2017年度秋季総合分科会(於:山形大学)

SUMMARY: In this talk, we discuss the uniformity property of the digits of special sequences: base-\(b\) expansion of smooth numbers and beta expansion of real numbers, where \(b\) is an integer greater than 1. In particular, we investigate asymptotic behavior of the number of nonzero digits as partial results of uniformity property.

For a real number \(x\geq 2\), we call a positive integer \(n\) \(x\)-smooth if every prime factor of \(n\) is at most \(x\). It is believed that the digits \(0,1,2\) in the ternary expansion of \(2\)-smooth numbers, namely, integers of the form \(2^m\) (\(m=0,1,2,\ldots \)), are uniform. However, this conjecture is unsolved. We introduce recent results for the uniformity property of the digits of general smooth numbers. In particular, we consider a problem on the number of nonzero digits suggested by Bugeaud.

Next, we consider the uniformity property of the digits in the beta expansion of real numbers. Beta expansion of real numbers is a generalization of base-\(b\) expansion of real numbers, which plays an important role in dynamical systems. A real number having uniform digits in its beta expansion is called a normal number. It is difficult to show the normality of a given real number. In this talk, we introduce recent results on the normality of the beta expansion of algebraic numbers. In particular, we consider the number of nonzero digits as partial results of normality.

SUMMARY: The numbers of faces of a triangulated manifold satisfy a certain symmetry, which is known as Klee’s Dehn–Sommerville equations. In this talk, I will show that Klee’s Dehn–Sommerville equations can be algebraically explained as the Matlis duality of certain quotients of Stanley–Reisner rings using Goto’s work on Buchsbaum rings.

SUMMARY: Braverman–Finkelberg–Nakajima gave a mathematically rigorous definition of the Coulomb branches of \(3d\) \(\mathcal {N}=4\) supersymmetric gauge theories. They are certain Poisson affine algebraic varieties and admit natural quantizations. In this talk we consider the quantized Coulomb branches associated with quiver gauge theories of Jordan type. We prove that they are isomorphic to the spherical parts of cyclotomic rational Cherednik algebras. This is a joint work with Hiraku Nakajima.

SUMMARY: Elastica is an ideal thin elastic rod. Jacob Bernoulli proposed the elastica problem 1691: to determine every shape of elastica in a plane mathematically. To solve the problem, Jacob, Daniel Bernoulli and Euler discovered mathematical facts related to the lemniscate integral, the elastic energy (the oldest harmonic map), variational method, elliptic integrals, the moduli of elliptic curves. Finally Euler completely solved the problem 1744.

As its generalization, I have been studying the statistical mechanics of elasitca to find the shape of DNA. Mathematically it means to investigate the geometrical structure of the loop space with the energy. It turned out that the structure is determined by the modified KdV flows and hyperelliptic functions including their moduli.

However, the hyperelliptic function theory is not sufficient to describe the generalized elastica problem whereas the shape of original elasitca is completely described by elliptic function theory. Thus I have been also studying a reconstruction of the Abelian function theory for decades with coauthors.

In this talk, after I will show the history of elastica (including its relation to lemniscate) and its generalization, I mention the recent progress of the reconstruction of the Abelian function theory and its application to the generalization of elastica.

SUMMARY: We introduce a kind of generalization of multiple zeta function to interpolate Arakawa–Kaneko and Kaneko–Tsumura multiple zeta functions. We show that the function is closely related to polynomials called \(t\)-MZVs, which interpolates multiple zeta and zeta-star values, and that its values at non-positive integers can be written as polynomials whose coefficients are linear combinations of multi-poly-Bernoulli numbers.

SUMMARY: The families of Le–Murakami relations of multiple zeta values and Aoki–Ohno relations of multiple zeta-star values are not equivalent with each other. In this talk, we present new \(\mathbb {Q}\)-linear relations among \(t\)-MZVs which interpolate Le–Murakami and Aoki–Ohno relations of height 1. In our proof, two different expressions of the value of the interpolated function between Arakawa–Kaneko and Kaneko–Tsumura multiple zeta functions, introduced by Ohno and the speaker, are used.

