Studio Phones Seminar continues as Kobe Studio Seminar from April 2013. Please refer to Kobe Studio Seminar for Mathematics.
Zhangjiashan Han bamboo book "Suanshu-shu" and Qin bamboo book "Shu" housed at Yuelu Academy are mathematical books in ancient China, whose texts and photographs were published several years ago.
Our procedure was to decipher again the letters from the photographs with the following investigation from mathematical and historical viewpoints. In this talk, the contents of two books will be introduced briefly and some issues will be compared with those of "The Nine Chapters on the Mathematical Art."
Hempel introduced the concept of distance of Heegaard splitting by using curve complex, and showed that there exist Heegaard splittings of closed orientable 3-manifolds with distance $>n$ for any integer $n$. In this talk, we construct pairs of curves with distance exactly $n$ for any integer $n$, and we show that there exist Heegaard splittings of 3-manifolds with distance exactly $n$.
We will introduce the concept of the dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given. Our methods not only recover some known results on the dimensions of derived categories in the sense of Rouquier, but also apply to various commutative and non-commutative noetherian rings.
Khovanov defined a bi-graded link homology whose Euler characteristic is Jones polynomial. Subsequently, Bar-Natan defined a cobordism category associated to Khovanov homology and showed a chain complex of cobordisms is a link invariant up to homotopy equivalence. In this talk, I'll introduce a cobordism category associated to Khovanov-Rozansky homology (a triple graded homology whose Euler characteristic is a specialized HOMFLY polynomial). If I have a time, I'll explain a relation of the cobordism category, a homotopy category of matrix factorizations and a category of Soergel bimodules.
Silting objects appeared in work of Keller-Vossieck on quiver representations. Following I.-Aihara (JLMS 2012), Koenig-Yang (arXiv:1203.5657) and Adachi-I.-Reiten, I will explain mutation theory of silting objects in triangulated categories as well as their relationship with t-structures and co-t-structures.
It is known that there is only trivial tilting module over selfinjective algebra. Hence it is difficult to give a derived equivalent algebra explicitly for selfinjective algebra. In this talk, I will explain some connections between selfinjective quivers with potentials and derived equivalences, and provide a derived equivalence class algebras.
I will start with a survay of well known properties of derived categories in algebraic geometry and Bridgeland's stability conditions. I also explain a Huybrechts' excellent article (arXiv:1009.4371). If we have enough time, I shall refer to relations with structures of Chow groups and some categorical problems in arithmetic geometry.
I. M. James introduce the characteristic of a CW-complex $S^q\cup e^n\cup e^{q+n}$. This is a relation between the homotopy class of the attaching map of $e^{q+n}$ and the ring structure of the singular cohomology. For a CW-complex $S^q\cup e^n\cup e^{q+n+k}$, we consider a relation between the homotopy class of the attaching map of $e^{q+n+k}$ and the ring structure of a generalized cohomology.
I will give a brief description of unstable homotopy groups of CW-complexes. Specially, for a CW-complex X and its subcomplex A, I will talk about methods of calculation of homotopy groups of X using by homotopy groups of A, relative homotopy groups of (X,A) and the CW-structure of the homotopy fiber of a pinching map A to a point in X.
We first introduce logical structures and discuss basic properties. Then we prove that strongly minimal structures are uncountably categorical. Specific examples of this result are "Vector spaces with fixed base field are isomorphic if and only if they have the same dimension defined by the linear dependence." and "Algebraically closed fields with characteristic p are isomorphic if and only if they have the same dimension defined by the algebraic dependence." This fact is a very simple version of Morley's categoricity theorem which is one of the most important classical result in model theory.
I will talk about an elementary proof of (local) Kronecker-Weber theorem. Other topics will include local class field theory and its applications.
In this talk we shall survey recent progress on the Bridgeland's stability conditions and it's wall-crossing. The material covered in this talk is based almost entirely on the two celebrated articles by T. Bridgeland. We shall treat special objects the bounded derived category of coherent sheaves on smooth projective curves and K3/abelian surfaces. If there is enough time, I will also explain some results on relations classical stability and Bridgeland's stability.
This is a report on joint work with T. Ochiai (Osaka University) on the relation between the characteristic ideal arising from Hida deformation and the 2-variable p-adic L-function.
