2018年度秋季総合分科会(於:岡山大学)

SUMMARY: Let \(G\) be a group and \(\Bbbk \) a commutative ring. A small category with a weak \(G\)-action is a pair \((\mathcal {C}, X)\) of a small category \(\mathcal {C}\) and a pseudofunctor \(X\) from \(G\) as a groupoid with one object \(*\) to the 2-cateory \(\Bbbk \)-Cat of small \(\Bbbk \)-categories sending \(*\) to \(\mathcal {C}\). We will explain the facts that equivalences between the 2-category of small \(\Bbbk \)-categories with weak \(G\)-actions the 1-morphism (resp. the 2-morphisms) of which are \(G\)-equivariant functors (resp. natural transformations compatible with \(G\)-equivariances) and the 2-category of G-graded \(\Bbbk \)-categories the 1-morphisms (resp. 2-morphisms) of which are weakly degree-preserving functors (resp. natural transformations compatible with degree-preserving structures) are given by 2-functors defined by orbit category constructions and by smash product constructions that are mutually quasi-inverses and that (when \(\Bbbk \) is a field) the derived equivalences defined on each 2-category are preserved under these constructions. We will also talk about some generalizations of these facts.

SUMMARY: In this talk, we introduce what we call a Schur multiple zeta function. This is a zeta-function analogue of Schur function and interpolates both the multiple zeta and multiple zeta-star functions of Euler–Zagier type. We first show some combinatorial relations among Schur multiple zeta functions coming from theory of Schur functions such as Jacobi–Trudi and Giambelli formulas and then give some explicit formulas for special values at positive integers of them such as 1–3 formulas, which are analogues of those for above-mentioned multiple zeta functions. The former is a joint work with Maki Nakasuji and Ouamporn Phuksuwan and the latter is with Henrik Bachmann.

SUMMARY: We shall overview how the notion of modulus enables us to generalize many aspects of motive theory. One of the earliest examples can be seen in Laumon’s generalization of the theory of Deligne \(1\)-motives. Another example is given by our work with Kahn and Saito on a modulus version of Voevodsky’s triangulated category of motives. Recently, in our joint work with Ivorra, we have introduced a Hodge theoretic counterpart that generalizes the classical notion of mixed Hodge structures. As an application, we generalize Kato–Russell’s construction of Albanese varieties with modulus to \(1\)-motives.

SUMMARY: A hyperplane arrangement \(\mathcal {A}\) is a finite set of linear hyperplanes in a fixed vector space. We may associate to an arrangement, the module of logarithmic vector fields \(D(\mathcal {A})\) which is a graded reflexive module. We say that an arrangement is free if \(D(\mathcal {A})\) is a free module. Free arrangements and logarithmic vector fields relates its algebraic structure with the topological and combinatorial aspects of arrangements. In particular, when \(\mathcal {A}\) is free, we can describe the topological Poincaré polynomial of the complement of \(\mathcal {A}\) in terms of the splitting type of \(D(\mathcal {A})\) by Terao’s factorization. However, to determine the freeness is difficult in general, and whether the freeness depends only on the combinatorial structure of an arrangement is an open problem, called the conjecture of Terao. We explain recent developments on this problem, mainly from the combinatorial point of view. Also, we will describe a recently found relation with logarithmic derivation modules and the cohomology ring of a regular nilpotent Hessenberg variety.

SUMMARY: The behavior of trace ideals and modules is explored in connection with the structure of the base ring and the ambient module. Firstly, over a commutative Noetherian ring, a characterization of a module for which every submodule is a trace module is given. Secondly, over an arbitrary commutative ring, correspondences between three sets, the set of trace ideals, the set of certain stable ideals, and the set of certain birational extensions of the base ring, are studied. The correspondences work very well, if the base ring is a Gorenstein ring of dimension one. As a conclusion, it is shown that with one extremal exception, the surjectivity of some correspondence characterizes the Gorenstein property of the base ring, provided the base ring is a Cohen–Macaulay local ring of dimension one.

SUMMARY: The purpose of this talk is to investigate the behavior of chains of Ulrich ideals, in a one-dimensional Cohen–Macaulay local ring, in connection with the structure of birational finite extensions of the base ring. The notion of Ulrich ideals is a generalization of stable maximal ideals, which J. Lipman started to analyze in 1971. Because Ulrich ideals are a very special kind of ideals, it seems natural to expect that, in the behavior of Ulrich ideals, there might be contained ample information on base rings, once they exist. We focus our attention on the one-dimensional case, clarifying the relationship between Ulrich ideals and the birational finite extensions of the base ring.

