2019年度秋季総合分科会(於:金沢大学)

SUMMARY: Vertex operator algebras are, shortly speaking, algebras of quantum fields and admit a lot of important infinite-dimensional graded representations. They appear as (chiral) symmetry algebras of 2d conformal field theory and these days also appear as a kind of invariants of 4d \(\mathcal {N}=2\) superconformal field theory. It is known that the characters of modules over vertex operator algebras with some finiteness conditions satisfy modular differential equations, which are linear ordinary differential equations invariant under the action of \(SL_2(\mathbb {Z})\). This allows us to study characters of modules using theory of differential equations and modular forms. In this talk, we recall modular differential equations and explain their application to study representations of vertex operator algebras. This talk is based on joint works with Tomoyuki Arakawa and Yuichi Sakai.

SUMMARY: The theory of integrally closed ideals in a two-dimensional regular local ring was developed by Zariski. One of the main results is the product theorem, which asserts that the product of any two integrally closed ideals in a two-dimensional regular local ring is again integrally closed. Since then, the theory has been attracting interest and has been generalized to more general situations. In this talk, I will talk about such a generalization in two different directions. First, I will discuss a possibility in higher dimensional regular local ring. Then, after a brief survey on a notion of integral closure of a module and a theory of integrally closed modules over a two-dimensional regular local ring developed by Kodiyalam, I will talk about a recent result on the ubiquity of indecomposable integrally closed modules of rank two with a monomial Fitting ideal.

SUMMARY: The Jacquet functor is one of the most basic and important functors in representation theory of \(p\)-adic groups. It is a local analogue of Siegel’s \(\Phi \) operator on Siegel modular forms, which is used to define Siegel cusp forms. In this talk, I will compute the Jacquet functors for irreducible tempered representations of symplectic groups \(\mathrm {Sp}(2n,F)\), where \(F\) is a \(p\)-adic group. To do this, one needs some sort of classification of irreducible representations of these groups. As such a classification, I use the local Langlands correspondence developed by Arthur.

SUMMARY: It is known that K3 surfaces admit no global derivations. However, if we allow K3 surfaces to have rational double point singularities (RDPs), then there exist many examples of K3 surfaces with global derivations, at least in small positive characteristics. Derivations \(D\) satisfying \(D^p = D\) (resp. \(D^p = 0\)) correspond to actions of the group scheme \(\mu _p\) (resp. \(\alpha _p\)), and the quotient morphism by such derivations are purely inseparable of degree \(p\). In the case of \(\mu _p\)-actions, we can show that the quotient is a K3 surface (with RDPs) if and only if the action is symplectic in the sense that the global 2-form is invariant under the action: This is an analogue of the result of Nikulin that the quotient of a K3 surface in characteristic \(0\) by a finite group action is a K3 surface (with RDPs) if and only if the action is symplectic. We also show that (in both cases of \(\mu _p\) and \(\alpha _p\)) the quotient singularities are related to the height of the K3 surface: This is peculiar in positive characteristic.

SUMMARY: For a transitive permutation group, if its permutation character is decomposed into the sum of irreducible characters with all their multiplicity 1, then the transitive permutation group is called multiplicity-free. An association scheme constructed by a transitive permutation group is commutative if and only if the transitive permutation group is multiplicity-free. As a generalization of the multiplicity-free condition, we introduce the concept of nearly multiplicity-free for imprimitive permutation groups and some relations to association schemes. In particular, we construct bases of matrix units for such association schemes.

