## 過去の記録

### 2019年06月20日(木)

#### 応用解析セミナー

16:00-17:30   数理科学研究科棟(駒場) 118号室

[ 講演概要 ]

#### 数理人口学・数理生物学セミナー

16:00-17:00   数理科学研究科棟(駒場) 056号室
Eric Foxall 氏 (University of Alberta)
Diffusion limit for the partner model at the critical value (ENGLISH)
[ 講演概要 ]
The partner model is a stochastic SIS model of infection spread over a dynamic network of monogamous partnerships. In previous work, Edwards, Foxall and van den Driessche identify a threshold in parameter space for spread of the infection and show the time to extinction of the infection is of order log(N) below the threshold, where N is population size, and grows exponentially in N above the
threshold. Later, Foxall shows the time to extinction at threshold is of order sqrt(N). Here we go further and derive a single-variable diffusion limit for the number of infectious individuals rescaled by sqrt(N) in both population and time, and show convergence in distribution of the rescaled extinction time. Since the model has effectively four variables and two relevant time scales, the proof features a succession of probability estimates to control trajectories, as well as an averaging result to contend with the fast partnership dynamics.

### 2019年06月19日(水)

#### 作用素環セミナー

16:45-18:15   数理科学研究科棟(駒場) 126号室

Type classification of extreme quantized characters

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 118号室

A pencil of Enriques surfaces with non-algebraic integral Hodge classes (TBA)
[ 講演概要 ]
The integral Hodge conjecture is the statement that the integral Hodge classes are algebraic on smooth complex projective varieties. It is known that the conjecture can fail in general. There are two types of counterexamples, ones with non-algebraic integral Hodge classes of torsion-type and of non-torsion type, the first of which were given by Atiyah-Hirzebruch and Kollar, respectively.

In this talk, we exhibit a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This construction relates to certain questions concerning rational points of algebraic varieties.

This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question on the universality of the Abel-Jacobi maps.

This is a joint work with John Christian Ottem.

### 2019年06月18日(火)

#### 統計数学セミナー

11:00-12:10   数理科学研究科棟(駒場) 052号室
Xiaohui Chen 氏 (University of Illinois at Urbana–Champaign)
Gaussian and bootstrap approximations of high-dimensional U-statistics with applications and extensions　※変更の可能性あり

[ 講演概要 ]
We shall first discuss the Gaussian approximation of high-dimensional and non-degenerate U-statistics of order two under the supremum norm. A two-step Gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed. Subject to mild moment conditions on the kernel, we establish the explicit rate of convergence that decays polynomially in sample size for a high-dimensional scaling limit, where the dimension can be much larger than the sample size. We also provide computable approximation methods for the quantiles of the maxima of centered U-statistics. Specifically, we provide a unified perspective for the empirical, the randomly reweighted, and the multiplier bootstraps as randomly reweighted quadratic forms, all asymptotically valid and inferentially first-order equivalent in high-dimensions.

The bootstrap methods are applied on statistical applications for high-dimensional non-Gaussian data including: (i) principled and data-dependent tuning parameter selection for regularized estimation of the covariance matrix and its related functionals; (ii) simultaneous inference for the covariance and rank correlation matrices. In particular, for the thresholded covariance matrix estimator with the bootstrap selected tuning parameter, we show that the Gaussian-like convergence rates can be achieved for heavy-tailed data, which are less conservative than those obtained by the Bonferroni technique that
ignores the dependency in the underlying data distribution. In addition, we also show that even for subgaussian distributions, error bounds of the bootstrapped thresholded covariance matrix estimator can be much tighter than those of the minimax estimator with a universal threshold.

Time permitting, we will discuss some extensions to the infinite-dimensional version (i.e., U-processes of increasing complexity) and to the randomized inference via the incomplete U-statistics whose computational cost can be made independent of the order.

#### PDE実解析研究会

10:30-11:30   数理科学研究科棟(駒場) 056号室
Piotr Rybka 氏 (University of Warsaw)
Ways to treat a diffusion problem with the fractional Caputo derivative
[ 講演概要 ]
The problem
$u_t = (D^\alpha u)_x + f$
augmented with initial and boundary data appear in model of subsurface flows. Here, $D^\alpha u$ denotes the fractional Caputo derivative of order $\alpha \in (0,1)$.

