2018年度年会(於:東京大学)

SUMMARY: A modular tensor category is usually defined as a semisimple ribbon category satisfying a certain non-degeneracy condition. Nevertheless, with motivation coming from CFT and TQFT, it is important and interesting to drop the semisimplicity assumption from the definition of a modular tensor category. Lyubashenko has formulated such a ‘non-semisimple’ modular tensor category and showed that, as in the semisimple case, a ‘non-semisimple’ modular tensor category yields an invariant of closed 3-manifolds and a projective representation of the surface mapping class groups. In this talk, while introducing several category-theoretical techniques that are important in the recent study of tensor categories, I will review recent developments of ‘non-semisimple’ modular tensor categories. I will, especially, mention the recent result that a ribbon finite tensor category is modular if and only if its Müger center is trivial. This criterion yields several new examples of modular tensor categories and factorizable (quasi-)Hopf algebras.

SUMMARY: In this talk, I would like to explain an approach to motivic cohomology in the étale topology with \({\mathbb Z}/p^n{\mathbb Z}\)-coefficients of an arithmetic scheme \(X\) which has good or semistable reduction at all primes dividing \(p\). We construct a complex of étale sheaves \({\mathbb Z}/p^n{\mathbb Z}(r)\) on \(X\) for \(r \geqq 0\) by gluing the \(r\)-fold tensor power of the locally constant sheaf \(\mu _{p^n}\) on \(X[p^{-1}]\) with a certain differential sheaf on the fiber over \(p\) via the boundary map of Galois cohomology groups due to K. Kato. In the good reduction case, the object \({\mathbb Z}/p^n{\mathbb Z}(r)\) is already considered by J. S. Milne and P. Schneider about 30 years ago. What I did on this object is that I defined it in the semistable reduction case and proved a global duality result for étale cohomology with coefficients in \({\mathbb Z}/p^n{\mathbb Z}(r)\), which had not been unknown even in the good reduction case. I will also talk about a few applications of \({\mathbb Z}/p^n{\mathbb Z}(r)\) to the study of algebraic cycles on arithmetic schemes.

SUMMARY: In this talk, I will explain the following aspects of level-zero representations of quantum affine algebras:

1) explicit combinatorial realization, by semi-infinite Lakshmibai–Seshadri paths, of crystal bases of Demazure submodules of level-zero extremal weight modules over quantum affine algebras;

2) explicit relation of graded characters of level-zero Demazure submodules with the specializations of nonsymmetric Macdonald polynomials at \(t = 0\) and \(t = \infty \);

3) algebro-geometric interpretation of graded characters of level-zero Demazure submodules via Borel–Weil–Bott type theorem for semi-infinite flag manifolds.

SUMMARY: Creating new trends of mutual development with combinatorics on convex polytopes, simplicial complexes, finite partially ordered sets and finite graphs and with statistics on contingency tables and experimental designs, commutative algebra on monomial ideals and binomial ideals is rapidly and dramatically growing by making the best use of modern techniques on, for example, Gröbner bases. In addition, plenty of fascinating problems remain unsolved. In the present draft, current streams of monomial ideals and binomial ideals will be surveyed quickly and their prospects will be predicted.

SUMMARY: Based upon machine-experiment we assert that, if Apery-like numbers X(p-r) with prime p is not ultimately constant then X(p-r) is congruent to (xp-em)/q (mod p), where e=1 or -1 and m is divisible by prime u with X(u-r)=0 (mod p), and that, in case X(p-r) has the ultimate constant “Aperi quotient” W(p) satisfies similar congruence.

SUMMARY: An amicable pair is a pair of distinct positive integers each of which is the sum of the proper divisors of the other. Gmelin (1917) conjectured that there is no relatively prime amicable pairs. Artjuhov (1975) and Borho (1974) proved that for any fixed positive integer \(K\), there are only finitely many relatively prime amicable pairs \((M,N)\) with \(\omega (MN)=K\), where \(\omega (n)\) denotes the number of the distinct prime factors of \(n\). Recently, Pollack (2015) obtained an upper bound \(MN<(2K)^{2^{K^2}}\) for such amicable pairs. In this talk, we improve this upper bound to \(MN<\frac {\pi ^2}{6}2^{4^K-2\cdot 2^K}\).

SUMMARY: Given an integer \(m\) and an odd prime \(P\), if the following equality \( (P-1)\sigma (a)=Pa-m \) holds then \(a\) is said to be a hyper perfect number with base \(P\) and translation parameter \(m\).

SUMMARY: Many mathematicians have researched the uniformity of the digit expansion of real numbers. In this talk we consider the beta expansion of algebraic numbers, which is a generalization of base-\(b\) expansion for a fixed integer \(b\geq 2\). For instance, Borel conjectured for each integral base-\(b\) that any algebraic irrational number has uniform digits in its base-\(b\) expansion. For the study of the uniformity, Bugeaud suggested to consider the number of digit exchanges in the beta expansion of algebraic numbers. In our main result, we considerably improve known results on the asymptotic behavior of the number of digit exchanges.

SUMMARY: We prove some arithmetic properties of the elliptic Dedekind sums introduced by Egami. Further, we also talk about some conjectures of the elliptic Dedekind sums.