SUMMARY: Huneke-Wiegand conjecture is, raughly saying, \(M \otimes _R \mathsf {Hom}_R(M, R)\) has torsion for any non-free torsion-free finitely generated module \(M\) over a commutative Noetherian ring \(R\). This conjecture is proved for hypersurface rings by Huneke and Wiegand and it is thought to be true for complete intersection rings. However, the conjecture is very much open and not known even in the case where \(M\) is an ideal. In this talk, I will give some equivalent statements of the conjecture and give another proof of Huneke–Wiegand theorem.

SUMMARY: In this talk, it is shown that the edge ring of a finite connected simple graph with a 3-linear resolution is a hypersurface.

SUMMARY: It is known that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope. A stronger question is to ask whether every \((0,1)\)-polytope is unimodularly equivalent to a facet of some reflexive polytope. A large family of \((0,1)\)-polytopes are the edge polytopes of finite simple graphs. In this talk, it is shown that, by giving a new class of reflexive polytopes, each edge polytope is unimodularly equivalent to a facet of some reflexive polytope. Furthermore, we extend the characterization of normal edge polytopes to a characterization of normality for these new reflexive polytopes.

SUMMARY: In this talk, given arbitrary integers \(d\) and \(r\) with \(d \geq 4\) and \(1 \leq r \leq d + 1\), a reflexive polytope \(\mathcal {P} \subset \mathbb {R}^d\) of dimension \(d\) with \({\rm depth\ } K[\mathcal {P}] = r\) for which its dual polytope \(\mathcal {P}^\vee \) is normal will be constructed, where \(K[\mathcal {P}]\) is the toric ring of \(\mathcal {P}\).

SUMMARY:  Let \(S\) be the polynomial ring in \(n\) variables over a field \(K\) and \(H\) a bipartite graph with \(n\) vertices. We denote by \(I(H)\) the edge ideal of \(H\). We are interested in the structure of \(H\) when \(S/I(H)\) is a “good” ring. In 2005, Hezrog and Hibi characterized Cohen–Macaulay bipartite graphs in terms of partially ordered sets. In 2007, Villarreal characterized unmixed bipartite graphs in terms of quasi-ordered sets.

 In this talk, we introduce the notion of semi-unmixed graphs and investigate semi-unmixed bipartite graphs in terms of partially ordered sets. As a application, we investigate sequentially Cohen–Macaulay bipartite graphs. Moreover, we calculate the regularity of semi-unmixed bipartite graphs.

SUMMARY: Let \( K\) be a field, \(H\) a finite distributive lattice, \({\cal R}_K[H]\) the Hibi ring defined over \( K\) on \(H\) and \(\omega \) the canonical ideal of \({\cal R}_K[H]\). Since \({\cal R}_K[H]\) is a Noetherian normal domain, and \(\omega \) is a divisorial ideal of \({\cal R}_K[H]\), we can consider the \(n\)-the power \(\omega ^{(n)}\) of \(\omega \) in \(D({\cal R}_K[H])\) for each \(n\in Z\). We call \(\omega ^{(-1)}\) the anticanonical ideal of \({\cal R}_K[H]\). In this talk, we introduce the notion “\(q^{(n)}\)-reduced sequence with condition N” and define a convex polytope for each \(q^{(\pm 1)}\)-reduced sequence with condition N. We show that the fiber cone of \(\omega \) (resp. \(\omega ^{(-1)}\)) is the sum of the Ehrhart rings defined by these convex polytopes and describe the analytic spread of \(\omega \) (resp. \(\omega ^{(-1)}\)) by the words of posets.

SUMMARY: Lyubeznik and Smith introduced for a commutative ring \(R\) with prime characteristic and an \(R\)-module \(M\) the notion of a ring of Frobenius operators \({\cal F}(M)\) which is a noncommutative graded ring. Enescu and Yau noticed that the complexity of \({\cal F}(E)\) is an important object to study, where \(E\) is the injective hull of the residue field of \(R\). They showed that the Frobenius complexities of the Segre product of polynomial rings with \(m\) and \(n\) variables over a field with characteristic \(p\) have limit \(m-1\) as \(p\to \infty \) if \(m>n\geq 2\). Page generalized this result that if a Hibi ring is anticanonical level and is not Gorenstein, then the limit of the Frobenius complexity of the Hibi rings (base field changed) converges to the analytic spread of the anticanonical module minus 1. In this talk, we report that this fact hold true for arbitrary Hibi rings which is not Gorenstein and the answer the question of Page affirmatively.

SUMMARY: Let \(I \subset S= K[x_1, \ldots , x_n]\) be a strongly stable ideal whose generators have degree at most \(d\). It is known that \(I\) admits the alternative polarization b-pol\((I) \subset K[x_{i,j} \mid 1 \le i \le n, 1 \le j \le d]\). We show that the Alexander dual ideal of b-pol\((I)\) is b-pol\((I^*)\) of some strongly stable ideal \(I^* \subset K[x_1, \ldots , x_d]\), after switching the variable \(x_{i,j} \longmapsto x_{j,i}\). This duality between \(I\) and \(I^*\) is in some sense known, but our construction has some advantage. We can described the Hilbert series of \(H_{\mathfrak m}^i(S/I)\) by the graded Betti numbers of \(I^*\). We also show that if \(S/I\) is Cohen–Macaulay then b-pol\((I)\) is a letterplace ideal in the sense of Fløystad.