The speaker recently introduced a generalization and a refinement of cell complexes, called cellular stratified spaces, which allow us to extend techniques used by combinatorialists to study posets and simplicial complexes to much more general structures such as topological categories. This talk is an exposition of cellular stratified spaces and related structures. We mainly focus on motivations and (possible) applications.
We review simplicial and cellular structures used in algebraic topology, such as simplicial complexes, cell complexes, simplicial sets, simplicial spaces, $\Delta$-sets, and cosimplicial spaces, from a historical perspective. Simplicial complexes played a fundamental role when Poincare tried to define homology. Cell complexes and CW complexes were introduced by Whitehead and form basic building blocks of classical algebraic topology. In modern algebraic topology, model categories equipped with simplicial or cellular structures are basic objects of study. Simplicial sets also provide a foundation of $(\infty,1)$-categories, from which Lurie developed derived algebraic geometry and extended topological quantum field theories. If time allows, we might be able to discuss these recent developments.
We give a higher order generalization of Fukaya's Morse homotopy theoretic approach to 2-loop Chern--Simons perturbation theory. In this talk, we construct a sequence of invariants of homology 3-spheres with values in a space of trivalent graphs (Jacobi diagrams) by using Morse homotopy theory.
The diffeomorphism type of a differentiable manifold can be studied by considering some smooth function on the manifold (Morse theory). I will review the basics of Morse theory and some related topics.
We give a higher order generalization of Fukaya's Morse homotopy theoretic approach to 2-loop Chern--Simons perturbation theory. In this talk, we construct a sequence of invariants of homology 3-spheres with values in a space of trivalent graphs (Jacobi diagrams) by using Morse homotopy theory.
This talk will be a brief introduction to the Model Theory. We will discuss a model theoretic viewpoint over mathematics and, especially, algebra.This talk assumes no prior knowledge of logic.
I will survey several results concerning topology of hyperplane arrangements with focus on the fundamental group. (Kew words: homotopy type, K(\pi, 1) space, minimality, fundamental group.)
Let $X$ be a definably compact definable $C^r$ manifold and $2 \le r <\infty$. Then the set of definable Morse functions $Def_{Morse}^r(X)$ is open and dense in the set $Def^r(X)$ of definable $C^r$ functions on $X$ with respect to the definable $C2$ topology.
An integrable system on a symplectic manifold of dimension 2n is an n-tuple of independent functions which are mutually Poisson commutative. Moment maps of torus actions on toric varieties are typical example of integrable systems. I would like to talk about integrable systems on the Grassmannians of two-planes, and their deformations into toric moment maps. We also discuss a relation to bending Hamiltonians on the moduli space of polygons in the Euclidean three-space. This is a joint work with K. Ueda.
We consider a certain ring for a finitely generated group, and introduce the results about correspondence of the ring and the quasi-isometric geometry of the finitely generated group.
This talk is a brief survey of pre-computable category, especially, for topological objects, invariant functions, and algebraic structures. On the background of our research, we are studying computer sciences, logics and linguistics. I also talk about some details of pre-computable category within the context of 2 and 3 dimensional computer graphics and general graphics.
"How many lines are there in the three space which meet all the four given lines ?" In 19th century, H.Schubert considered this problem in an insightful but not rigorous way. He invented a symbolic "calculation'' for the conditions on lines as follows: [intersecting a given line]^{\cap 4} = [lying on a given line] \cup [lying on a given line], and obtained the answer two. In fact, the "algebra'' of the conditions on lines is isomorphic to the ring of the symmetric polynomials called Schur polynomials. D.Hilbert asked for a rigorous foundation for the above calculus as the 15th problem in his 1900 lecture and now Schubert’s quiz can be rephrased in terms of cohomology, or equivalently, intersection theory of a Grassmaniann manifold. In this talk, I will briefly review the basics of Schubert calculus with a focus on the correspondence of several algebras occurring in this subject including the above one.
I will explain about Zhu's modular invariance property of characters of modules for rational vertex operator algebras.
In the 70's, Jorgensen and Thruston proved that for any V>0 there exists a finite collection of manifolds X_1,...,X_n so that any complete hyperbolic 3-manifold of volume at most V is obtained by filling some X_i. A well-know "folk theorem" of Thruston says that there exists a constant K so that X_i can be triangulated using at most KV tetrahedra. We will first motivate this theorem by describing two applications. The purpose of this talk is providing a proof of Thurston's theorem. The proof follows an outline that appeared in the litrature, but as remarked by Benedetti and Petronio, it requirs control over the intersection between the Voronoi cells and the thin and thick parts of the manifolds (the terms will be explained in the talk). We will show how we control these intersections. Most of the work is elementary and done in hyperbolic 3-space. I will make an effort to make it accessible to students familiar with the upper half space model.