SUMMARY: We express the Riemann zeta function \(\zeta \left ( s \right )\) of argument \(s=\sigma +i\tau \) with imaginary part \(\tau \) in terms of three absolutely convergent series. The resulting simple algorithm allows to compute, to arbitrary precision, \(\zeta \left ( s \right )\) and its derivatives using at most \( C \left (\epsilon \right )\left |\tau \right |^{\frac {1}{2}+\epsilon }\) summands for any \(\epsilon >0\), with explicit error bounds. It can be regarded as a quantitative version of the approximate functional equation. The numerical implementation is straightforward. The approach works for any type of zeta function with a similar functional equation such as Dirichlet \(L\)-functions.

SUMMARY: In 2015, Li and Radziwiłł proved an approximate functional equation of the second moment of the Riemann zeta function \(\zeta (s)\) on vertical arithmetic progressions on Re\((s)=1/2\). Using this formula, they could show that there is an at least \(1/3\) proportion of points on arithmetic progressions on Re\((s)=1/2\) such that \(\zeta (s)\) does not vanish. We are interested in finding the proportion of points on the line such that two consecutive values of \(\zeta (s)\) differ. For this purpose, we need an approximate functional equation for the fourth moment of \(\zeta (s)\) of Li and Radziwiłł’s form. In this talk, we introduce this approximate functional equation we obtained.

SUMMARY: We consider the existence of horizontal lines in critical strip on which a Dirichlet \(L\)-function takes its extreme values uniformly. This is an extension of the result for the Riemann zeta-function that was shown by Ramachandra and Sankaranarayanan in 1991. In addition, as an application of this theorem, we also obtain the estimate of the sum of derivatives of Dirichlet \(L\)-functions and certain Dedekind zeta-functions over the non-trivial zeros.

SUMMARY: The study of zeros of zeta-functions is a classical topic in analytic number theory. The Riemann hypothesis assert that any nontrivial zeros of the Riemann zeta-function are located on the critical line. On the other hand, the Hurwitz zeta-function has zeros off the critical line in general. Then the upper and lower bounds for the number of such zeros has been considered. In this presentation, we obtain an asymptotic formula on the number of zeros of the Hurwitz zeta-function on the right side of the critical line, applying a certain density function related to the value-distribution of the Hurwitz zeta-function.

SUMMARY: Let \(k\geq 2\) and \(m\geq 1\) be integers. Suryanarayana and Sitaramachandra Rao studied the number of \(k\)-free integers \(n\leq x\) satisfying \((n,m)=1\). We shall reconsider error terms in their formula.

SUMMARY: Let \(\zeta _{2}(s_{1},s_{2})\) be the double zeta-function of Euler–Zagier type \(\zeta _{2}(s_{1},s_{2})=\sum _{m=1}^{\infty }\sum _{n=m+1}^{\infty }\frac {1}{m^{s_{1}}n^{s_{2}}}\). We shall give several bounds of an error of approximate formula of \(\zeta _{2}(s_{1},s_{2})\).

SUMMARY: We consider a family of integral operators arising from zeta functions, and state an equivalence condition of the Riemann hypothesis in terms of operators.

SUMMARY: For any cocompact subgroup of \(PSL(2,\mathbb R)\), and its finite dimensional unitary representation \(\rho \) not containing the trivial representation, the Selberg zeta function \(Z(s,\rho )\) is defined by the Euler product, which is regular at \(s=1\). It is known that the Euler product is absolutely convergent in \(\mathrm {Re}(s)>1\). In this talk we show convergence of the Euler product for \(\mathrm {Re}(s)\ge 3/4\), under assuming the analog of the Riemann hypothesis for the Selberg zeta function.

SUMMARY: Kaplansky conjectured that if two positive-definite real ternary quadratic forms have perfectly identical representations over \(\mathbb Z\), they are equivalent over \(\mathbb Z\) or constant multiples of regular forms, or is included in either of two families parametrized by \(\mathbb Q\). Firstly, the result of an exhaustive search for such pairs of integral quadratic forms is presented, in order to provide a concrete version of the Kaplansky conjecture. The obtained list contains a small number of non-regular forms that were confirmed to have the identical representations up to 3,000,000. Secondly, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over \(\mathbb Q\), their constant multiples are equivalent over \(\mathbb Q\). This was motivated by the question why the other families were not detected in the search.

SUMMARY: A problem that has been discussed in crystallography is introduced. When we consider a crystal structure as a periodic point set in \(R^n\), this problem is equivalent to the determination of the periodic points set from its average theta series. We prove that the theta series can almost determine the smallest set that contains the difference set of the periodic point set and is invariant by the action of the automorphism group of the period lattice. This result explains how the problem is reduced to a problem about integral representations of quadratic forms.

SUMMARY: Finite multiple zeta values (FMZVs) were defined first by Kaneko and Zagier and studied by many mathematicians. In this talk, we introduce 2-colored rooted trees, which are some combinatorial objects, and define FMZVs associated with 2-colored rooted trees. We will show that they can be regarded as common generalizations of FMZVs and finite Mordell–Tornheim multiple zeta values defined first by Kamano. Moreover, we will explain that with a mild assumption, FMZVs associated with 2-colored rooted trees can be written as a sum of usual FMZVs. As a corollary, we will give another proof of the shuffle relation among FMZVs, which was first proved by Kaneko and Zagier.