SUMMARY: In this talk, we discuss the function defined by Ito and prove some relations of multiple series, which can be regarded as a generalization of Mordell–Tornheim multiple zeta values.

SUMMARY: Let \(0<a\leq 1, s\in \mathbb {C}\), and \(\zeta (s,a)\) be the Hurwitz zeta-function. Recently, T. Nakamura showed that \(\zeta (\sigma ,a)\) does not vanish for any \(0<\sigma <1\) if and only if \(1/2\leq a \leq 1\). In this talk, we show that \(\zeta (\sigma ,a)\) has precisely one zero in the interval \((0,1)\) if \(0<a<1/2\). Moreover, we reveal the asymptotic behavior of this unique zero with respect to \(a\).

SUMMARY: Higher Mahler measures (HMM) and zeta Mahler measures (ZMM) are two kinds of generalization of classical Mahler measures, which were introduced by N. Kurokawa, M. Lalín and H. Ochiai and by H. Akatsuka, respectively. In this talk, we present a formula for limiting values of HMM and an analytic continuation of ZMM.

SUMMARY: The theta function is given by the series \( \theta _3(\tau ):=\sum _{n=-\infty }^{\infty }e^{i\pi n^2\tau }, \) which converges for \(\tau \) in the complex upper half-plane \(\mathbb {H}\). In this talk we give algebraic independence results for the values of \(\theta _3(\tau )\). For example, the three values \(\theta _3(\tau )\), \(\theta _3(n\tau )\), and \(D\theta _3(\tau )\) are algebraically independent over \(\mathbb {Q}\) for any \(\tau \in \mathbb {H}\) such that \(q=e^{i\pi \tau }\) is an algebraic number, where \(n\geq 2\) is an integer and \(D:=(\pi i)^{-1}{d}/{d\tau }\) is a differential operator. This is a joint work with Carsten Elsner.

SUMMARY: We give a sufficient condition for zeros of certain modular forms on Fricke groups of level 2 or 3.

SUMMARY: A modular linear differential equation is one of tools to see the relation between elliptic modular forms and characters of specific vertex operator algebras (VOAs). In this talk, we give the relation between modular forms of certain fractional weights and characters of minimal models (the simple Virasoro VOA). Furthermore, we also give the order of a modular linear differential equation which has such modular forms as solutions.

SUMMARY: We will give the generators of the \(M_*^{even}(\Gamma _2)\)-module of vector-valued Siegel modular forms of type \((k,2)\) over \(\mathbb Z_{(p)}\). This gives an example of the positive solution to more general problem whether the module of vector-valued modular forms of arbitrary degree is finitely generated over the ring of modular forms for \(\mathbb Z_{(p)}\).

SUMMARY: We define typical spherical functions \(\omega (x;s)\) on the space \(X_n\) of quaternion hermitian forms over a \(p\)-adic field of size \(n\), by Poisson transform of certain relative invariants on \(X_n\). They can be regarded as generating functions of local densities of integral representations of quaternion hermitian forms. We study functional equations of \(\omega (x; s)\) with respect to \(S_n\) acting on \(s \in \mathbb {C}^n\), and give an explicit formula of \(\omega (x;s)\) by the method that the author gave in a general context. The situation is similar to the space of sesquilinear forms, but the main term of the explicit formula is written by a series of symmetric Laurent polynomial of different type from usual Hall–Littlewood polynomials.

SUMMARY: Rank-2 Drinfeld modules are a function-field analogue of elliptic curves. It is natural to investigate similarities and differences between rank-2 Drinfeld modules and elliptic curves. An explicit formula of a supersingular polynomial for elliptic curves was given by Max Deuring. We show an explicit formula of a supersingular polynomial for rank-2 Drinfeld modules.