We offer three approaches:
1) from the point of view of semigroups;
2) from the point of view of the theory of viscosity solutions;
3) from the point of view of numerical simulations.

This is a joint work with T. Namba, K. Ryszewska, V. Voller.

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

Filtered instanton homology and the homology cobordism group (JAPANESE)
[ 講演概要 ]
We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y×R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in ’90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.

### 2019年06月17日(月)

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室
Andrei Pajitnov 氏 (Universite de Nantes)
Inoue surfaces and their generalizations (English)
[ 講演概要 ]
In 1972 M. Inoue constructed complex non-algebraic surfaces that proved very important for classification of surfaces via the Enriques-Kodaira scheme. Inoue surface is the quotient of H ¥times C by action of a discreet group associated to a given matrix in SL(3, Z). In 2005 K. Oeljeklaus and M. Toma generalized Inoue’s construction to higher dimensions. Oeljeklaus-Toma manifold is the quotient of H^s ¥times C^n by action of a discreet group, associated to the maximal order of a given algebraic number field.
In this talk, I will give a brief overview of these works and related results. Then I will discuss a new generalization of Inoue surfaces to higher dimensions. The manifold in question is the quotient of H ¥times C^n by action of a discreet group associated to a given matrix in SL(2n+1, Z). This is joint work with Hisaaki Endo.

### 2019年06月13日(木)

#### 情報数学セミナー

16:50-18:35   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]
21世紀に入り、公開鍵暗号が発展した新しい暗号概念として完全準同型暗号と関数型暗号が研究されるようになった．これら暗号では，暗号は単に秘匿性を保証するだけではなく，秘匿性を保証したまま様々な演算や高度なデータ検索を可能とする．いわば，暗号化したままクラウド計算やビックデータ検索を行うといったことが可能となる．

### 2019年06月12日(水)

#### 代数学コロキウム

17:00-18:00   数理科学研究科棟(駒場) 056号室

On extension of overconvergent log isocrystals on log smooth varieties (Japanese)
[ 講演概要 ]
Kを混標数の完備な非アルキメデス付値体とし，kをその剰余体とする．
Kedlayaおよび志甫の研究により，k上の滑らかな代数多様体Xとその上の単純正規交叉因子Zについて，X ¥setminus Z上の過収束アイソクリスタルのうちZの周りである種のモノドロミーを持つものは，XにZから定まる対数的構造を入れた対数的代数多様体上の収束対数的アイソクリスタルに延長できることが知られている．

### 2019年06月11日(火)

#### 解析学火曜セミナー

16:50-18:20   数理科学研究科棟(駒場) 128号室
Antonio De Rosa 氏 (クーラン数理科学研究所)
Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)
[ 講演概要 ]
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)

### 2019年06月06日(木)

#### 情報数学セミナー

16:50-18:35   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]

### 2019年06月05日(水)

#### 代数学コロキウム

17:30-18:30   数理科学研究科棟(駒場) 056号室

Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)
[ 講演概要 ]
Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.

### 2019年06月04日(火)

#### PDE実解析研究会

10:30-11:30   数理科学研究科棟(駒場) 056号室
Giuseppe Mingione 氏 (Università di Parma)
Recent progresses in nonlinear potential theory (English)
[ 講演概要 ]
Nonlinear Potential Theory aims at studying the fine properties of solutions to nonlinear, potentially degenerate nonlinear elliptic and parabolic equations in terms of the regularity of the give data. A major model example is here given by the $p$-Laplacean equation
$$-\operatorname{div}(|Du|^{p-2}Du) = \mu \quad\quad p > 1,$$
where $\mu$ is a Borel measure with finite total mass. When $p = 2$ we find the familiar case of the Poisson equation from which classical Potential Theory stems. Although many basic tools from the classical linear theory are not at hand - most notably: representation formulae via fundamental solutions - many of the classical information can be retrieved for solutions and their pointwise behaviour. In this talk I am going to give a survey of recent results in the field. Especially, I will explain the possibility of getting linear and nonlinear potential estimates for solutions to nonlinear elliptic and parabolic equations which are totally similar to those available in the linear case. I will also draw some parallels with what is nowadays called Nonlinear Calderón-Zygmund theory.