SUMMARY: In my recent study, I introduced a combinatorial object called 2-colored rooted tree and finite multiple zeta value (FMZV) associated with it to generalize FMZV of Euler–Zagier type and Mordell–Tornheim type simultaneously. In this talk, we introduce multiple zeta function (MZF) associated with 2-colored rooted trees and give some analytical properties. In particular, we give a conjecture on the singularities of MZFs associated 2-colored rooted trees and a new example of MZF satisfying this conjectures.

SUMMARY: An identity involving symmetric sums of regularized multiple zeta-star values of harmonic type was proved by Hoffman. In this talk, we prove an identity of shuffle type. In the proof, we meet Bell polynomials appearing in the study of set partitions.

SUMMARY: In the spherical principal series representation of a \(p\)-adic group, we consider the space of Iwahori-fixed vectors, which has a natural basis and the so-called Casselman basis both indexed by the Weyl group. The latter is defined by using the intertwining integrals. We are interested in the transition matrix of these bases. In order to describe the matrix, we introduced a deformation of the Kazhdan–Lusztig \(R\)-polynomials, and proved certain functional equations and a duality formula.

SUMMARY: We consider a quasi-split classical group \(G\) over a \(p\)-adic field \(F\). By the local Langlands correspondence for \(G\), which is recently established by Arthur, we have a natural partition of the set of irreducible smooth representations of \(G(F)\) into finite sets which are parametrized by \(L\)-parameters. On the other hand, Gross and Reeder defined a some special class of supercuspidal representations which they call the simple supercuspidal representations. In this talk, I will explain a result on an explicit description of the local Langland correspondence for simple supercuspidal representations of quasi-split classical groups.

SUMMARY: In this talk, we consider \(p\)-adic functions interpolating Dedekind–Rademacher sums and their reciprocity formula. The results are natural generalizations of the ones due to Rosen and Snyder.

SUMMARY: The Mazur–Tate refined conjecture connects arithmetic invariants of modular forms with associated \(L\)-functions by using Mazur–Tate elements, which are certain elements of group rings of Galois groups and regarded as analogues of Stickelberger elements. In this talk, we discuss its rank-part, which compares the rank of Selmer groups with the order of zeros of Mazur–Tate elements, and our main result is as follows. Under some assumptions, we prove it for higher weight modular forms, generalizing our previous proof for elliptic curves.

SUMMARY: We prove that Nekovář and Nizioł’s syntomic cohomology and log rigid syntomic cohomology are isomorphic for a strictly semistable log scheme having a nice compactification and for non-negative twist. Key points of the proof are a generalization of Große–Klönne’s log rigid cohomology theory and the compatibility of crystalline and rigid Hyodo–Kato maps on Frobenius eigenspaces.

SUMMARY: The Brauer group of varieties has various application to algebraic geometry and number theory. Chernousov and Guletskii studied the 2-torsion part of the Brauer group of elliptic curves, especially its explicit generators represented by norm residue symbols and their relations. Using their method, we study the 3-torsion part of the Brauer group of diagonal cubic curves. In this presentation, we will explain our result in the case of Fermat curves of degree 3.

SUMMARY: The notion of isoclinism was introduced by P. Hall in 1940 to classify finite \(p\)-groups. We show that this notion also plays an important role to classifying Galois groups of number fields.

SUMMARY: Let \(k\) be a field, \(G\) be a finite group and \(G\) act on the rational function field \(k(x_g:g\in G)\) by \(k\)-automorphisms defined by \(h\cdot x_g=x_{hg}\) for any \(g,h\in G\). Define \(k(G)=k(x_g:g\in G)^G\). Noether’s problem asks whether \(k(G)\) is rational over \(k\). The unramified cohomology groups \(H_{\rm nr}^i(\mathbb {C}(G),\mathbb {Q}/\mathbb {Z})\) are obstructions to the rationality of \(\mathbb {C}(G)\). Theorem 1. Let \(G\) be a group of order \(3^5\). Then \(H_{\rm nr}^3(\mathbb {C}(G),\mathbb {Q}/\mathbb {Z})\neq 0\) if and only if \(G\) belongs to the isoclinism family \(\Phi _7\). Moreover, if \(H_{\rm nr}^3(\mathbb {C}(G),\mathbb {Q}/\mathbb {Z})\neq 0\), then \(H_{\rm nr}^3(\mathbb {C}(G),\mathbb {Q}/\mathbb {Z})\simeq \mathbb {Z}/3\mathbb {Z}\). Theorem 2. Let \(G\) be a group of order \(p^5\) where \(p=5\) or \(p=7\). Then \(H_{\rm nr}^3(\mathbb {C}(G),\mathbb {Q}/\mathbb {Z})\) \(\neq 0\) if and only if \(G\) belongs to the isoclinism family \(\Phi _6\), \(\Phi _7\) or \(\Phi _{10}\). Moreover, if \(H_{\rm nr}^3(\mathbb {C}(G),\mathbb {Q}/\mathbb {Z})\neq 0\), then \(H_{\rm nr}^3(\mathbb {C}(G),\mathbb {Q}/\mathbb {Z})\simeq \mathbb {Z}/p\mathbb {Z}\). Theorem 3. Let \(G\) be a group of order \(243\). Then \(\mathbb {C}(G)\) is \(\mathbb {C}\)-rational if and only if \(G\) belongs to the isoclinism family \(\Phi _i\) where \(1 \le i \le 6\) or \(8 \le i \le 9\).