SUMMARY:  Tomography is the field that reconstruct a three-dimensional object from its two-dimensional cuts. Let \(f\) be a function on \(\mathbb {Z}^n\), and \(w\) be a finite subset of \(\mathbb {Z}^n\). Discrete tomography reconstructs the function \(f\) from the data \(f_{w+p}=\sum _{x\in w+p}f(x),p\in \mathbb {Z}^n.\) This problem is proved by F. Hazama to be described completely by the zero locus of a certain polynomial in \(n\) variable associated with \(w\). The purpose of this talk is to apply his result to the zero-sum arrays when the window \(w\) has the form \(w=(s_0,s_1),(s_1,s_2),\cdots ,(s_{n-2},s_{n-1}),(s_{n-1},s_{0})\), where \(s_{i}\in \mathbb {Z}(0\leq i \leq n-1).\) Furthermore we describe the way how one can find the rational zero-sum arrays for \(w\).

SUMMARY: Given an integer \(m\) and an odd prime, if the following equality \( \overline {P}\sigma (a)=Pa-m \) is satisfied then any natural number \(a\) is said to be subperfect number with base \(P\), translation parameter \(m\).

SUMMARY: Let \(K\) be a number field and let \(\mathcal {O}_K\) be its ring of integers. We regard an \(m\)-tuple of ideals of \(\mathcal {O}_K\) as a lattice point in \(K^m\). We say that a lattice point \((\mathfrak {a}_1, \mathfrak {a}_2,\ldots ,\mathfrak {a}_m)\) is relatively \(r\)-prime, if there exists no prime ideal \(\mathfrak {p}\) such that \(\mathfrak {a}_1, \mathfrak {a}_2,\ldots ,\mathfrak {a}_m\subset \mathfrak {p}^r\).

We study the distribution of relatively \(r\)-prime lattice points in \(K^m\) with their components having norm less than \(x\). We show some results for abelian extensions or extensions with small degree by using the subconvexity bounds of Dedekind zeta functions on the critical line.

SUMMARY: We introduce a cubic analogue of the Pell’s equations associated with the simplest cubic fields and write down all integer solutions of this system explicitly by using special values of complete symmetric polynomials.

SUMMARY: We describe the ramification in cyclic extensions arising from the Kummer theory of the Weil restriction of the multiplicative group. This generalizes the classical theorem by Hecke describing the ramification of Kummer extensions.

SUMMARY: Let \(n\) and \(m\) be natural numbers greater than one. In this talk we construct an infinite family of imaginary quadratic fields \(\mathbb {Q}(\sqrt {D})\) such that both \(\mathbb {Q}(\sqrt {D})\) and \(\mathbb {Q}(\sqrt {mD})\) have ideal classes of order \(n\).

SUMMARY: In this talk, concerning Greenberg’s conjecture, we will show the existence of certain infinite families of real quadratic fields whose Iwasawa \(\lambda \)-invariant of the cyclotomic \(\mathbb {Z}_p\)-extension is equal to \(0\).

SUMMARY: In this talk, for certain rational polytopes, we consider power sums attached to Dirichlet characters for the integer points. The main result is expressed by making use of generalized multiple Dedekind sums attached to Dirichlet characters.

SUMMARY: We study an error term in a certain divisor problem related to the derivatives of the Riemann zeta-function. In particular, we obtain an analogue of Chowla–Walum formula in an error term of an asymptotic formula for \(\sum _{n\leq x}\sum _{d|n}d^{a}(\log n)^k (\log n/d)^l\). Moreover we get an upper bound for the error term.

SUMMARY: We state a criterion of when a Hibi ring is non-Gorenstein and almost Gorenstein in terms of the combinatorial structure of the poset which defines the Hibi ring.

SUMMARY: I proved that the twinned order polytope associated with partially ordered sets \(P\) and \(Q\) is normal if \(P\) and \(Q\) have a common linear extension (joint work with Takayuki Hibi (Osaka University)). In this talk, I generalize this theorem.