SUMMARY: Let \(p\) be a prime. Suppose that \(P\) is a finite abelian \(p\)-group of type \(m=(m_1,m_2,\cdots )\) with \(m_1\geq m_2\geq \cdots \) and \(\sum m_i=s\). Define nonnegative integers \(u\) and \(v\) by \(u=\max \{m_1,[(s+1)/2]\}\) and \(v=\min \{s-m_1,[s/2]\}\). For each nonnegative integer \(n\), let \(h_n(P)\) denote the number of homomorphisms from \(P\) to the symmetric group \(S_n\) on \(n\) letters. Except for the case where \(p=2\) and \(u+\delta _{v0}\leq v+1\) or \(p=3\) and \(u=v\geq 1\), there exist \(p\)-adic analytic functions \(f_r(X)\) for \(r=0,\,1,\,\cdots ,\,p^{u+1}-1\) and a polynomial \(g(X)\in {\mathbb {Z}}[X]\) such that for any nonnegative integer \(y\), \(h_{p^{u+1}y+r}(P)=p^{\{\sum _{j=1}^up^j-(u-v)\}y}f_r(y)\prod _{j=1}^yg(j)\) and \({\mathrm {ord}}_p(h_{p^{u+1}y+r}(P)) =\{\sum _{j=1}^up^j-(u-v)\}y+{\mathrm {ord}}_p(f_r(y))\). If \(p=2\), \(\lambda _3=0\), and \(u=v\geq 1\) or if \(p=3\) and \(u=v\geq 1\), then \(h_n(P)\) has analogous properties.

SUMMARY: The modular isomorphism problem—which is open for more than 60 years—asks whether \(\mathbb {F}_pG \cong \mathbb {F}_pH\) implies \(G \cong H\) for finite \(p\)-groups \(G\) and \(H\). In this talk, we introduce a new class of finite groups and provide a criterion (sufficient condition) for the problem from adjoint and counting homomorphisms. New proofs for the theorems by Deskins and Passi–Sehgal are provided.

SUMMARY: Let \(\mathcal O\) be a complete discrete valuation ring of characteristic zero with residue class field of characteristic \(p>0\). Let \({\mathcal O}G\) be the group ring of a finite group \(G\) over \(\mathcal O\). Suppose that \(L\) is a virtually irreducible \({\mathcal O}G\)-lattice with vertex \(Q\) and \(p'\)-rank \(Q\)-source. Then the tensor product of an almost split sequence terminating in a Scott \({\mathcal O}G\)-lattice with vertex \(Q\) and \(L\) is the direct sum of an almost split sequence terminating in \(L\) and a split sequence.

SUMMARY: The complete cohomology of a Frobenius algebra is introduced by Nakayama, which is an analogy to Tate cohomology of a finite group. Recently, Wang discovered a Batalin–Vilkovisky (BV) structure on the complete cohomology for a symmetric algebra. In this talk, we show that there exists a BV structure on the complete cohomology for a Frobenius algebra whose Nakayama automorphism is diagonalizable.

SUMMARY: In this talk, we study a relationship between tilting modules with finite projective dimension and dominant dimension with respect to injective modules as a generalization of results of Crawley-Boevey–Sauter, Nguyen–Reiten–Todorov–Zhu and Pressland–Sauter. As an application, we give a characterization of relative Auslander algebras in terms of such tilting modules.

SUMMARY: It is known that an algebra with radical square zero is stable equivalent to a certain hereditary algebra. By comparing indecomposable \(\tau \)-rigid modules between both algebras, we give a characterization of \(\tau \)-tilting finite algebras with radical square zero in terms of the separated quivers. This is an analog of a famous characterization of representation-finite algebras with radical square zero due to Gabriel.

SUMMARY: For a tagged triangulation \(T\) of a marked surface \((S,M)\) of rank \(n\), we study \(g\)-vector cones associated with support \(\tau \)-tilting modules of the Jacobian algebra defined from \(T\). We show that the closure of the union of \(g\)-vector cones associated with all support \(\tau \)-tilting modules is equal to \(\mathbb {R}^n\). As an application, if \((S,M)\) is a closed surface with exactly one puncture, the exchange graph of support \(\tau \)-tilting modules has precisely two connected components. Otherwise, it is connected.

SUMMARY: In representation theory of algebras, endomorphism algebras play an important role. In particular, the endomorphism algebra of a generator has good homological properties. In this talk, I give representation finiteness of a gendo-symmetric algebra, which is the endomorphism algebra of a generator over a symmetric algebra.

SUMMARY: We explore the subject on the weakly Iwanaga–Gorenstien property of gendo algebras. Here, a gendo algebra means the ENDOmorphism algebra of a Generator over an algebra. We state that if a given algebra is representation-finite, then its gendo algebra is weakly Iwanaga–Gorenstein with finite CM type.