SUMMARY: We explain how to compute unramified cohomology group \(H^3_{\rm nr}(\mathbb {C}(G),\mathbb {Q}/\mathbb {Z})\) of degree three using Saltman–Peyre method and GAP.

Some algorithms are available from https://www.math.kyoto-u.ac.jp/~yamasaki/Algorithm/UnramDeg3/.

SUMMARY: It is a joint work with M. Toyoizumi and Y. Deguchi. The aim of our study is to give another algorism to determine the singularity or non-singularity of an integral matrix. Our method can give the result on large order integral matrices within a practical time.

SUMMARY: We establish a relation between the Sprague–Grundy function of \(p\)-saturations of Welter’s game and the degrees of the ordinary irreducible representations of symmetric groups. We present a theorem on these degrees, and using this theorem we obtain an explicit formula for the Sprague–Grundy function of \(p\)-saturations of Welter’s game.

SUMMARY: We propose a construction of the Burnside ring of an essentially finite category admitting an epi-mono factorization and enough coequalizers. The main result of our talk is a vast generalization of the embedding theorem of the classical Burnside ring in its ghost ring, with finite cokernel of obstructions. This unifies many constructions and results relative to similar rings, such as various generalizations of the classical Burnside ring (monomial Burnside ring, section Burnside ring, crossed Burnside ring, slice Burnside ringetc), but also some of a seemingly different nature (such as the Möbius algebra of a poset).

SUMMARY: In this talk, I will discuss the structure of the unit group of the partial Burnside ring relative to the set of parabolic subgroups of a finite reducible Coxeter group of type A.

SUMMARY: We describe the multiplicity of the irreducible components of tensor products in even numbers for spin representations.

SUMMARY: For symmetric plane partitions we have generating functions which can be nicely factored, such as the size generating function, half-the-size generating function and Gansner–Nakada’s generating function that respects the diagonal sums. In this talk we show two conjectural formulas which refine the above nice generating functions. Pfaffian expressions for those formulas are also given.

SUMMARY: The \(\mathcal W\)-algebras are vertex algebras defined by the generalized Drinfeld–Sokolov reductions. Using the Wakimoto representations of affine Lie algebras, we describe the explicit formulae of the screening operators for the \(\mathcal W\)-algebras with generic level. As applications, we show that the \(\mathcal W\)-algebras of type \(A\) have the “coproduct” structures related to affine Yangians.

SUMMARY: The notion of affine highest weight category introduced by Kleshchev generalizes the notion of highest weight category and axiomatizes certain homological structures of some non-semisimple abelian categories of Lie theoretic origin. In this talk, we see the existence of a special kind of tilting module in an affine highest weight category with a large categorical center. As an application, we can prove that a block of the BGG category of \(\mathfrak {gl}_{m}(\mathbb {C})\) is embedded fully faithfully into the module category of finite-dimensional modules over the degenerate affine Hecke algebra of \(GL_{n}\) by the Arakawa–Suzuki functor.

SUMMARY: For a Dynkin quiver \(Q\) (of type ADE), Hernandez–Leclerc defined a good monoidal subcategory \(\mathcal {C}_{Q}\) inside the category of finite-dimensional modules over the quantum loop algebra \(U_{q}(L\mathfrak {g})\) based on its relationship with the Auslander–Reiten quiver of \(Q\). By using the geometric construction of \(U_{q}(L\mathfrak {g})\)-modules with quiver varieties due to Nakajima, we see that a completion of the category \(\mathcal {C}_{Q}\) has a structure of affine highest weight category. As an application, we can prove that Kang–Kashiwara–Kim’s generalized quantum affine Schur–Weyl duality functor gives an equivalence of monoidal categories between the category of finite-dimensional modules over the quiver Hecke (KLR) algebra associated to \(Q\) and Hernandez–Leclerc’s category \(\mathcal {C}_{Q}\).

SUMMARY: In 1941, Brauer–Nesbitt established a characterization of a block with trivial defect group as a block \(B\) with \(k(B) = 1\). In 1982, Brandt established a characterization of a block with defect group of order two as a block \(B\) with \(k(B) = 2\). These correspond to the cases when the block is Morita equivalent to the one-dimensional algebra and to the non-semisimple two-dimensional algebra respectively. In this talk, we redefine \(k(A)\) to be the codimension of the commutator subspace \(K(A)\) of a finite-dimensional algebra \(A\) and show analogous statements for arbitrary finite-dimensional algebras.

SUMMARY: Let \(b\) be a block ideal of the finite group algebra \(kG\) with a defect group \(S\) which is extraspecial of order \(p^3\) and of exponent \(p\). Let \(i\) be a source idempotent. We shall examine the module structure of the source algebra \(ikGi\) of the block ideal \(b\) and show that the image of the tranfer map \(t:H^*(S,k)\rightarrow H^*(S,k)\) induced by \(ikGi\) coincides with the cohomology ring of the block \(b\) wih respect to the source idempotent \(i\).