SUMMARY: It is fashionable among the study on convex polytopes to classify the lattice polytopes with a given \(\delta \)-polynomial. As a basic challenges toward the classification problem, we achieve the study on classifying lattice simplices with a given \(\delta \)-polynomial of the form \(1+t^{k+1}+\cdots +t^{(v-1)(k+1)}\), where \(k \geq 0\) and \(v >0\) are integers. The lattice polytope with the above \(\delta \)-polynomial is necessarily Gorenstein. A complete classification is already known, when \(v\) is prime. In this talk, we will give a complete classification when \(v\) is either \(p^2\) or \(pq\), where \(p\) and \(q\) are prime integers with \(p \neq q\).

SUMMARY: Let \(K\) be a field, and let \(S\) be the polynomial ring in \(n\) variables over \(K\). Let \(\Delta \) be a simplicial complex with \(n\) vertices and \(H\) be a bipartite graph with \(n\) vertices. We denote by \(I_{\Delta }\) the Stanley–Reisner ideal of \(\Delta \) and 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 (seq. gen. CM for short) rings. In this talk, we investigate conditions of \(\Delta \) when \(S/I_{\Delta }\) is a seq. gen. CM ring. We see that \(S/I_{\Delta }\) is a seq. gen. CM ring if all pure skeletons of \(\Delta \) are generalized Cohen–Macaulay. As an application, we give examples of a bipartite graph \(H\) such that \(S/I(H)\) is a seq. gen. CM ring.

SUMMARY: A reduced Gröbner basis of an ideal \(I\) gives a flat deformation from \(I\) to its initial ideal. Therefore, when we give a monomial ideal \(J\) and its minimal generator \(\mathscr {C}\), we are interested in the moduli space of reduced Gröbner bases whose set of leading monomials is \(\mathscr {C}\). In this seminar, we show that there exists such a moduli space as a scheme or an ind-scheme, and we give its relation with the Hilbert scheme and a characterization of its singularity.

SUMMARY: We introduce the notion of the strong Rees property (SRP) for \(\mathfrak {m}\)-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal \(\mathfrak {m}\) has its property if the associated graded ring \(G\) of m satisfies depth \(G\) is greater than 1. As its application, we characterize two-dimensional excellent normal local domains so that \(\mathfrak {m}\) is a \(p_g\)-ideal, which is related to Takahashi–Dao’s question.

SUMMARY: Let \(R\) be a commutative Noetherian ring. The notion of localization (resp. colocalization) functors in the derived cateogry \(D(R)\) is a natural generalization of left (resp. right) derived functors of completion (resp. section) functors. In this talk, we report several results about localization and colocalization functors. As an application, we can show that Grothendieck type vanishing theorem holds for colocalization functors. Moreover, by using localization functors, it is possible to give a simple proof of a classical theorem due to Raynaud and Gruson, which states that the projective dimension of a flat \(R\)-module is at most the Krull dimension of \(R\).

SUMMARY: By virtue of Balmer’s celebrated theorem, the classification of thick tensor ideals of a tensor triangulated category \(\mathcal {T}\) is equivalent to the topological structure of its Balmer spectrum \(\mathsf {Spc \mathcal {T}}\). Motivated by this theorem, we discuss connectedness and noetherianity of the Balmer spectrum of a right bounded derived category of finitely generated modules over a commutative ring.

SUMMARY: Pseudo-reflections of linear representations of groups can be extended to the affine group actions on Krull domains over an algebraically closed field \(K\). Let \(G\) be an affine algebraic group over \(K\) with a reductive identity component \(G^{0}\) acting regularly on a Krull \(K\)-domain \(R\). Let \(T\) be an algebraic closed subtorus of \(G\) and suppose that \({\mathcal Q}(R)^{T}= {\mathcal Q}(R^{T})\) of quotient fields. We will show: If \(G\) is the centralizer of \(T\) in \(G\), then the pseudo-reflections of the action of \(G\) on \(R^{T}\) can be lifted to those on \(R\). This seems to be the best possible result for the lifting of pseudo-reflections on the invariant ring of \(R\) of a normal connected subgroup to those on \(R\).