SUMMARY: For finite-dimensional simple Lie superalgebras (or supergroups), all irreducible representations can be constructed in an analogous way as the ordinary (non-super) case. However, in general, it is hard to describe its characters. Over an algebraically closed field of characteristic zero, V. Kac determined characters of irreducible representations for “typical” weights. In this talk, we will extend Kac’s result to Chevalley supergroups of type I defined over an arbitrary field.

SUMMARY: Let \(Sp_{2n}\) be a symplectic group and \(B\subset Sp_{2n}\) be a Borel subgroup. It is known that Schubert subvarieties of flag variety \(Sp_{2n}/B\) have singular points. The combinational formula of multiplicities of points on Schubert varieties in symplectic Grassmannian is already known (Ghorpade and Raghavan 2006, Ikeda and Naruse 2009). We were able to obtain a combinatorial formula of multiplicities of points on Schubert varieties of symplectic flag variety. That is an extension of the case of symplectic Grassmannian. This research is a joint work with David Anderson and Minyoung Jeon.

SUMMARY: A Nakashima–Zelevinsky polytope is a rational convex polytope whose lattice points give a polyhedral realization of a highest weight crystal basis. This polytope can be realized as a Newton–Okounkov body of a flag variety, and it induces a toric degeneration. In this talk, we give a recursive construction of a specific class of Nakashima–Zelevinsky polytopes by using Kiritchenko’s Demazure operators on polytopes. From this construction, it follows that polytopes in this class are all lattice polytopes. We also give a geometric application to the normal toric variety associated with a Nakashima–Zelevinsky polytope.

SUMMARY: We investigate the Jordan–Hölder property (JHP) in exact categories. First, we introduce a new invariant of exact categories, the Grothendieck monoids, show that (JHP) holds if and only if the Grothendieck monoid is free, and give some numerical criterion. Next, we apply these results to the representation theory of algebras. In most situation, (JHP) holds precisely when the number of projectives is equal to that of simples. We study examples in type A quiver in detail by using combinatorics on symmetric groups.

SUMMARY: Let \(A\) be a finite-dimensional algebra. We introduce a category \(\mathcal {S}_{m,n}(A)\) consisting of diagrams of monomorphisms between finitely generated \(A\)-modules. We then show that \(\mathcal {S}_{m,n}(A)\) has Auslander–Reiten sequences, and constuct the Auslander–Reiten translation in \(\mathcal {S}_{m,n}(A)\).

SUMMARY: In commutative ring theory, Knörrer’s periodicity theorem plays a crucial role to study maximal Cohen–Macaulay modules over hypersurfaces, and matrix factorizations are essential ingredients to prove the theorem. In order to study noncommutative hypersurfaces, which are important objects in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization and show noncommutative graded versions of Eisenbud’s theorem and Knörrer’s periodicity theorem.

SUMMARY: This talk is based on joint work with Alex Chirvasitu and S. Paul Smith. We introduce a new method to construct 4-dimensional Artin–Schelter regular algebras as normal extensions of 3-dimensional ones. When this is applied to a 3-Calabi–Yau algebra, it produces a flat family of 4-dimensional Calabi–Yau central extensions parametrized by a projective space. The construction is explicit and gives a rich source of new 4-dimensional regular algebras.

SUMMARY: Let \(A=A(\alpha , \beta )\) be a graded down-up algebra with \({\rm deg}\,x=1,\,{\rm deg}\,y=n\geq 1\) and \(\beta \neq 0\). The aim of our talk is to give the dimension formula of the Hochschild cohomology groups \({\rm HH}^{i}(\nabla A)\) of the Beilinson algebra \(\nabla A\) of \(A\). Our result implies that the structure of the bounded derived category \(\mathsf {D^b}(\mathsf {tails}\,A)\) of the noncommutative projective scheme \(\mathsf {tails}\,A\) of \(A\) is different depending on whether \(\begin{pmatrix} 1&0 \end{pmatrix} \begin{pmatrix} \alpha &1 \\ \beta &0 \end{pmatrix}^n \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is zero or not.