SUMMARY: We shall be talking about Brauer indecomposabilities of the Scott modules. The Scott module is an indecomposable \(p\)-permutation \(kG\)-module which contains the trivial \(kG\)-module (where \(G\) is a finite group and \(k\) is an algebraically cloased field of characteristic \(p>0\)). This plays an imporant role to try to prove so-called global-local conjecture in the representation theory of finite groups.

SUMMARY: We shall be talking on the projective cover of the trivial module for the group algebra \(kG\) where \(G\) is a finite group and \(k\) is a field of characteristic \(p>0\). Especially we will be interested in the case when \(p\) is odd and \(G\) is a finite simple group of Lie type defined over a finite field of the same characteristic \(p\).

SUMMARY: We will be talking on the positions of simple modules for finite group algebras in connected components of the stable Auslander–Reiten quivers to which the simple modules belong.

SUMMARY: Stability is the appropriate notion of linearity in homotopical algebra, which supersedes Abelianness in classical algebra. Thus, stable homotopical algebra is useful e.g., for refining the method of derived categories. While homotopical algebra can conveniently be formulated in a (\(\infty ,1\))-category, to be simply called “category” here, stable homotopical algebra can be understood as algebra in a “stable” such. We discuss the theory of filtration of a stable category, which is useful for controlling behaviour of limits in stable homotopical algebra, and can be applied e.g., for the study of the Koszul duality and higher Morita categories. Examples of a filtration include a t-structure, and a natural filtration on the category of filtered objects in a stable category. Both may come with a compatible symmetric monoidal structure in practice.

SUMMARY: The noncommutative projective scheme associated to an AS-regular algebra is considered as a noncommutative projective space, and has been studied deeply and extensively in noncommutative algebraic geometry. In this talk, we will characterize a \(k\)-linear abelian category \(\mathcal C\) such that \(\mathcal C\) is equivalent to the noncommutative projective scheme associated to some AS-regular algebra.

SUMMARY: In this talk, we determine the automorphism group of an elliptic curve \(E\) in \(\mathbb {P}^{2}\) depending on its \(j\)-invariant \(j(E)\). By using the automorphism \(\sigma \) of \(E\), we calculate the defining relations of a \(3\)-dimensional quadratic AS-regular algebra corresponding to the pairs \((E,\sigma )\). By this calculations, we find a counterexample to the conjecture that any \(3\)-dimensional quadratic AS-regular algebra \(\Lambda \) corresponding to an elliptic curve is isomorphic to a twist \(A^{\varphi }\) of a Sklyanin algebra \(A\) by \(\varphi \in {\rm Aut}\,A\).

SUMMARY: It is known that for any finite dimensional algebra \(\Lambda \) of finite global dimension and any finitely generated \(\Lambda \)-module \(M\), if \({\rm Ext}^{\geq 1}_\Lambda (M,\Lambda )=0\), then \(M\) is projective. Let \(Q\) be the following quiver: \[ \circ \to \circ \to \cdots \circ \to \circ \to \] and \(K\) an algebraically closed field. Using the compactness theorem of Mathematical Logic, we prove that for any finitely dimensional \(K\)-representation \(\mathcal K\), if \({\rm Ext}^{1}_{KQ}(\mathcal K, KQ)=0\), then \(\mathcal K\) is projective. It is also proved that, under Martin’s Axiom (which is a combinatorial statement consistent with Axiomatic Set Theory), there exists a non-projective \(KQ\)-module \(M\) such that \({\rm Ext}^{1}_{KQ}(\mathcal K, KQ)=0\).

SUMMARY: For a bound quiver algebra satisfying the condition that the every oriented cycles in the quiver are vanished in the algebra, Fernádez and Platzeck determined the bound quiver algebra which is isomorphic to the trivial extension algebra. In this paper, we consider a Hochschild extension algebra which is a generalization of a trivial extension algebra. Our aim is to determine the bound quiver algebras which are isomorphic to Hochschild extension algebras of some finite dimensional self-injective Nakayama algebras.

SUMMARY: In this talk, we determine the Batalin Vilkovisky algebra structure on the Hochschild cohomology of self-injective Nakayama algebras over an algebraically closed field.

SUMMARY: Mizuno gave an isomorphism of lattices from a Coxeter group of Dynkin type to the set of torsion-free classes in the module category of the corresponding preprojective algebra. Combining it with my bijection on semibricks, we obtain a bijection from the Coxeter group to the set of semibricks over the preprojective algebra. My aim is to explicitly describe the semibrick associated to each element in the Coxeter group in this bijection. In this process, a combinatorial notion “canonical join representations” introduced by Reading, is very useful. I observed that the canonical join representation of an element in the Coxeter group gives the decomposition of the corresponding semibrick into bricks. I will talk about such theoretic strategies to determine the semibrick.