SUMMARY: Let \(A=\bigoplus _{i=0}^{c}A_{i}\), \(A_{c}\neq 0\), be a graded Artinian algebra. We say that \(A\) has the strong Lefschetz property if there exists an element \(L\in A_{1}\) such that the multiplication map \(\times L^{c-2i}:A_{i}\to A_{c-i}\) is bijective for each \(i\in \{0,1,\ldots , \lfloor \frac {c}{2}\rfloor \}\). In this presentation, we consider an algebra constructed from a graph. The algebra is defined to be the quotient algebra of the ring of the differential polynomials by the annihilator of \(F_{\Gamma }\), where \(F_{\Gamma }\) is the weighted generating function for the spanning trees in \(\Gamma \). For \(n\leq 5\), we show the strong Lefschetz property for the algebra corresponding to the complete graph with \(n\) vertices.

SUMMARY: We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules.

SUMMARY: Let \(A\) be a non-negatively graded differential (DG) algebra over a commutative ring \(R\), and \(B=A\langle X | dX = t\rangle \) be an extended DG algebra by the adjunction of a variable \(X\) of positive even variable \(n\) which kills a cycle \(t\) in \(A\). Let \(N\) be a semi-free DG \(B\)-module. The aim of this talk is to explain the following our results; \((1)\) If \(N\) is bounded below and \(\mathrm {Ext}^{n+1}_B(N,N)=0\), then \(N\) is liftable to \(A\), i.e., there is a DG \(A\)-module \(M\) such that \(N\cong B\otimes _AM\). \((2)\) If \(N\) is liftable to \(A\) and \(\mathrm {Ext}^{n}_B(N,N)=0\), then the lifting of \(N\) is unique up to DG \(A\)-isomorphisms.

SUMMARY: R. Ware gave the following problem in his paper;
Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), 233–256.

Let \(R\) be a ring and \(P\) a projective right \(R\)-module with unique maximal submodule \(L\), then \(L\) is the largest submodule of \(P\).

We give the affirmative answer to this Ware’s problem.

To solve this problem, we generalize Nakayama–Azumaya Lemma for any projective modules.

SUMMARY: We treat a certain class of basic algebras which contains preprojective algebras of type A, Nakayama algebras, and generalized Brauer tree algebras. We provide a necessary condition for that an algebra share the same support \(\tau \)-tilting poset with a given algebra \(A\) in this class. Furthermore, we see that this necessary condition is also a sufficient condition if \(A\) is either a preprojective algebra of type A, a Nakayama algebra, or a generalized Brauer tree algebra.

SUMMARY: Cluster algebras are commutative algebras with a distinguished set of generators, which are called cluster variables. By Laurent phenomenon, any cluster variable is expressed by a Laurent polynomial of the initial cluster variable. An explicit formula for the Laurent polynomials of cluster variables is called a cluster expansion formula. We give a cluster expansion formula for cluster algebras defined from triangulated surfaces by using perfect matchings of angles. Moreover, they correspond bijectively with perfect matchings of snake graphs, perfect matchings of bipartite graphs, and minimal cuts of quivers with potential.

SUMMARY: Path algebras are one of the most classical and important classes of algebras. In this talk, we discuss torsion pairs for path algebras. In particular, we review a close relationship with some elements of the Coxeter group, called sortable elements, and explain how to parametrize the torsion pairs by these objects via preprojective algebras.

SUMMARY: In this talk, we study a relationship between silting objects and bounded \(t\)-structures. Let \(A\) be a finite dimensional algebra over a field. Then there exists an injective map from silting objects of the bounded homotopy category of finitely generated projective \(A\)-modules to bounded \(t\)-structures on the bounded derived category of finitely generated \(A\)-modules. However, the map is not necessarily bijective (e.g., \(A\) is the path algebra of the Kronecker quiver). We give a characterization of the injective map being bijective.