SUMMARY: We study an Euler–Fermat type theorem for matrices.

SUMMARY: A module \(M\) is square free if whenever its submodule is isomorphic to \(N^2 = N \oplus N\) for some module \(N\), then \(N = 0\). We introduce the dual concept “d-square free”; a module \(M\) is d-square free if whenever its factor module is isomorphic to \(N^2 = N \oplus N\) for some module \(N\), then \(N = 0\). This property is not closed under submodules and essential extensions in general. The main purpose is to study rings whose d-square free modules are all closed under submodules and essential extensions.

SUMMARY: We have studied about group algebras of non-noetherian groups and showed that they are often primitive if base groups have non-abelian free subgroups. Our main method was two edge-colored graph theory. In general our method using these graphs seems to be effective for a group algebra of a group with a non-abelian free subgroup. But there exist some non-Noetherian groups with no non-abelian free subgroups such as a Thompson group F. In this talk, we introduce an improvement of our graph theory and its application to a problem on a group algebra of a Thompson group F.

SUMMARY: The \(m\)-free arrangement is a generalization of the free arrangement where \(m\) is a nonnegative integer. Holm asked whether all arrangements are \(m\)-free for \(m\) large enough. In a recent work by Abe and the speaker, counter examples are given for the question when the dimension of vector space is grater than three. However the question is still open when \(m\) is three. In this talk I show that \(3\)-arrangements are \(m\)-free for \(m\) large enough and determine \(m\)-exponents in that cases.

SUMMARY: The radius of subcategories of abelian categories was introduced by Dao and Takahashi in 2014 as an analogue of the dimension of triangulated categories. We focus on the category consisting of totally reflexive R-modules G(R) and we find an upper bound of the radius of G(R) when R is a residue class ring of a Noetherian local ring.

SUMMARY: Let \(R\) be a Cohen–Macaulay local ring and \(Q\) be an ideal of \(R\) generated by a regular sequence on \(R\). Due to M. Auslander, S. Ding, and Ø. Solberg, the Auslander–Reiten conjecture holds for \(R\) if and only if it holds for \(R/Q\). In the former part of this talk, we study the Auslander–Reiten conjecture for the ring \(R/Q^\ell \) in connection with that for \(R\). As a corollary, the Auslander–Reiten conjecture holds for determinantal rings with some conditions. In the latter part, we study the existence of Ulrich ideals and generalize the result of J. Sally. We finally show that the Auslander–Reiten conjecture holds if there is an Ulrich ideal whose residue ring is a complete intersection.

SUMMARY: The purpose of this talk is to investigate the structure and ubiquity of Ulrich ideals in a hypersurface ring. In a Cohen–Macaulay local ring \((R, \rm {m})\), an \(\rm {m}\)-primary ideal \(I\) is called an Ulrich ideal in \(R\) if there exists a parameter ideal \(Q\) of \(R\) such that \(I \supsetneq Q\), \(I^2 = QI\), and \(I/I^2\) is \(R/I\)-free. Even for the case of hypersurface rings, there seems known only scattered results which give a complete list of Ulrich ideals, except the case of finite CM-representation type and the case of several numerical semigroup rings. Therefore, in this talk, we focus our attention on a hypersurface ring which is not necessarily finite CM-representation type.

SUMMARY: Torsion in tensor products of modules has been well studied by several authors with relation to Auslander–Reiten conjecture. Such a study is started by Auslander and he proved that over a regular local ring, if the tensor product of finitely generated modules is torsion-free, then these modules are Tor-independent. Three decades later, Huneke–Wiegand generalized this result for hypersurface local rings. The aim of this talk is to prove a generalization of these result using \(n\)-Tor-rigid modules.

SUMMARY: Let \(P\) be a finite poset, \(C(P)\) the chain polytope of \(P\), \(E_K[C(P)]\) the Ehrhart ring of \(C(P)\) over a field \(K\) and \(\omega \) the canonical ideal of \(E_K[C(P)]\). In this talk, we show that the positive and negative symbolic powers of \(\omega \) are identical with the ordinary powers of \(\omega \).