This is a joint work with Kazuhiro Ichihara. We will introduce simplicial complexes by using various invariants and local moves on knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In this talk, we will study simplicial complexes defined by using the Alexander-Conway polynomial and the crossing change or the Delta-move. We will show that these simplicial complexes are Gromov hyperbolic.
Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve.We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions.Furthermore, we give data set which determine the conjugacy class of a root. As a consequence, we can find the minimum degree and the maximum degree, and show that the degree must be odd. Also, we give Dehn twist expression of the root of degree 3
Every smooth compact n-manifold is decomposed into some copies of n-disks (= handles) according to certain rules. I would like to introduce handle decompositions of manifolds (especially knot complements) and demonstrate how to draw them.
Quandle is an algebraic structure which was introduced by David Joyce in 1982, motivated by knot theory and conjugation of a group. In this talk I will introduce several aspects of quandles and related topics; quandle homology and relation with group homology, rack space, (twisted) Alexander polynomial, etc.
In recent years, several people have suggested the various kinds of Iwasawa Main Conjecture formulated in the language used in their own field of study. In this talk, I begin with a brief review on classical Iwasawa theory, including the classical Main Conjecture. Then I move on to the p-adic modular forms, L-functions, and the Main Conjecture. The main theme of this talk is to discuss Bertini-type theorem and characteristic invariants which also serve as strategic tools to study similar invariants in toplogy and algebra.
The homology and cohomology theory of a quandle is useful to study knotted surfaces in 4-space. We define the length of a 3-cocycle of a quandle equipped with a quandle-set, and calculate the lengths of some 3-cocycles. We give lower bounds of the triple point number of a surface-knot by the length of a quandle 3-cocycle.
Our result is to determine all quandle homology groups of Alexander quandle of prime order. The proof is obtained from the calculations of quandle 'co'homology group with the generators (cocycles) by means of calculus over positive characteristic, which is inspired by T.Mochizuki's computations of third degree cocycles. This talk outlines the proof, and our goal is to illustate why 'the higher degree cocycles are constructed from lower degree ones, and linear independently generate the quandle cohomology group'.
I explain about similarity between Alexander-Fox Theory from Knot Theory, and Iwasawa Theory from Number Theory.
I will talk about a new class of Noetherian rings defined by the Frobenius map and discuss its basic properties. The talk will also focus on some open questions.
Iwasawa Main Conjecture (IMC) in two variables was proposed by several people and proved by K.Rubin in the classical case. More recently, T.Ochiai formulated IMC over some type of deformation spaces and he proved it in the case when the deformation space is the Iwasawa algebra.His idea is to take sufficiently many hypersurfaces and analyze the problem on them.I will talk about the so-called Bertini type theorem to generalize IMC to more general deformation spaces.
In this talk, we introduce an overview of a proof of Godel's incompleteness theorem. First we see the details of arithmetization of natural number theory, then we see how Godel sentence is constructed. Then we see the arithmetical hierarchy and its "circular structure" in Global sense. If we have enough time, we also try to see some facts about non-standard model of Arithmetic.
We propose a new data analytical tool for directed networks by using category theory. We develop a category theoretical treatment of directed networks in order to obtain functional networks for real networks. By applying our method to concrete data on real information processing biological networks, we find a distinguishing global structure of functional networks. We discuss a possibility of a new hypothesis on network motifs based on our theory and data analysis. We also present a glimpse of calculations when our method is thought of an abstract graph transformation.
Gorenstein rings have nice homological properties to investigate the category of finitely generated modules. In this talk, I will give some basic properties of Cohen-Macaulay modules over Gorenstein rings.
Gorenstein rings have nice homological properties to investigate the category of finitely generated modules. In this talk, I will give some basic properties of Cohen-Macaulay modules over Gorenstein rings.
I will give an introduction to algebraic geometry. We will see several aspects of algebraic curves, being as conscious to the general theory as possible. If time permits, I will explain the Mathieu group, the first generation of the Happy Family of finite simple sporadic groups.
This is an introduction to exterior differential systems. I will talk the classical theory given by E.Cartan.