SUMMARY: Let \(A\) be an algebra over a field \(K\). For the real-valued Grothendieck group \(K_0(\mathsf {proj\,} A)_\mathbb {R}\) of the projective module category \(\mathsf {proj}\, A\) and the real-valued Grothendieck group \(K_0(\mathsf {mod}\, A)_\mathbb {R}\) of the module category \(\mathsf {mod}\, A\), there exists a non-degenerate \(\mathbb {R}\)-bilinear form called Euler form. Each \(\theta \in K_0(\mathsf {proj}\, A)_\mathbb {R}\) gives a semistable subcategory \(\mathcal {W}_\theta \) of \(\mathsf {mod}\, A\). \(\mathcal {W}_\theta \) is an abelian subcategory of \(\mathsf {mod}\, A\), so its simple objects are bricks. In this talk, I set \(\varTheta _S:= \{ \theta \in K_0(\mathsf {proj}\, A)_\mathbb {R} \mid S \in \mathcal {W}_\theta \}\) for each brick \(S\), and consider the chamber structure of an Euclidean space \(K_0(\mathsf {proj}\, A)_\mathbb {R}\) with the walls given by \(\varTheta _S\) for all bricks \(S\).

SUMMARY: Lin and Xi introduced Auslander–Dlab–Ringel (ADR) algebras of semilocal modules as a generalization of original ADR algebras. In this talk, we prove that ADR algebras of semilocal modules are left-strongly quasi-hereditary algebras. As an application, we give a tightly upper bound for global dimension of an ADR algebra. Moreover, we describe characterizations of original ADR algebras to be strongly quasi-hereditary algebras which are a special class of left-strongly quasi-hereditary algebras.

SUMMARY: We give complete characterization of two-sided Harada rings using weak co-\(H\)-sequences.

SUMMARY: Classification of AS-regular algebras is one of the main interests in non-commutative algebraic geometry. Recently, a complete list of superpotentials (defining relations) of all 3-dimensional AS-regular algebras which are Calabi–Yau was given by Mori–Smith (the quadratic case) and Mori–Ueyama (the cubic case), however, no complete list of defining relations of all \(3\)-dimensional AS-regular algebras has not appeared in the literature. So the purpose of this research is to give a complete list of defining relations of all \(3\)-dimensional quadratic AS-regular algebras, to classify them up to isomorphism, and up to graded Morita equivalence in terms of their defining relations. In this talk, for the case that the point scheme is an elliptic curve, we give classifications up to isomorphism, and up to graded Morita equivalence in terms of their defining relations.

SUMMARY: Richard P. Stanley has conjectured that chromatic symmetric functions are complete invariants for trees. Vesselin Gasharov has conjectured that chromatic symmetric functions of claw-free graphs are \(s\)-positive. In this talk, we consider the analog for trivially perfect graphs and cographs.

SUMMARY: The double Grothendieck polynomials introduced by Lascoux–Schützenberger represent the equivariant K-theory Schubert classes for the type A flag varieties. Their explicit formulas have been known for vexillary (2143 avoiding) permutations in the form of determinants or set-valued tableaux. Recently Anderson–Chen–Tarasca (2017) obtained the determinant formula in the case of 321 avoiding permutations. Motivated by this result, we obtained their tableau formula too. I will report on this new formula with its combinatorial proof.

SUMMARY: Hessenberg varieties are subvarieties of a full flag variety. Its topology makes connection with other research areas such as geometric representation of Weyl groups, the quantum cohomology of flag varieties, hyperplane arrangements, and the chromatic quasisymmetric functions in graph theory. In this talk, we consider polynomials which determine the fundamental relation in the cohomology of a regular nilpotent Hessenberg variety, and I will explain that each of the polynomials can be written as an alternating sum of certain Schubert polynomials.

SUMMARY: We construct all relative invariants of prehomogeneous vector spaces associated with valued quivers of type \(\mathbb {B}\).

SUMMARY: We obtain a summation formula of a generalization of the Fibonacci numbers which is special values of the homogenous complete symmetric polynomials. We also give an expression formula of our Fibonacci numbers.

SUMMARY: It is well-known that the center of a Chevalley groups (or a split and connected reductive algebraic group) can be described in terms of its root datum. In this talk, we generalize the result to the super situation. We give an explicit description of centers of Chevalley supergroups.

SUMMARY: Categorifying the symmetric or skew Howe representation of \(U_q(\mathfrak {gl}_m)\), we construct a 2-representation of the quiver Hecke (KLR) algebra of \(U_q(\mathfrak {gl}_m)\). In the skew case, the 2-representation is realized in a category of matrix factorizations (a joint work with Mackaay). In the symmetric case, the 2-representation is realized in a bimodule category over a deformation of Webster algebra of type \(A_1\) (a joint work with Khovanov, Lauda, and Sussan). As a consequence, we obtain a braid group action on the homotopy category of each of the above categories.

SUMMARY: In this talk I will give a candidate notion to unify exact categories and triangulated categories. This talk is partly based on a joint work with Yann Palu. If the time permits, I will also introduce recent developments on this class of categories.