SUMMARY: Let \(P\) be a finite poset, \(O(P)\) (resp. \(C(P)\)) the order polytope (resp. chain polytope) of \(P\), \(E_K[O(P)]\) (resp. \(E_K[C(P)]\)) the Ehrhart ring of \(O(P)\) (resp. \(C(P)\)) over a field \(K\) and \(\omega _{E_K[O(P)]}\) (resp. \(\omega _{E_K[C(P)]}\)) the canonical ideal of \(E_K[O(P)]\) (resp. \(E_K[C(P)]\)). In our previous work, we characterized the generators of \(\omega _{E_K[O(P)]}\).

In this talk, we characterize the generators of \(\omega _{E_K[C(P)]}\). As a corollary, we show that if \(E_K[C(P)]\) is level, the so is \(E_K[O(P)]\). We exhibit an example that shows the converse does not hold true. We also show that, as in the case of \(\omega _{E_K[O(P)]}\), the degrees of the generators of \(\omega _{E_K[C(P)]}\) are consecutive integers.

SUMMARY: In 1987, Hibi introduced a class of commutative rings associated to finite partially ordered sets, which are called Hibi rings. Hibi rings are normal Cohen–Macaulay domains and Koszul. Moreover, Stanley showed that the Hilbert functions of Hibi rings coincide with some counting functions of \(P\)-partitions. In this talk, from the theory of (left) enriched \(P\)-partitions, which are introduced and studied by Stembridge and Petersen, we introduce enriched Hibi rings. In particular, we show that enriched Hibi rings are normal Gorenstein domains and Koszul, and their Hilbert functions coincide with some counting functions of left enriched \(P\)-partitions.

SUMMARY: Let \(S\) be the polynomial ring over a field \(K\) and \(I \subset S\) a homogeneous ideal. Let \(h(S/I,\lambda )\) be the \(h\)-polynomial of \(S/I\) and \(s = \deg h(S/I,\lambda )\) the degree of \(h(S/I,\lambda )\). We are interested in finding a natural class of finite simple graphs \(G\) for which \(S/I(G)\), where \(I(G)\) is the edge ideal of \(G\), satisfies \(s - r = d - e\), where \(r = {\rm reg}(S/I)\), \(d = \dim S/I\) and \(e = {\rm depth} S/I\). Let \(a(S/I(G)) = s - d\) be the \(a\)-invariant of \(S/I\). One has \(a(S/I(G)) \leq 0\). In this talk, by showing the fundamental fact that every Cameron–Walker graph \(G\) satisfies \(a(S/I(G)) = 0\), a classification of Cameron–Walker graphs \(G\) for which \(S/I(G)\) satisfies \(s - r = d - e\) will be exhibited.

SUMMARY: Let \(G\) be a finite simple graph on the vertex set \(V(G) = \{x_1, ..., x_n\}\) and \(I(G) \subset K[V(G)]\) its edge ideal, where \(K[V(G)]\) is the polynomial ring in \(x_1, ..., x_n\) over a field \(K\) with each \(\deg x_i = 1\) and where \(I(G)\) is generated by those squarefree quadratic monomials \(x_{i}x_{j}\) for which \(\{x_i, x_j\}\) is an edge of \(G\). In the present paper, given integers \(1 \le a \le r\) and \(s \ge 1\), the existence of a finite connected simple graph \(G = G(a, r, s)\) with \(\mathrm {im}(G) = a\), \(\mathrm {reg}(R/I(G)) = r\) and \(\deg h_{K[V(G)]/I(G)} (\lambda ) = s\), where \(\mathrm {im}(G)\) is the induced matching number of \(G\) and where \(h_{K[V(G)]/I(G)} (\lambda )\) is the \(h\)-polynomial of \(K[V(G)]/I(G)\).

SUMMARY: Given an Apery-like numbers \(X(n)\) the author had once a cojecture that there hold that \(X(p-r)\) with \(r=2, 3,\ldots \) and \(p\) primes are congruent to \((xp - em)/q\) mod \(p\), where \(e=-1\) or \(1\) and both \(m\) and \(q\) are indepenet of \(p\). If this is true then one can deduce that \(em\) are congruent to \(-q X(p-r)\) mod \(p\), which provides us with a situation of the Chinese remainder theorem. And we get a weapon for finding \(m,q\).