SUMMARY: Let \(K\) be an algebraically closed field. For a positive integer \(s\), we consider a self-injective special biserial algebra \(\Lambda _{s}\) obtained by a circular quiver with \(s\) vertices and \(2s\) arrows. This algebra \(\Lambda _{s}\) is a Koszul self-injective special biserial algebra for \(s\geq 1\), but is not a weakly symmetric algebra for \(s\geq 3\). Our purpose in this talk is to show that, for \(s\geq 3\), \(\Lambda _{s}\) satisfies the fineiteness condition (Fg) introduced by Erdmann–Holloway–Taillefer–Snashall–Solberg.

SUMMARY: Ringel introduced a special class of quasi-hereditary algebras called right-strongly quasi-hereditary algebras, motivated by Iyama’s finiteness theorem of representation dimensions of artin algebras. In this talk, we give characterizations of these algebras in terms of heredity chains and right rejective subcategories. As an application, we prove that any artin algebra of global dimension at most two is always right-strongly quasi-hereditary.

SUMMARY: Let \(A\) be an algebra over an algebraically closed field \(\Bbbk \) with a basic set \(A_0\) of primitive idempotents, which we regard as a \(\Bbbk \)-category with the object set \(A_0\). We denote by \(\hat {A}\) the repetitive category of \(A\), whose object set is given by \(\{x^{[i]}=(x,i) \mid x\in A_0, i\in \mathbb {Z} \}\) with the Nakayama automorphism \(\nu \) of \(\hat {A}\) sending \(x^{[i]}\) to \(x^{[i+1]}\). We set \(A^{[0]}\) to be the full subcategory of \(\hat {A}\) consisting of objects \(x^{[0]}\) with \(x\in A_0\), and \(1^{[0]} \colon A\to \hat {A}\) the embedding sending \(x\) to \(x^{[0]}\) for all \(x\in A_0\).

Let \(n\in \mathbb {Z}\). Then we show that an algebra of the form \(\hat {A}/\langle \phi \rangle \), where \(\phi \) is an automorphism of \(\hat {A}\) such that \(\phi (A^{[0]}) = A^{[n]}\) is isomorphic to an algebra of the form \(\hat {A}/\langle \widehat {\phi _0} \nu _A^n \rangle \), where \(\widehat {\phi _0}\) is an automorphism of \(\hat {A}\) naturally induced from \(\phi _0 :=(1^{[0]})^{-1}\phi \nu _{A}^{-n}1^{[0]}\) if \(eAe = \Bbbk \) for all \(e\in A_0\).

SUMMARY: Let \(k\) be a commutative field. For a given finite dimensional \(k\)-algebra, it is known that an idempotent induces a recollement of module categories. In this talk, we generalize this construction for a Krull–Schmidt \(k\)-linear category equipped with a nice duality. As an application, we provide another proof of Auslander–Reiten duality.

SUMMARY: In this talk, we give a sufficient condition related to 2-cocycles for Hochschild extension algebras of bound quiver algebras by the standard duality module to be symmetric.

SUMMARY: Let \(T\) be a Hochschild extension algebra of a finite dimensional algebra \(A\) over a field \(K\) by the standard duality \(A\)-bimodule \({\rm Hom}_K(A,\,K)\). We determine the ordinary quiver of \(T\) if \(A\) is a self-injective Nakayama algebra by means of the \(\mathbb {N}\)-graded \(2\)nd Hochschild homology group \(HH_2(A)\) in the sense of Sköldberg.

SUMMARY: In my talk, we introduce a certain condition satisfied many infinite groups and show that the group algebra KG of a group G satisfying the condition is primitive for any field K.