SUMMARY: We determine the ring structure of the Hochschild cohomology \(HH^*(\Bbb {Z} G)\) of the integral group ring of the semidihedral group \(G\) of order \(8\ell \) for arbitrary integer \(\ell \geq 2\) by giving the precise description of the integral cohomology ring \(H^*(G, \Bbb {Z})\).

SUMMARY: In this talk, I give a generalization of Matsuda’s theorem and some results. In particular, I give isomorphism between partial Burnside rings of different groups. Moreover, I consider the relationship between an image of Frobenius–Wielandt homomorphism, a partial Burnside ring, and a structure of a group.

SUMMARY: Let \(k\) be a field of prime characteristic \(p\) and \(G\) a finite group. We shall give a theorem on direct summands of a source algebra of a block ideal of the group algebra \(kG\). Let \(b\) be a block ideal of \(kG\) with \(P\) as a defect group. If an element \(x\) in \(G\) satisfies some conditions, which cannot be written down here although, then the \((kP,kP)\)-bimodule \(k[PxP]\) is isomorphic with a direct summand of the source algebra of the block ideal with the multiplicity congruent to \(1\) modulo \(p\).

SUMMARY: We derive explicit formulas for the discriminants of classical quasi-orthogonal polynomials, as a full generalization of the result of Dilcher and Stolarsky (2005). We consider a certain system of Diophantine equations, originally designed by Hausdorff (1909) as a simplification of Hilbert’s solution (1909) of Waring’s problem, and then create the relationship to quadrature formulas and quasi-Hermite polynomials. We reduce these equations to the existence problem of rational points on a hyperelliptic curve associated with discriminants of quasi-Hermite polynomials, and thereby show a nonexistence theorem for solutions of Hausdorff-type equations.

SUMMARY:   Tomography is the field that reconstructs a three-dimensional object from its two-dimensional cuts. Let \(f\) be a function on \(\mathbb {Z}^n\), and \(\mathbf {w}\) be a finite subset of \(\mathbb {Z}^n\). Discrete tomography reconstructs the function \(f\) from the data \(f_{\mathbf {w}+p}=\sum _{x\in \mathbf {w}+p}f(x),p\in \mathbb {Z}^n.\) This problem is proved by F. Hazama to be described completely by the zero locus of a certain polynomial in \(n\) variable associated with \(\mathbf {w}\). The purpose of this talk is to apply his result to the zero-sum arrays when the window \(\mathbf {w}\) has the form \(\mathbf {w}=\{(0,1),(0,0),(1,0),\cdots ,(n-2,0),(n-1,0)\}\). Furthermore we describe the way how one can find the rational zero-sum arrays for \(\mathbf {w}\).

SUMMARY: In this talk, by making use of a modular matrix, we consider a generalization of the \(p\)-adic reciprocity formula for \(p\)-adic Dedekind sums due to Snyder.

SUMMARY: We show mean square theorems of the double zeta function.

SUMMARY: The speaker studied on a function \(S(t)\) which is related to the zero distribution of the Riemann zeta-function. This function has a well-known estimate under the assumption of the Riemann Hypothesis. The speaker has proved that this assumption can be weaken and is going to report the result in this talk.

SUMMARY: The poly-Bernoulli numbers and its relative are defined by the generating series using the polylogarithm series, and we call them type \(B\) and \(C\), respectively. As a generalization of these poly-Bernoulli numbers, we introduce hook Schur type Bernoulli numbers, which has relation with Schur multiple zeta functions of hook type. We obtain the relation between hook Schur type poly-Bernoulli numbers of type \(B\) and that of type \(C\). Furthermore, we define a generalization of Arakawa–Kaneko multiple zeta functions and Kaneko–Tsumura type multiple zeta functions to hook Schur type, and obtain their expression using hook Schur type Bernoulli numbers.

SUMMARY: Ihara, Kaneko, and Zagier defined two regularizations of multiple zeta values and proved the regularization theorem that describes the relation between those regularizations. We show that the regularization theorem can be generalized to polynomials whose coefficients are regularizations of multiple zeta values and that specialize to symmetric multiple zeta values defined by Kaneko and Zagier.

SUMMARY: In 2001, J. M. Borwein, D. J. Broadhurst and J. Kamnitzer proved a formula including one multiple zeta value and values of multiple polylogarithms at \(e^{\frac {\pi }{3}i}\). We show that this formula can be regarded as the formula including one multiple zeta value and values of log-sine integrals at \(\pi /3\). Moreover, we introduce some applications of this formula to the theory of multiple zeta values.

SUMMARY: Here, given an integer \(m\), \(a\) is said to be a super perfect number with translation parameter \(m\), if \( \sigma (\sigma (a)+m)=2a+m .\) For \(m=-28,-18,-14\), structure of super perfect numbers with translation parameters \(m\) are investigated in detail.