SUMMARY: We study the number of integer solutions \((x,y,l)\) of an equation \(F(x,y)=\Pi _K(l)\), where \(F(x,y)\) is a homogeneous polynomial with integer coefficients and \(\Pi _K(l)\) is a generalized factorial function over number fields. We show a sufficient condition for the finiteness of solutions for \(F(x,y)=\Pi _K(l)\). As a corollary, we obtain the finiteness of solutions for \(P(x)=l!\), where \(P\) is an irreducible polynomial with \(\deg P\ge 2\) or satisfies some condition. This corollary solves the generalized Brocard–Ramanujan problem partially.

SUMMARY: The generating function of Beatty sequence \(\{[k\omega ]\}_{k\geq 1}\) for real irrational \(\omega \) is called Hecke–Mahler series. We also consider exponential-type Hecke–Mahler series \(\sum _{k=1}^\infty z^{[k\omega ]}\) for positive irrational \(\omega \). In this talk, we study the algebraic independence of not only the values of the Hecke–Mahler series or the exponential-type Hecke–Mahler series but also its derivatives at any nonzero distinct algebraic numbers inside the unit circle.

SUMMARY: We derive new explicit formulas for the Dirichlet series of the Liouville and Moebius functions.

SUMMARY: R. R. Hall studied the mean square of the \(k\)th derivative of Hardy’s \(Z\)-function and obtained an asymptotic formula with the error \(O\left (T^{3/4}(\log T)^{2k+1/2}\right )\), as \(T\to \infty \). We show that this error term is estimated by \(O\left (T^{1/2}(\log T)^{2k+1}\right )\).

SUMMARY: In this talk, we discuss a relation between the prime numbers and the distribution of zeros of the Riemann zeta-function under the Riemann Hypothesis. The speaker recently showed a formula for the logarithm of the Riemann zeta-function and its iterated integrals. By using the formula, he obtained some results which are related with the present theme and a value distribution of the Riemann zeta-function. The speaker is going to introduce the formula and these results in this talk.

SUMMARY: It is the famous open problem whether or not the values of the Riemann zeta-function on the critical line is dense in the complex plane. We considered an analogue problem for the function \(\int _{0}^{t} \log \zeta (1/2 + i\beta )d\beta \) and obtained a result that the values of this function is dense in the complex plane under the Riemann hypothesis. In this talk, we will discuss the problem for the function of iterated integral of \(\log \zeta (\sigma + it)\) over the vertical line and explain the above result.

SUMMARY: In this talk, we study the discrete value-distribution of Artin \(L\)-functions associated with cubic fields. We prove that discrete mean values of the Artin \(L\)-functions are represented by integrals involving a density function which can be explicitly described. As an application, we obtain an asymptotic formula of the counting function for a certain family of cubic fields.

SUMMARY: Iterated log-sine integrals which are defined as iterated integrals of (generalized) log-sine integrals was introduced to study on multiple zeta values. In this talk, we give an evaluation of iterated log-sine integrals by multiple polylogarithms and multiple zeta values.

SUMMARY: Recently, Onozuka gave the asymptotic expansion of the multiple zeta-function at non-positive integers. In this talk, we show that coefficients of the asymptotic expansion are evaluated inductively.

SUMMARY: In this talk,we introduce certain integrals, regarded as two parameter deformations of multiple zeta values, and investigate their properties. In particular, we consider two parameter generalizations of the harmonic and shuffle product formulas, which are fundamental relations for multiple zeta values.

SUMMARY: It is known that there exists a series expression of symmetric multiple zeta value of harmonic type. In this talk, we give a series expression of symmetric multiple zeta value of shuffle type. By using this series expression, we give another proof of the shuffle relation for symmetric multiple zeta values.

