. This is joint work with Maria Colombo and Luca Spolaor.

#### Seminar on Mathematics for various disciplines

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

### 2019/02/12

#### Tuesday Seminar on Topology

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

Representations of knot groups (ENGLISH)

**Anastasiia Tsvietkova**(Okinawa Institute of Science and Technology, Rutgers University)Representations of knot groups (ENGLISH)

[ Abstract ]

We describe a new method of producing equations for the representation variety of a knot group into (P)SL(2,C). Unlike known methods, this does not involve any polyhedral decomposition or triangulation of the link complement, and uses only a link diagram satisfying a few mild restrictions. This results in a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. This is a joint work with Kathleen Peterson (Florida State University).

We describe a new method of producing equations for the representation variety of a knot group into (P)SL(2,C). Unlike known methods, this does not involve any polyhedral decomposition or triangulation of the link complement, and uses only a link diagram satisfying a few mild restrictions. This results in a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. This is a joint work with Kathleen Peterson (Florida State University).

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