Beginning with a short introduction to symbolic and numeric computation, we discuss interaction between computer science and low-dimensional topology, especially, algorithms and programming paradigms. In this talk, we give an extension language designed specifically to symbolic manipulaton of pure functions by using low-level structures of Mathematica .
Using Cerf theory Kirby proved that two framed links represent diffeomorphic closed 3-manifolds if and only if they are related to each other by a sequence of "Kirby moves" on framed links. A rough sketch of Kirby's proof is explained along his original paper.
Based on the analogy between Galois groups and Knot groups, we discuss on analogy of the theory of Dehn suregery for knots.
integral points of elliptic curves have long history in number theory and algebraic geometry. In this talk I will introduce the analogue of integral points in the elliptic surface case, originally defined by T. Shioda. To do so, I will explain elliptic surfaces, their Mordell-Weil lattices, related sphere packings, lattice theory and group theory. The purpose is to overview the subject.
We discuss interaction between Teichmuller spaces and hoomorphic Lefschetz fibrations.
Beginning with a short introduction to 4-manifolds, we discuss various properties of Lefschetz fibrations, especially, signature of Lefschetz fibrations and substitution technique for positive relations.
The speaker shall define 3-dimensional lens spaces in some ways and shall compute various topological invariants. We will consider whether we can investigate the Dehn surgery problem from the values.
The speaker shall define 3-dimensional lens spaces in some ways and shall compute various topological invariants. We will consider whether we can investigate the Dehn surgery problem from the values.
In this talk, we will survey properties of the standard models of train tracks.
In 1991, R. Stanley assigned a commutative ring $A_P$ to a finite simplicial poset $P$ for a combinatorial purpose. Recently, M. Masuda et al. studied $A_P$ as the equivariant cohomology ring of a torus manifold. In this talk, we give a concise description of the dualizing complex of $A_P$, and show an application.
In this talk I will explain how combinatorics of words can be used in Schubert calculus, i.e. reducing the problem to calculate topological invariants for Schubert varieties in terms of subword combinatorics of Coxeter groups. There are recurrence relations for canonical classes and structure constants, and explicit formulas for the solutions are given for some special cases.
It is well known that Russell paradox shows that the naive set theory implies a contradiction within classical logic. However, in 1970's, it turned out that the theory is consistent within many non-classical logics. In this talk, we introduce the case of fuzzy logic. One of the most interesting aspect of such set theory is that it forgives the very strong form of circular definitions. However this makes difficult to construct its models. In this talk we briefly introduce about them.
A virtual link is an extended concept of a link in S^3 which can be realized as a diagram on S^2 by introducing virtual crossings. Geometrically, a virtual link can be considered as a link in the product space of an oriented closed surface and the closed interval. So we can study virtual links geometrically. I will introduce some geometric methods of Virtual Knot Theory.
In this talk, I give a brief introduction of the Teichmueller theory. Especially, we will discuss the complex analytic structure of the Teichmueller space.
In this talk, I give a brief introduction of the Teichmueller theory. Especially, we will discuss the complex analytic structure of the Teichmueller space.
The detail of the proof that Ford-like polygons for a given Fuchsian group are generic for almost all points will be discussed in this talk.
I will explain some basic properties of limit sets of Kleinian groups.
About symbolic computation.
The Gelfand-Cetlin system is a completely integrable system on a flag manifold of type A, whose moment polytope is known as the Gelfand-Cetlin polytope. We show that the Gelfand-Cetlin system can be deformed into a moment map on a toric variety corresponding to the Gelfand-Cetlin polytope. We also discuss an application to disk counting problem.
Juhasz has defined sutured Floer homologies for sutured manifolds. We will talk about its definition, calculation, and applications.
For a given knot K in the 3-sphere , D. Gabai has given a method for constructing a taut finite depth foliations on the exterior of such that the minimal genus Seifert surface for K is a leaf of the foliation and the restriction of the foliation to the boundary is a foliation by circles. In this talk, we apply the method to 5_2 knot and see the behavior of leaves of the foliation.
In this talk, we will show that the hyperbolic volume of a hyperbolic knot is a quandle cocycle invariant. Further we will show that it completely determines invertibility and positive/negative amphicheirality of hyperbolic knots.
In this talk, we will show that the hyperbolic volume of a hyperbolic knot is a quandle cocycle invariant. Further we will show that it completely determines invertibility and positive/negative amphicheirality of hyperbolic knots.