SUMMARY: We shall consider the radii of circles which are tangental to each othe and to the quadratic curves. These radii are derived from ‘the symmetric recurrent formula’. The solutions of Pell’s equation and every other Fibonacci sequence are also derived from the symmetric recurrent formula. We shall research the characters of the symmetric recurrent formula and express the radii of circles in the quadratic curves by the symmetric recurrent formula uniformly. And more we shall express the solutions of Pell’s equation and the every other Fibonacci sequence as the radii of the circles in the hyperbola.

SUMMARY: This short talk gives that the graph of an increasing positive integer sequence approximated by a function whose second derivative goes to zero faster than or equal to \(1/x^\alpha \) for some \(\alpha >0\), contains arbitrarily long arithmetic progressions. As a corollary, it follows that the graph of the sequence of the integer parts of \(\{n^{a}\}_{n=1}^{\infty }\) contains arbitrarily long arithmetic progressions for every \(1\le a<2\). We also prove that the graph of the same form sequence does not contain any arithmetic progressions of length \(3\) for every \(a\ge 2\).

SUMMARY: We investigate the cardinality of subsets of the matrix ring over certain residue ring.

SUMMARY: Let \(X\) be an abelian surface, and \(D^{(n)}\) be the sum of \(n\) distinct theta divisors having normal crossings. A set of defining relations of the fundamental group of \(X-D^{(n)}\) is determined.

SUMMARY: We consider a Diophantine equation about the factorial function over number fields. It is not known whether or not there exist infinitely many solutions of it. We give a necessary and sufficient condition for the existence of trivial solution and show the finiteness of trivial solutions. In addition we give an explicit upper bound for trivial solutions.

SUMMARY: We show that, for an abelian variety defined over a \(p\)-adic field \(K\) which has potential good reduction, its torsion subgroup with values in the composite field of \(K\) and a certain Lubin–Tate extension over a \(p\)-adic field is finite.

SUMMARY: We consider a question about congruences for the Fourier coefficients of vector valued Siegel modular forms of type \((k,2)\), which was answered by Sturm in the case of an elliptic modular form and by Choi–Choie–Kikuta, Poor–Yuen and Raum–Richter in the case of scalar valued Siegel modular form.

SUMMARY: It is well known that modular linear differential equations (MLDEs) appear as one of tools in studies related to supersingular elliptic curves and classifications of characters of vertex operator algebras (VOAs). In some cases, MLDEs give a certain correspondence between modular forms and characters of VOAs. In this talk, we determine the properties of coefficients of MLDEs of any order. Furthermore, we give a general expression of MLDEs under the natural assumption for the ring structure of (quasi)modular forms.

SUMMARY: For an even positive integer \(\ell \), \(d_{\ell }'\) denotes the smallest integer \(d\) such that the minimal period of the simple continued fraction expansion of \(\sqrt {d}\) is equal to \(\ell \), where \(d\) runs through non-square positive integers with \(d\) congruent to \(2\) or \(3\) modulo \(4\). We can observe some characteristic phenomena from numerical data; especially, for each even positive integer \(\ell \) less than or equal to \(83552\), the class number of a real quadratic field \({\mathbb Q}(\sqrt {d_{\ell }'})\) is equal to \(1\). In this talk, we investigate partial quotients of the continued fraction expansions of \(\sqrt {d}\), and then consider some properties of that of \(\sqrt {d_{\ell }'}\).

SUMMARY: The aim of this talk is to give a lower bound for the class number of real quadratic fields \({\mathbb Q}(\sqrt {d})\) of minimal type such that the primary symmetric part of the simple continued fraction expansion of \(\sqrt {d}\) is of ELE type. By applying this lower bound to a sequence \(\langle 2,\ldots ,2,2,1\rangle \) of pre-ELE type, we get an infinite family of real quadratic fields with non-trivial class number.

SUMMARY: Let \(\mathbb {F}_q\) be the finite field with \(q\) elements. For a monic \(m \in \mathbb {F}_q[T]\), let \(h_m\) be the class number of the \(m\)th cyclotomic function field. The goal of this talk is to determine the \(p\)-divisibility of \(h_m\) when \(\deg m=2\).

SUMMARY: A post-critically finite rational map \(\phi \) of prime degree \(p\) yields a sequence of finitely ramified iterated extensions of number fields, and sometimes provides an arboreal Galois representation with a \(p\)-adic Lie image. In this talk, regarding such sequences by \(\phi (x)=x^2-2\) as analogues of \(\mathbb {Z}_{2}\)-extensions, we study the size of \(2\)-part of ideal class groups along the sequences or \(2\)-adic Lie extensions.