SUMMARY: This talk is based on joint work with Yuji Yoshino. Let \(R\) be a commutative Noetherian ring. We denote by \(\mathcal {D}\) the unbounded derived category of \(R\). An exact functor \(\lambda :\mathcal {D}\to \mathcal {D}\) is called a localization functor if there is a morphism \(\eta :{\rm id}_{\mathcal {D}}\to \lambda \) such that \(\lambda \eta \) is invertible and \(\lambda \eta =\eta \lambda \). This notion was introduced by A. K. Bousfield in his topological work (1979). In this talk, we give a concrete way to compute localization functors on \(\mathcal {D}\) by using the notions of cosupport and Čech complexes. As an application, we can obtain a functrial way to construct pure-injective resolutions for complexes of flat \(R\)-modules and complexes of finitely generated \(R\)-modules.

SUMMARY: The singularity category \(D_{sg}(R)\) of a commutative Noetherian ring \(R\) is a triangulated category which measures singularity of \(R\). Two commutative Noetherian rings \(R\) and \(S\) are said to be singularly equivalent if their singularity categories are equivalent as triangulated categories. Singular equivalence have deeply been studied in non-commutative setting and various examples are known, while in commutative setting, only a few examples of singular equivalence are known. The aim of this talk is to give a necessary condition for singular equivalence by using singular loci. The key tool to prove our main result is the support theory for triangulated categories without tensor structure.

SUMMARY: I will talk on syzygies of (maximal) Cohen–Macaulay modules over one dimensional Cohen–Macaulay local rings. We compare these modules to Cohen–Macaulay modules over the endomorphism ring of the maximal ideal. After this comparison, we give several characterizations of almost Gorenstein rings in terms of syzygies of Cohen–Macaulay modules.

SUMMARY: Reduction of ideals introduced by Northcott and Rees plays an important role in the study of local rings, especially the multiplicity theory of ideals. This notion is extended to the multigraded modules and used in the study of asymptotic properties of them. In this talk, I will give a result about existence of certain complete reductions of multigraded modules. By applying the result to multi-Rees algebras of finitely many ideals, we obtain a result on normality of monomial ideals, which extends and improves several known results on this topic.

SUMMARY: Let \(R\) be a Noetherian local ring and \(M\) be a nonzero finitely generated \(R\)-module. The notion of almost Gorenstein rings is one of the generalization of the notion of Gorenstein rings. The purpose of this talk is to explore the question of when the idealization \(R \ltimes M\) of \(M\) is an almost Gorenstein local rings. Although this problem was investigated by S. Goto, R. Takahashi, and N. Taniguchi, it is still open.

SUMMARY: The notion of a generalized Gorenstein local ring (GGL ring for short) is one of the generalizations of Gorenstein rings. Similarly for almost Gorenstein local rings, the notion is given in terms of a certain specific embedding of the rings into their canonical modules. In this talk, we give a characterization of GGL rings in terms of their canonical ideals and related invariants.

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 \(H\). In 2003, N. T. Cuong and L. T. Nhan introduced the notion of sequentially generalized Cohen–Macaulay. \(H\) is called sequentially generalized Cohen–Macaulay when so is \(S/I(H)\).

In this talk, we consider properties of sequentially generalized Cohen–Macaulay bipartite graphs. In particular, we investigate the behavior of edges of \(H\) when \(\textrm {depth}(S/I(H)) \le 3\). Consequently, we give a characterization of sequentially generalized Cohen–Macaulay graphs of essential dimension 3.

SUMMARY: Gorenstein Fano polytopes form one of the distinguished classes of lattice polytopes. Especially normal Gorenstein Fano polytopes are of interest. In this talk, we will give a new class of normal Gorenstein Fano polytopes arising from perfect graphs.

SUMMARY: Normality or the integer decomposition property (IDP) is one of the most important properties on lattice polytopes. In fact, many authors have been studied the properties from view-points of combinatorics, commutative algebra and algebraic geometry. In this talk, we discuss when a Cayley sum is normal. Moreover, we consider when a Cayley sum is level.

SUMMARY: Let \(S = K[x_1, \ldots , x_n]\) denote the polynomial ring in \(n\) variables over a field \(K\) with each \(\deg x_i = 1\) and \(I \subset S\) a homogeneous ideal of \(S\) with \(\dim S/I = d\). The Hilbert series of \(S/I\) is of the form \(h_{S/I}(\lambda )/(1 - \lambda )^d\), where \(h_{S/I}(\lambda ) = h_0 + h_1\lambda + h_2\lambda ^2 + \cdots + h_s\lambda ^s\) with \(h_s \neq 0\) is the \(h\)-polynomial of \(S/I\). It is known that, when \(S/I\) is Cohen–Macaulay, one has \(\mathrm {reg}(S/I) = \deg h_{S/I}(\lambda )\), where \(\mathrm {reg}(S/I)\) is the (Castelnuovo–Mumford) regularity of \(S/I\). In this talk, given arbitrary integers \(r\) and \(s\) with \(r \geq 1\) and \(s \geq 1\), a monomial ideal \(I\) of \(S = K[x_1, \ldots , x_n]\) with \(n \gg 0\) for which \(\mathrm {reg}(S/I) = r\) and \(\deg h_{S/I}(\lambda ) = s\) will be constructed.