SUMMARY: In this talk, we will discuss multiple harmonic q-series evaluated at roots of unity. The motivation to study these series comes from recent results on the connection of finite multiple zeta values (FMZV) and symmetrized multiple zeta values (SMZV). We start by giving a small introduction into the theory of multiple zeta values and then discuss their finite analogues, which were introduced by Kaneko and Zagier. After this, we introduce the notion of finite multiple harmonic q-series at roots of unity and show that these specialize to the FMZV and the SMZV through an algebraic and analytic operation, respectively. This talk is based on joint work with Y. Takeyama and K. Tasaka.

SUMMARY: For a positive definite binary quadratic form \(f\), the theta function with a congruence condition is defined as a restricted sum by the congruence condition of the usual theta function associated to the quadratic form. By forming a certain linear combination of these theta functions, we can construct a modular form \(\Theta \) on \(\Gamma _0 (N)\). We compute the first terms of the Fourier expansions of the modular form \(\Theta \) at any cusps by means of the Gauss sums associated with the quadratic form defined by Springer. It turns out that they can be expressed in terms of classical Gauss sums under a certain mild condition.

SUMMARY: We will partly classify spaces of characters of vertex operator algebras with central charges 8 and 16 whose spaces of characters are 3-dimensional and each space of characters forms a basis of the space of solutions of a third order monic modular linear differential equation with rational indicial roots.

SUMMARY: An iterated tower of number fields is constructed by adding preimages of a base point by iterations of a rational map. A certain basic quadratic rational map defined over the Gaussian number field yields such a tower of which any two steps are relative bicyclic biquadratic extensions. Regarding such towers as analogues of a basic \(\mathbb Z_2\)-extension, we examine the parity of the class numbers (and the \(2\)-ideal class numbers) along the towers, with some examples.

SUMMARY: A basic \(2\)-adic Lie extension of a number field is constructed as an iterated tower by a conjugate of Joukowski map. If the number field is totally real, the unramified Iwasawa module over the \(2\)-adic Lie iterated extension is conjecturally pseudo-null under Greenberg’s conjecture for all intermediate cyclotomic \(\mathbb Z_2\)-extensions. The pseudo-nullity is also considered with some examples.

SUMMARY: Let \(f: X \to S\) be a surjective morphism of finite type between connected locally Noetherian normal schemes whose geometric generic fiber \(X_{\overline {\eta }}\) is connected. Conditions that the sequence of the étale fundamental groups \(\pi _{1}(X_{\overline {\eta }},\ast ) \rightarrow \pi _{1}(X,\ast ) \rightarrow \pi _{1}(S,\ast )\rightarrow 1\) becomes exact have been studied, for example, in SGA1. In this talk, I give a sufficient (respectively, necessary and sufficient) condition that the sequence becomes exact in the case where \(f\) is flat or \(S\) is regular (respectively, \(S\) is a hyperbolic curve over a field of characteristic \(0\)) which is written in terms of algebraic stacks.

SUMMARY: Given a positive integer \(m\), if positive integers \(a\) and \(A\) satisfy \(A=\sigma (a)+m\) and \(\sigma (A)=2a+m\), then \(a\) is said to be a super perfect number with translation parameter \(m\), \(A\) its partner.

If \(a=2^e\) then \(A \) are primes. Given a positive integers \(m\), if positive integers \(a\) and \(A\) satisfy \(A=\sigma (a)-m\) , \(\sigma (A)=2a-2m+1\) then \(a\) is said to be a Mersenne perfect number.

If \(a\) is prime then \(A=2^e\) . The converse is true.

SUMMARY: By using the first Hochschild cohomology \(H^{1}(A, {\rm M}_n(k)/A)\), we can describe when the orbit morphism \(P \mapsto PAP^{-1}\) from \({\rm PGL}_n\) to the moduli of subalgebras of the full matrix ring is smooth. We also calculate Hochschild cohomology \(H^{i}(A, {\rm M}_3(k)/A)\) for several \(k\)-subalgebras \(A\) of \({\rm M}_3(k)\).