SUMMARY: Quantization is studied from a viewpoint of Galois theory, in which the field in mathematics to which a partition function or a wave function in physics belongs, be the algebraic extension of the field in mathematics to which an action and fields of a system in physics belong. This viewpoint was proposed by Y. Nambu three decades ago. Here, choosing quantum mechanics (one dimensional field theory) as an example, it is shown that the different Galois extension corresponds to the different quantization scheme in physics. Although one type of Galois’ extension reproduces the usual quantization scheme, there exist other schemes of quantization, if we follow Galois and Nambu.

SUMMARY: Many mathematicians have researched the transcendence of the values of power series at algebraic point \(\alpha \). In particular, Bailey, Borwein, Crandall, and Pomerance gave remarkable criterion for transcendence in the case where \(\alpha =1/2\). Consequently, we deduce that \(\sum _{n=1}^{\infty } 2^{-\lfloor n^{\log n}\rfloor }\) is transcendental, which cannot be proved by early methods. Their result was generalized for the case of \(\alpha =\beta ^{-1}\), where \(\beta \) is a Pisot or Salem number. In this talk, we investigate partial results on the transcendence of the values of power series in the case where \(\beta \) is a more general algebraic integer.

SUMMARY: We describe the moduli of 3-dimensional subalgebras of the full matrix ring of degree 3.

SUMMARY: Let \({\rm Mold}_{n, d}\) be the moduli of \(d\)-dimensional subalgebras of the full matrix ring of degree \(n\) over \(\Bbb Z\). We describe the dimension of the Zariski tangent space \(T_{x}{\rm Mold}_{n, d}\) and the smoothness of \({\rm Mold}_{n, d} \to {\Bbb Z}\) at \(x\) by using Hochschild cohomology.

SUMMARY: Let \(G\) be a finite subgroup of \({\rm SL}(n,\mathbb {C})\), then the quotient \(\mathbb {C}^n/G\) has a singularity. \({\rm Hilb}^G(\mathbb {C}^n)\) is known to be related to crepant resolutions of the quotient singularity. If \(n=2\) or \(3\), \({\rm Hilb}^G(\mathbb {C}^n)\) is crepant resolutions. But when n is greater than or equal to four, it is not in generally cases. We will show the existence of a crepant resolutions for series of finite abelian subgroup of \({\rm SL}(4,\mathbb {C})\) via \({\rm Hilb}^G(\mathbb {C}^4)\).

SUMMARY: Let \(R = \oplus _{k\geq 0}R_k\) be a \(n\)-dimensional normal graded ring with an algebraically closed field \(R_0\), and \( x_1, \; ... \;, x_s\) be a minimal homogeneous generator \( x_1, \; ... \;, x_s\) of the homogeneous maximal ideal \(R_+\). We will discuss a condition on the reducedness of \(x_i\) in terms of \(\deg (x_i)\).

Theorem. (1) Let \(i \in \{ 1, \; ... \;,s \}\). If \(c_i := \gcd (\deg (x_1), \; ... {}_\wedge ^i ... \;, \deg (x_s)) \geq 2\), then \(R/x_iR\) is reduced. (2)Let \(i,j \in \{ 1, \; ... \;,s \}\) and \(i\neq j\). If \(c_i \geq 2\) and \(c_j \geq 2\), then \(x_i,x_j\) is a part of parameter system.

SUMMARY: Given a smooth projective variety on a number field and an endomorphism on it, we would like to know how the height of a point grows by iteration of the action of the endomorphism. When the endomorphism is polarized, Call and Silverman construct the canonical height, which is an important tool for the calculation of growth of heights. In this talk, we will give a generalization of the Call–Silverman canonical heights for not necessarily polarized endomorphisms, ample canonical heights, and propose an analogue of the Northcott finiteness theorem as a conjecture. We will see that the conjecture holds when the variety is an abelian variety or a surface.

SUMMARY: Hyperplane arrangement i.e., a finite set of hyperplanes relates to many mathematics and is studied widely. Representative example is the braid arrangement and a fundamental group of its complement coincides with the pure braid group. In 1989 Manin and Schechtman defined the discriminantal arrangement which is a generalization of braid arrangement. So far its combinatorial structure is studied mainly when original arrangement is in a Zariski open set in the space of general position arrangements. In this talk the speaker would talk about study of combinatorics of discriminantal arrangement when the original arrangement is not in the Zariski open set, and a related topic, Pappus’s theorem.

SUMMARY: Generalized Zariski Cancellation Problem asks when \(V\times _{k}\mathbb {A}^{1} \simeq W\times _{k}\mathbb {A}^{1}\) implies \(V \simeq W\). In general, this is not true, and many of conunterexamples for this problem are constructed as principal \(\mathbb {G}_{a}\)-bundles over integral schemes of finite type over \(\mathbb {C}\), so-called “prevariety”. In this talk, we show that for varieties \(V\), \(W\) which has a principal \(\mathbb {G}_{a}\)-bundle structure over a prevariety \(X\), \(Y\), respectively, if a prevariety \(Y\) has a dominant morphism to a variety with nonnegative logarithmic kodaira dimension, then \(V\times \mathbb {A}^{1} \simeq W\times \mathbb {A}^{1}\) if and only if \(X \simeq Y\). To prove this, We slightly generalize Fujita–Iitaka’s cancellation theorem (1977) and a generalized version of it by T. Nishimura (2017).