SUMMARY: Let \(K\) be an algebraically closed field of positive characteristic. We set \(S(t)=K[x_1,y_1] \# \cdots \# K[x_t,y_t]\) and \(R(r,s)=K[x_1,\ldots ,x_{r+1}] \# K[y_1,\ldots ,y_{s+1}]\). In this talk, we will compute generalized F-signatures of all modules belonging to the FFRT system of each of \(S(t)\) and \(R(r,s)\).

SUMMARY: Given integers \(k\) and \(m\) with \(k \geq 2\) and \(m \geq 2\), let \(P\) be an empty simplex of dimension \((2k-1)\) whose \(\delta \)-polynomial is of the form \(1+(m-1)t^k\). In this talk, the necessary and sufficient condition for the \(k\)-th dilation \(kP\) of \(P\) to have a regular unimodular triangulation will be presented.

SUMMARY: Let \(R\) be a Hibi ring and \(\omega \) the canonical ideal of \(R\). We denote the \(n\)-th power of \(\omega \) in \(D(R)\), the group of divisorial ideals of \(R\), by \(\omega ^{(n)}\) for any integer \(n\). \(R\) is by definition, anticanonical level if all the generators of \(\omega ^{(-1)}\) have the same degree. In this talk, we analyze the structure of \(\omega ^{(n)}\) for any integer \(n\) and state a criterion of anticanonical level property of \(R\).

SUMMARY: The Mallows–Sloane bound is the inequality which estimates the minimum distance by the code length for a divisible self-dual code. Analogous inequalities for formal weight enumerators are completed.

SUMMARY: An \(m\)-free hyperplane arrangement is a generalization of a free arrangement. There are rich researches for free arrangements, but the behavior of \(m\)-freeness has not been well analyzed yet when \(m>2\). Some basic questions remain open. In particular, Holm asked the following: (1) Does \(m\)-free imply \((m+1)\)-free for any arrangement? (2) Are all arrangements \(m\)-free for \(m\) large enough? In this talk, we characterize \(m\)-freeness for product arrangements and show that all localizations of an \(m\)-free arrangement are \(m\)-free. From these results, we give counter examples to Holm’s questions.

SUMMARY: Hessenberg varieties are subvarieties of a flag variety. This subject makes connections with many research areas such as geometric representation theory, quantum cohomology of the flag variety, chromatic quasisymmetric functions of graph theory, and hyperplane arrangemants. In this talk, I will explain the connection between Hessenberg varieties and hyperplane arrangemants. More concretely, we show that a certain graded ring derived from the logarithmic derivation module of an ideal arrangement is isomorphic to the cohomology ring of a regular nilpotent Hessenberg variety, and the Weyl group invariant subring of the cohomology of a regular semisimple Hessenberg variety. This is joint work with Takuro Abe, Mikiya Masuda, Satoshi Murai, and Takashi Sato.

SUMMARY: Let \(X\) be an affine \(G\)-variety, where \(G\) is a reductive algebraic group. The invariant Hilbert scheme parametrizes closed \(G\)-subschemes of \(X\) whose coordinate rings have a prescribed decomposition as \(G\)-modules. One of the main usage of the invariant Hilbert scheme, together with so-called the Hilbert–Chow morphism, is to study singularities of affine quotient varieties. In this talk, we study Popov’s \(SL(2)\)-varieties by means of the invariant Hilbert scheme.

SUMMARY: Spherical twists along spherical objects are autoequivalences of a triangulated category defined by Seidel and Thomas as a categorical analogue of Dehn twists along simple closed curves. Spherical twists share many properties with Dehn twists. On the other hand, there is a classical result by Humphries which states that if a collection of simple closed curves admits a “complete partition” and does not bound a disk then the group generated by the Dehn twists along them is isomorphic to the free product of free abelian groups. In this talk, we give a categorical analogue of Humphries’ argument.

SUMMARY: We define “non-commutative Kähler projective space”. We describe fully the “infinitesimal deformation case” and discuss “finite (non-perturbative) deformation case”. We then examine the cohomology in the “infinitesimal defomation case” from a view point of commutative algebraic geometry.

SUMMARY: We give basic concepts of semialgebraic varieties. We will treat a semialgebraic subset of a real algebraic variety as a certain kind of abstract locally ringed space. We study a relation with semialgebraic varieties and complex algebraic varieties.

SUMMARY: We will prove a theorem of Ax–Lindemann type for complex semi-abelian varieties as an application of a big Picard Theorem proved by the author in 1981, and then apply it to prove a theorem of the classical Manin–Mumford Conjecture for semi-abelian varieties, which was proved by M. Raynaud 1983, M. Hindry 1988, \(\ldots \), and Pila–Zannier 2008 by a different method from others, which is most relevant to ours.

SUMMARY: A curve is called superspecial if its Jacobian is isomorphic to a product of supersingular elliptic curves. The purpose of this study is to enumerate superspecial curves of genus \(g\) over \(\mathbb {F}_q\) for a given \(g\) and for a given \(q\). In 2016, the speakers gave an algorithm to enumerate superspecial curves of genus \(g = 4\) over \(\mathbb {F}_q\) with \(q > 5\). By executing the algorithm on a computer algebra system Magma, they also enumerated superspecial curves of genus \(4\) over \(\mathbb {F}_{25}\) and \(\mathbb {F}_{49}\). In this talk, we present an improved algorithm, which works for any finite field \(\mathbb {F}_q\) with \(q \geq 5\), and classifies the isomorphism classes of superspecial curves of genus \(4\). By our implementation of the improved algorithm over Magma, we newly enumerate superspecial curves of genus 4 over prime fields \(\mathbb {F}_{p}\) for \(p\leq 11\).