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

Gluck twist on branched twist spins (JAPANESE)
[ 講演概要 ]
Branched twist spin とは４次元球面上の円作用の特異点集合として定義される埋め込まれた２次元球面であり，スパン結び目やツイストスパン結び目などの２次元結び目の一般化となっている．Gluck は４次元多様体内の２次元結び目に沿った向きを保つ手術は微分同相類を除いて２種類のみであることを示しており，自明でない手術を Gluck twist と呼ぶ．一般に Gluck twist が全空間の微分同相を保つかどうかは知られていないが，Pao によって branched twist spin に沿った Gluck twist は 再び４次元球面と微分同相になることが知られている．本講演では，Pao の結果の別証明として円作用を用いて４次元球面の分解を与え，各ピースが Gluck twist を通してどのように変化するかを説明する．また，２次元結び目に注目したとき，Gluck twist によって branched twist spin は再び branched twist spin になることを証明する．

### 2019年05月29日(水)

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 118号室

Minimal log discrepancies of 3-dimensional non-canonical singularities (English)
[ 講演概要 ]
Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov.
This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.

#### 代数学コロキウム

17:00-18:00   数理科学研究科棟(駒場) 056号室

On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)
[ 講演概要 ]
PEL型志村多様体のp進整数環上の整モデルは, Abel多様体と付加構造のモジュライ空間として定義される. その幾何的特殊ファイバーのうち, 超特異Abel多様体に対応する点からなる閉部分スキームを超特異部分という. 超特異部分の構造の明示的な記述は, arithmetic intersectionをはじめとする整数論への応用をもつことが知られている.

### 2019年05月28日(火)

#### 諸分野のための数学研究会

10:30-11:30   数理科学研究科棟(駒場) 056号室

[ 講演概要 ]
これまでに独自に開発した細胞内温度イメージング法を用いて、細胞内に時空間的な不均一な温度変動があることを発見した。この温度変化は同じ体積の水と比較すると著しく大きな値であり、細胞内に特殊な熱移動機構が予想された。本研究では人工熱源を用いた際の細胞内の熱移動に関する定量的計測と熱拡散方程式への近似についてその限界と展望を紹介したい。

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
R. Inanc Baykur 氏 (University of Massachusetts)
Exotic four-manifolds via positive factorizations (ENGLISH)
[ 講演概要 ]
We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.

### 2019年05月27日(月)

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室

Gluing construction of K3 surfaces (Japanese)
[ 講演概要 ]
Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of an elliptic curve embedded in a surface whose normal bundle satisfies "Diophantine condition" in the Picard variety. By applying this theorem, we construct a K3 surface by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of anti-canonical curves of blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the resulting K3 surface is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction. (Based on joint work with Takato Uehara)

### 2019年05月24日(金)

#### 談話会・数理科学講演会

15:30-16:30   数理科学研究科棟(駒場) 002号室

「ポスト量子」暗号と格子暗号 (日本語)
[ 講演概要 ]

### 2019年05月23日(木)

#### 情報数学セミナー

16:50-18:35   数理科学研究科棟(駒場) 128号室

ビットコインの問題点とその他の仮想通貨、ブロックチェーン (Japanese)
[ 講演概要 ]

### 2019年05月22日(水)

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室

Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic

[ 講演概要 ]
In characteristic zero, cotangent bundle of n(>1)-dimensional smooth projective varieties does not contain a big line bundle. This is a part of Bogomolov vanishing and this vanishing plays an important role in the proof of Miyaoka-Yau inequality. In positive characteristic, it is known that Bogomolov vanishing does not hold. There exists a general type surface whose cotangent bundle contains an ample line bundle. So, it is natural to ask when Bogomolov type vanishing holds in positive characteristic. In this talk, I discuss Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic.

### 2019年05月21日(火)

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Maria de los Angeles Guevara 氏 (大阪市立大学)
On the dealternating number and the alternation number (ENGLISH)
[ 講演概要 ]
Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.

### 2019年05月20日(月)

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室

Cohomology and normal reduction numbers of normal surface singularities (Japanese)
[ 講演概要 ]
The normal reduction number of a normal surface singularity relates the maximal degree of the generators of associated graded algebra for certain line bundles on resolution spaces. We show fundamental properties of this invariant and formulas for some special cases. This talk is based on the joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.