SUMMARY: A non-hyperelliptic fibered surface \(f\colon S\to B\) of genus \(4\) is called Eisenbud–Harris special (resp. Eisenbud–Harris general) if the general fiber of \(f\) has a unique trigonal pencil (resp. exactly two trigonal pencils). It is known that the slope equality holds for Eisenbud–Harris general fibrations of genus \(4\). In this talk, I will explain that the slope equality also holds for relatively minimal Eisenbud–Harris special fibrations of genus \(4\).

SUMMARY: We study the several properties on indecomposable vector bundles of rank two on a weighted projective line \(\mathbb {X}\) of type (2, 2, 2, 2, 2; \(\lambda _1\), \(\lambda _2\)). Since \(\mathbb {X}\) has the negative Euler characteristic, all vector bundles are not necessarily semi-stable. First, we check the stability of indecomposable vector bundles of rank two. Next, we check the exceptionality of all of them in the category of coherent sheaves coh(\(\mathbb {X}\)) and the exceptionality of some of them in the stable category of vector bundles \(\underline {\rm {vect}}(\mathbb {X})\).

SUMMARY: Let \(E\) be an obstructed stable sheaf on some elliptic surfaces with Kodaira dimension one, and we consider its deformation space over Artin rings. Suppose the number of multiple fibers is relatively few. If \(E_{\eta }\) has no sub line bundle with fiber degree zero (Case I), then \(E\) is a canonical singularity of moduli of stable sheaves.

SUMMARY: Let \(E\) be an obstructed stable sheaf on some elliptic surfaces with Kodaira dimension one, and we consider its deformation space over Artin rings. Suppose the number of multiple fibers is relatively few. If \(E_{\eta }\) has a sub line bundle with fiber degree zero, but not decomposable (Case II), then the defining equation of the deformation space of \(E\) is not always determined by degree-two terms.

SUMMARY: We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be Fano or weak Fano in terms of the building set.

SUMMARY: We give a necessary and sufficient condition for the nonsingular projective toric variety associated to the graph cubeahedron of a finite simple graph to be Fano or weak Fano in terms of the graph.

SUMMARY: The closure of a torus orbit in the flag variety of type A is known to be normal, so that it is a toric variety. When the orbit is generic, its closure is known to be a permutohedral variety which is smooth. In this talk we introduce the notion of a generic orbit in a Schubert variety and give a criterion of the smoothness of its closure in terms of graphs associated to permutations.

SUMMARY: For the geometrical sectional genus \(g(X,L)\) of a polarized variety \((X,L)\) has the well-known upper bound established by Fujita. Polarized varieties are called Castelnuovo varieties if \(g(X,L)\) attains the upper bound, which are a higher-dimensional extension of extremal curves. In this talk, we consider polarized toric varieties, and provide some properties of toric Castelnuovo varieties. From the viewpoint of the theory of polytopes, these properties give the formulae for the volume and the boundary volume of the polytope associated to a Castelnuovo variety.

SUMMARY: We shall report \(k\)-Fano varieties, as was studied by de Jong–Starr and Araujo–Castravet, with appropriately large peudo-index are covered by rational \(k\)-folds. We shall also report a similar claim for weak \(k\)-Fano varieties, as was studied by Taku Suzuki, with slightly larger pseudo-index.

SUMMARY: For an embedded Fano manifold \(X\), we introduce chains of higher order families of lines and define a new invariant \(S_X\) as the maximal length of such chains. This invariant \(S_X\) is related to the dimension of covering linear spaces. Our goal is to classify Fano manifolds \(X\) which have large \(S_X\).

SUMMARY: This talk is a sequel to the previous one at the MSJ annual meeting (March, 2018), in which the author gave an analogue of Miyaoka–Yau type inequality for three-dimesnional extremal contraction \((X,C) \to (Z,o)\) of type (IIA) with regading to the associated third Chen class, and also showed that the existence of a kind of Harder–Narasimhan (HN) filtrarion about \(c_2,c_3\) appearing in \((X,C)\), is a condition for such a analogous inequality with the third Chern class to be held. In this talk, the author will talk about an explicit formulation about such HN-filtrarion, and moreover, will give a realization about the pencil structure associated to \((X,C)\) by the moduli space of a coherent system (Le Potier, Trautmann) associated to \((X,C)\) based on such filtration.

SUMMARY: We define non-commutative Kähler projective varieties. Some cohomological features are examined.