SUMMARY: A nonsingular projective curve \(C\) over a field \(K\) of positive characteristic is called superspecial if its Jacobian is isomorphic to a product of supersingular elliptic curves. In 2016 and 2017, Kudo and Harashita enumerated nonhyperelliptic superspecial curves of genus 4 over \(\mathbb {F}_{25}\), \(\mathbb {F}_{49}\) and \(\mathbb {F}_{11}\). By implementing our algorithm over Magma, we determined the structure of the automorphism group of every nonhyperelliptic superspecial curve of genus 4 over \(\mathbb {F}_{11}\).

SUMMARY: We consider the embedded topology of plane curves in the complex projective plane, as in knot and link theory. In this talk, we introduce a new invariant, called the splitting graph, to distinguish the embedded topology of plane curves. This invariant is a generalization of the splitting number, which is an invariant not determined by the fundamental group of the complement of a plane curve. By using the splitting graph, we distinguish the embedded topology of plane curves consisting of one smooth curve and three lines.

SUMMARY: In this talk we introduce some recent study of rational points of elliptic surfaces done from a geometric point of view, and apply it to construct interesting examples of plane curve arrangements of low degree which give rise to candidates for Zariski pairs. The constructed arrangements can be distinguished topologically by studying the arithmetic properties of the rational points used in the construction. Previously known examples of Zariski pairs consisting of cubic-line arrangements all involved flex tangent lines, but our new examples do not contain any flex tangents.

SUMMARY: In the classification problem of Poisson structures, log symplectic structures which are generically symplectic Poisson structures with reduced degeneracy divisor is one of the most important class. Lima and Pereira studied log symplectic structures with simple normal crossing degeneracy divisor in the case that the variety is a Fano variety of Picard number 1 and they discovered a characterization of projective spaces. My research consider the case that the variety is a product of Fano varieties of Picard number 1. We will extend the result of Lima and Pereira and also give a bettter characterization of projective spaces.

SUMMARY: Taku Suzuki (arXiv.16060.9350) gave some sufficient condition for a smooth Fano manifold to be covered by rational \(N\)-folds. Results of this sort might be useful for future investigation of “higher connectivity” properties in the Morel–Voevodsky Motivic homotopy theory. Motivated by this, I shall report that a slight relaxation of the condition in this Suzuki’s theorem is possible.

SUMMARY: The multiplicity of a point on a variety is a fundamental invariant to estimate how bad the singularity is. It is introduced in a purely algebraic context. On the other hand, we can also attach to the singularity the log canonical threshold, which is introduced in a birational theoretic context. In this talk, we show bounds of the multiplicity by functions of this birational invariant for a singularity of locally a complete intersection. As an application, we obtain the affirmative answer to Watanabe’s conjecture on the multiplicity of canonical singularity of locally a complete intersection up to dimension 32.

SUMMARY: The minimal model theory is a fundamental method to classify higher-dimensional algebraic varieties. Today the theory is not completed. It consists of the minimal model conjecture and the abundance conjecture. On the other hand, the non-vanishing conjecture is also an important open problem in the minimal model theory. In fact, the minimal model conjecture and the abundance conjecture implies the non-vanishing conjecture, and it is known by Birkar that the non-vanishing conjecture implies the minimal model conjecture. In this talk I focus on the minimal model conjecture and the simplest case of the non-vanishing conjecture, that is, the non-vanishing conjecture for smooth varieties. I explain that the non-vanishing conjecture for smooth varieties implies the minimal model conjecture.

SUMMARY: Let \(X\) be a smooth complex projective variety of dimension \(n\), and let \(L\) be an ample line bundle on \(X\). Then the pair \((X,L)\) is called a polarized manifold. In my short talk, I will talk about some problems related with the dimension of the global sections of adjoint bundles for polarized manifolds.

SUMMARY: An ACM bundle on a polarized algebraic variety is defined as a vector bundle whose intermediate cohomology vanishes. We are interested in ACM bundles of rank one with respect to a very ample line bundle on a K3 surface. In this session, we give a necessary and sufficient condition for a non-trivial line bundle \(M\) on \(X\) with \(|M|\neq \emptyset \) and \(M^2\geq L^2-6\) to be an ACM and initialized line bundle with respect to \(L\), for a given K3 surface \(X\) and a very ample line bundle \(L\) on \(X\).

SUMMARY: Threefold semistable extremal neighborhoods, as of type (IIA), have important properties as rationality criterion of Q-conic bundles, or the existence of semistable flips by division algorithm by S. Mori. In this talk, the author reports his study about inducing an analogue of Miyaoka–Yau type inequality for threefold extremal contractions of type (IIA) with special regarding to the third Chern classes associated to the filtration of bi-anti-canonical divisors used in local-to-global deformation in the proof of [Mori1988], based on (non-)normal hyperplane section case of threefold extremal contractions of type (IIA) [Mori–Prokhorov 2016, 2017].