SUMMARY: For a germ of a variety in positive characteristic and a non-zero ideal sheaf on the variety, we can define the F-pure threshold of the ideal by using Frobenius morphisms, which measures the singularities of the pair. In this talk, I will show that the set of all F-pure thresholds with fixed embedding dimension satisfies the ascending chain condition. This is a positive characteristic analogue of the “ascending chain condition for log canonical thresholds” in characteristic 0, which was recently proved by Hacon, McKernan, and Xu.

SUMMARY: We proved Bertini type theorems in mixed characteristic case. As an application, we find sufficiently many normal Cartier divisors from normal arithmetic schemes.

SUMMARY: We study automorphism groups of projective plane curves over the complex number field. Recently, Harui gave a classification of automorphism groups of smooth curves. For each group \(G\) in the classification, we can ask which curves have \(G\) as their automorphism groups. Especially, we consider the projective plane curves whose automorphism group is the alternative group \({A}_{6}\) that is embedded in \(PGL(3,\mathbb {C})\), called the Valentiner group. The invariant ring of the Valentiner group and the geometric properties of some invariant curves were studied by Wiman. We use this to find all \(d\) such that there exist nonsingular or irreducible curves of degree \(d\) whose automorphism group is the Valentiner group.

SUMMARY: We can classify toric manifolds as algebraic varieties in terms of the associated fans, but we do not know the classification of toric manifolds as differentiable manifolds. On this topic, the problem whether toric manifolds can be distinguished as differentiable manifolds in terms of cohomology rings is well studied. In this talk, we will talk about some results on this topic in the case of toric Fano manifolds.

SUMMARY: It is conjectured that \(\mathbb {Q}\)-Gorenstein (qG-)deformation equivalence classes of locally qG-rigid class TG orbifold del Pezzo surfaces with Euler characteristic \(n\) and singular locus \(\mathcal {B}\) are in one-to-one correspondence with mutation equivalence classes of Fano polygons with singularity content \((n,\mathcal {B})\). In this talk, for the classification of qG-deformation equivalence classes, we will classify all Fano polygons with singularity content \((0,\{\frac {1}{r}(1,s_1),\ldots ,\frac {1}{r}(1,s_k)\})\), where \(1 \leq s_i < r\) is coprime to \(r\).

SUMMARY: We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.

SUMMARY: Any \(3\)-dimensional affine normal quasihomogeneous \(SL(2)\)-variety, which was shown by Popov to be parameterized by two numbers, has an equivariant resolution of singularities given by an invariant Hilbert scheme. The main purpose of this talk is to provide a necessary and sufficient condition on the parameter for the invariant Hilbert scheme to be the minimal resolution of such an \(SL(2)\)-variety.

SUMMARY: A criterion for the existence of higher-uniruledness=lower-rationality properties, which consistute a hierachy interpolating uniruledness and unirationality, is given. This criterion is stated in terms of some numerical condition involving the Chern classes of the tangent bundle, and the proof make use of generalized Bott towers.

SUMMARY: The complex moduli space of a Calabi–Yau manifold is naturally a Kähler manifold with the Weil–Petersson metric. In light of the mirror duality, we expect that the Kähler moduli space carries an equivalently rich geometric structure. In this talk, I will introduce our program to develop Weil–Petersson geometry on the Kähler moduli space via the stability conditions of triangulated categories. This is a joint work with Yu-Wei Fan and Shing-Tung Yau.

SUMMARY: For \(3\)-dimensional (semistable) extremal neighborhood \((X,C)\), according to terminologies of [S. Mori 1988], the author reported an analogue of Miyaoka–Yau type inequality with the associated \(c_3\) (Mar., 2018), and also reported the related Reider type theorem by the moduli space of the associated coherent systems (Sep., 2018), and moreover reported such a type inequality in which \(c_2,c_3\) have coefficients (Mar., 2019) under the case \(C\) is not necessary irreducible nor reduced,with aim to characterize Mukai–Umemura \(3\)-folds. In this talk, based on these studies, for the case \(C\) is not necessary irreducible nor reduced, the author will report the studies about local-to-global automorphisms of \((X,C)\) related to such a Miyaoka–Yau type inequality with having coefficients of \(c_2,c_3\) by inducing Higgs sheaves structure on \((X,C)\) according to [Y. Miyaoka 2009].