SUMMARY: Let \(\mathcal O\) be a complete discrete valuation ring of characteristic zero with residue class field \(k= {\mathcal O} /\pi {\mathcal O}\) of characteristic \(p>0\). Let \({\mathcal O}G\) be the group ring of a finite group \(G\) over \(\mathcal O\). Suppose that \(P\) is a \(p\)-subgroup of \(G\) and \(Q \ (\not = \{ 1_{G} \})\) is a proper normal subgroup of \(P\). We show that there exists an indecomposable \({\mathcal O}G\)-lattice \(X\) with vertex \(P\) such that all vertices of the direct summands of a \(kG\)-module \(X/\pi X\) are contained in \(Q\).

SUMMARY: We will determine the ring structure of the Hochschild cohomology \(HH^*({\Bbb Z}G)\) of the integral group ring of a split metacyclic 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: A 2-block of a finite group having a Klein four hyperfocal subgroup has the same number of irreducible Brauer characters as the corresponding 2-block of the normalizer of the hyperfocal subgroup.

SUMMARY: We study some properties of incomplete (restricted and associated) Fubini numbers. In particular, they have the natural extensions of the original Fubini numbers in the sense of determinants. We also introduce modified incomplete (restricted and associated) Bernoulli and Cauchy numbers and study characteristic properties.

SUMMARY: We introduce the loop generators corresponding to rotary operations in the \(mn-1\) puzzle. They are very useful to solve the \(mn-1\) puzzle and suitable to explain algorithms to solve it because the number of them is just \(m-1\) and the rotary operation is easy to manipulate. We show that God’s number for the 8 puzzle in the loop generators is 16, and an experimental result using a factorization algorithm for 547 test instances of the 15 puzzle is reported. The result teaches us that the length of the solution by the loop generators is not long compared with optimal solutions using single-tile moves.

SUMMARY: Kimura and Takano introduced confluent hypergeometric systems associated with centralizers of regular elements of \(\mathfrak {gl}(n,\mathbb {C})\). We introduce hypergeometric systems associated with principal nilpotent \(p\)-tuples and show to deform integrands of solutions of this systems to that of Aomoto–Gel’fand systems.

SUMMARY: Let \(K\) be a field, \(n\) be a natural number and \(G\) be a finite subgroup of \(GL(n;\mathbb {Z})\). In this talk, we will consider the rationality problem for purely monomial group actions, which asks whether the fixed field \(K(x_1,\ldots ,x_n)^G\) under the purely monomial action of \(G\) is rational over \(K\).

SUMMARY: A Weyl arrangement is the hyperplane arrangement defined by a root system. R. P. Stanley gave a characterization of the freeness of the Weyl subarrangements of type \(A\) in terms of simple graphs. The Weyl subarrangements of type \(B\) can be represented by signed graphs. Any characterization of the freeness of them has not been known. However, characterizations of the freeness for a few restricted classes are known. In this talk, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type \(B\) under certain assumption.

SUMMARY: Let \(B_n\) be the Coxeter group of type \(B\). In 1978, L. Geissinger and D. Kinch presented the concept of Young subgroups of \(B_n\) and showed that there exists a \(\mathbb {Z}\)-basis of the character ring \(R(B_n)\) of \(B_n\) consisting of Young characters, which forces \(R(B_n)\) to be isomorphic to the partial Burnside ring \(\Omega (B_n,\widehat {{\mathcal U}}_n)\) relative to the set \(\widehat {{\mathcal U}}_n\) of Young subgroups of \(B_n\). The linear \(\mathbb {C}\)-characters of \(B_n\) are identified with reduced Lefschetz invariants which are units of \(\Omega (B_n,\widehat {{\mathcal U}}_n)\).

SUMMARY: Let us denote a finite field with 4 elements by \(\mathbb {F}_4\). We will introduce an additive code over \(\mathbb {F}_4\) of length \(n\) defined by an additive subgroup of \(\mathbb {F}_4^n\). It is known that every self-dual additive code can be represented by the adjacency matrix of a simple undirected graph. Danielsen and Parker (2006) classified all self-dual additive codes over \(\mathbb {F}_4\) for lengths up to \(12\) by using graphs on up to \(12\) vertices. In this talk, we give a classification of codes having the largest minimum weight among the constructed additive codes from some graphs.

SUMMARY: Formal weight enumerators were introduced by M. Ozeki in 1997. In this talk, we propose an algorithm for the search of similar polynomials and show some examples.

SUMMARY: Zeta functions for codes were introduced by I. Duursma in 1999 and were generalized to other invariant polynomials by the present author. One of the famous problems is whether extremal weight enumerators satisfy the Riemann hypothesis. In this talk, we give an example of an extremal invariant polynomial (not being related to a code) not satisfying the Riemann hypothesis.

SUMMARY: Let \(V^\natural \) be the Moonshine vertex operator algebra which is a holomorphic vertex operator algebra of central charge \(24\) whose full automorphism group is the Monster simple group. We prove that for primes \(p=3,5,7,13\), \(V^\natural \) is constructed from the Leech lattice vertex operator algebra by a \({\mathbb Z}_p\)-orbifold construction.

SUMMARY: Untwisted/twisted affine Lie algebras are well understood and have a lot of applications not only in mathematics but also in theoretical physics. On the other hand, infinite-dimensional “Lie groups” constructed from given affine Lie algebras (á la C. Chevalley), which we shall call affine Kac–Moody groups, seems to be less understood. D. Peterson and V. Kac (1983) mentioned and Y. Chen (1996) proved that an untwisted affine Kac–Moody group can be realized as a central extension of an algebraic loop group.

In this talk, we generalize the result to all twisted cases. This is a joint work with J. Morita (University of Tsukuba) and A. Pianzola (University of Alberta).

SUMMARY: We study a compacification of quotient spaces by groupooid by the associated equivalent etale cohomology. As a corollary, we give certain type of a rigid Pic-torsor which appears in the studies of unirationality of supersingular K3 surfaces or the related moduli spaces by Ekedahl–Hyland–Shepherd–Barron, or Liedtke.

SUMMARY: In this talk, we will explain some properties of coordinates in polynomial rings by introducing some concepts which are weaker than coordinates. In particular, in the polynomial ring in two variables over an algebraically closed field of characteristic zero, we show some relations between polynomials and their fibers on an affine plane.

SUMMARY: Let \(k\) be a field of characteristic zero, \(R\) a finitely generated \(k\)-domain with field of fractions \(K\neq R\), and \(K[\mathbf {x}]=K[x_1,\ldots ,x_n]\) the polynomial ring in \(n\) variables over \(K\). In this talk, we discuss non-finite generation of the \(R\)-algebras \(B\cap R[\mathbf {x}]\) for \(K\)-subalgebras \(B\) of \(K[\mathbf {x}]\). This problem is closely related to Hilbert’s fourteenth problem and its generalization by Zariski. One of our results implies that, if \(n\geq 3\), then every non-trivial \(\mathbf {G}_a\)-action on the affine \(n\)-space over \(R\) can be ‘converted’ into a \(\mathbf {G}_a\)-action on the affine (\(n+1\))-space over \(R\) with non-finitely generated invariant ring.

SUMMARY: Let \(U\) be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar Conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck’s étale fundamental group of \(U\). In this talk, I will consider a purely inseparable analogue of this problem, formulated in terms of Nori’s profinite fundamental group scheme. I will give a partial answer to it.

SUMMARY: In this talk, we introduce the notion of stable \(k\)-rigid objects in a triangulated category for \(k\geq 1\). We then describe some properties of stable \(k\)-rigid objects in a higher cluster tube. As a main result, we completely determine the structures of stable \(2\)-rigid objects in a higher cluster tube.

SUMMARY: We give a explicit description of gluing stability conditions on geometric ruled surfaces by introducing gluing perversity. Moreover, we describe a destabilizing wall of skyscraper sheaves on ruled surfaces by deformation of stability conditions glued from \(\widetilde {GL^{+}}(2,\mathbb {R})\)-translates of the standard stability condition on the base curve.

SUMMARY: Ishii–Uehara classifies pure sheaves on the fundamental cycle of the Kleinian singularity \(A_n\). The classification is an analogue of Grothendieck’s classification of vector bundles on projective lines. We study the classification of pure sheaves on the other Kleinian singularities.

SUMMARY: A smooth projective plane curve which satisfies the inequality \(|{\rm Aut}(C)|>d^2\) is well known. In this talk, we consider a smooth projective plane curve which satisfies the equality \(|{\rm Aut}(C)|=d^2\).

SUMMARY: The field of meromorphic functions on the sigma divisor of a hyperelliptic genus 3 curve is described in terms of the gradient of its sigma function. Solutions of corresponding families of polynomial dynamical systems in \(\mathbb {C}^4\) with two polynomial integrals are constructed as an application. These systems were introduced in the work of V. M. Buchstaber and A. V. Mikhailov on the base of commuting vector fields on the symmetric products of algebraic curves.

SUMMARY: A fibered surface whose general fiber is a smooth plane curve is called a plane curve fibration. In this talk, I will show that relative invariants for plane curve fibrations can be localized at a finite number of fiber germs and a certain equality between local invariants, which is called a slope equality, holds for these fibrations. As a corollary, we can define a local signature for plane curve fibrations.

SUMMARY: In 1978, A. H. Durfee conjectured that for a hypersurface surface singularity, six times its geometric genus does not exceed its Milnor number, which is nowadays called Durfee’s strong conjecture. In this talk, I will show that an inequality among invariants of hypersurface surface singularities holds and then Durfee’s strong conjecture is true for such a singlarity with non-negative topological Euler number of the exceptional set of the minimal resolution. For the proof, we use the method of invariants of plane curve fibrations.

SUMMARY: We prove that for every smooth prime Fano threefold \(V\), the Hilbert scheme of smooth connected curves on \(V\) contains a generically non-reduced irreducible component of Mumford type.

SUMMARY: In this talk, we introduce higher order minimal families \(H\) of rational curves associated to Fano manifolds \(X\). We show that \(H\) is also a Fano manifold if the Chern characters of \(X\) satisfy some positivity conditions. We also provide a sufficient condition for Fano manifolds to be covered by higher rational manifolds.

SUMMARY: A log symplectic structure is a generically symplectic Poisson structure with a reduced degeneracy divisor. The hypothesis that a Poisson structure is generically symplectic ensures that degeneracy locus become a divisor and then it is an anti-canonical divisor. Thus the diversity of generically symplectic Poisson structures measures positivity of anti-canonical class in some sense. The past few years, classification of log symplectic structures on higher-dimensional Fano varieties attracts considerable attention in this view point. In this talk, I will explain classifications of log symplectic structures with simple normal crossing degeneracy divisor on blowing-up of projective spaces along a linear subspace and give two new examples.

SUMMARY: For a dominant rational self-map of an algebraic variety, one can define an invariant, called (first) dynamical degree, which measures the asymptotic behavior of the iterates of the map. On the other hand, when the variety is defined over a number field, one can associate to an orbit an invariant using Weil height function, called arithmetic degree, which measures the arithmetic complexity of the orbit. Kawaguchi and Silverman conjectured that the first dynamical degree and the arithmetic degree of a Zariski dense orbit coincide. We show that the arithmetic degree is less than or equals to the dynamical degree. We also prove that the conjecture is true for endomorphisms on surfaces. This is partially joint work with Kaoru Sano and Takahiro Shibata.

SUMMARY: We classify the boundary components of the central stream for a Newton polygon consisting of two slopes, where one slope is less than 1/2 and the other slope is greater than 1/2. By our method, we can enumerate boundary components much faster than the enumeration by using the criterion obtained by Moonen–Wedhorn. Moreover, this method is expected to have many applications. Central streams and boundary components are described in terms of truncated Dieudonné modules of level one (abbreviated as \(\rm DM_1\)’s). To study specializations of \(\rm DM_1\)’s, we introduce a combinatorial tool which plays an important role in our proof.