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## JARCS 2017 Australian-Japanese Real and Complex Singularities Workshop

Last updated: September 20, 2017

### 1 Takao Akahori (Hyogo University, Japan): Partially integrable almost CR structuresand CR Hamilonian flows

Abstract: Let N be a complex manifold. LetM be a real hypersurface of N, which is strongly pseudo convex. Then, over M, a CR structure is induced. For this CR structure, the deformation theory is successfully developed and the notion of CR Hamiltonian ows is introduced. While if N is an almost complex manifold, over M, an almost CR structure is induced(not a CR structure). Especially,we assume that our almost CR structure is partially integrable. In this talk, we see that even in this case, we have similar results.

### 2 Hirokazu Anai (Fujitsu Labs Ltd / Kyushu University, Japan): Automatic NaturalLanguage Mathematical Problem Solving by Real Quantifier Elimination

Abstract: In this talk I present a logic-based architecture for solving mathematical problems written in natural language. At the core of the system is a computer algebra solver that handles any formula in the language of real closed field, what we call real quantifier elimination. The natural language processing (NLP) module translates the problem text into a logical form on the basis of linguistic analysis of the text. We aim at an end-to-end problem solving system through the synthesis of the advances in linguistics, NLP, and computational mathematics. Some empirical results on Japanese university entrance examination problems will be shown.

### 3 Michael Barwick (Sydney University, Australia): Tjurina numbers and severalequisingularity relations

Abstract: Working on the classical problem of classifying function germs (${ℂ}^{2},0\right)\left(ℂ,0\right)$, Morse stability is an equisingularity condition under which two jets are equivalent if a smooth deformation can be constructed connecting them which also preserves topological type and the structure of the truncated polars. This talk gives a summary of previous work and then provides examples to show that the Tjurina number (which is an analytic invariant of the singularity) is not preserved by a Morse stable deformation.

### 4 Alex Fish (Sydney University, Australia): Unwinding Spirals

Abstract: We show that there is no bi-Lipschitz homeomorphism of R2 that maps a spiral with a sub-exponential decay of winding radii to an unwound arc. This result is sharp as shows an example of a logarithmic spiral.

### 5 Toshi Fukui (Saitama University, Japan): Arc spaces, motivic measure and Lipschitzproperty on real algebraic varieties

Abstract: We discusswe the motivic measure of the arc spaces of real algebraic varieties with respect to a homeomorphism with reasonable properties concerning arc-analycity and jacobian. We show an improvement of the change of variables formula, a version of inverse mapping theorem and Lipschitz version of inverse mapping theorem.

### 6 Michael Harre (Sydney University, Australia): Singularity Theory in Economics: Thebaby and the bathwater

Abstract: In 2007 Barkley Rosser wrote an article questioning whether the baby had been thrown out with the bathwater in economics by turning away from catastrophe theory. However, the history of ”critical economies” is not quite as simple as it first appears. This talk traces one branch of this history, a branch due to Nobel prize winner Gerard Debreu, showing how he formally argued that singularity theory was not significant to most of economic theory. This has had practical consequences, following the Global Financial Crisis in 2007-08 economics has been criticised for not having recognised the risks of such a crisis occurring, risks due to an economy being close to a ”critical state”. In particular, critics have pointed to the mathematisation of economics: the focus on economic theory and theorem proving at the expense of key economic applications. Some practical problems in economics based on singularity theory will be outlined and I will conclude with some comments on our work on the risks in Australias housing market.

### 7 Adam Harris (University of New England, Australia): Stable embedding of isolatedsurface singularities

Abstract: If X is a Stein surface with isolated singularity at the origin, when do the holomorphic functions which define an embedding in complex Euclidean space extend holomorphically to complex deformations of the regular part of X ? Deformations for which this holds are said to be stably embeddable, and correspond to parametric families of surfaces in which the geometric genus of each fibre is invariant. However, such a criterion assumes a priori knowledge of the fibres obtained from a given deformation of X. The question examined in this talk is how one may characterize stably embeddable deformations intrinsically, i.e., purely in terms of the complex structure of X itself.

### 8 Shihoko Ishii (Tokyo Woman’s Christian University): Singularities with respectMather-Jacobian discrepancies in arbitrary characteristic

Abstract: In the talk I will introduce Mather-Jacobian discrepancy which is a variant of “usual discrepancy”. “Usual discrepancy” plays important roles in Minimal Model Program. While Mather-Jacobian discrepancy appears recently and well described by jet-schemes. In the talk I will also show the usefulness of Mather-Jacobian discrepancies in the study of positive characteristic by making use of jet-schemes.

### 9 Satoshi Koike (Hyogo University of Teacher Education, Japan): Characterisations of$V$-sufficiency and${C}^{0}$-sufficiencyof relative jets

Abstract: The Kuiper-Kuo theorem is well-known in the singularity theory, as a result giving sufficient conditions for $r$-jets to be $V$-sufficient and ${C}^{0}$-sufficient in ${C}^{r}$ functions or ${C}^{r+1}$ functions. The converse is also proved by J. Bochnak - S. Lojasiewicz. Generalisations of the criteria for $V$-sufficiency and ${C}^{0}$-sufficiency to the mapping case are established by T.-C. Kuo and J. Bochnak - W. Kucharz, respectively. In this talk we consider the problems of sufficiency of jets relative to a given closed set. We give several equivalent conditions to the relative Kuo condition, and discuss characterisations of $V$-sufficiency and ${C}^{0}$-sufficiency of relative jets, correspoding to the above non-relative results. Applying the results obtained in the relative case, we construct examples of polynomial functions whose relative $r$-jets are $V$-sufficient in ${C}^{r}$ functions and ${C}^{r+1}$ functions but not ${C}^{0}$-sufficient in ${C}^{r}$ functions and ${C}^{r+1}$ functions, respectively. This is a joint work with Karim Bekka.

### 10 Tzee-Char Kuo (The University of Sydney, Australia): An Old Man’s MathematicalStories

Abstract: A finite, disjoint, union of short stories: 1962 Chicago lecture series “How To Do Research” (Mac Lane, Calderon, Halmos,...,.); Impact of Category Theory on Steenrod; Steenrod’s correction of a “mistake” of Cartan-Eilenberg; Chevalley and Thom; Thom vs. Grothendick; Universal stratification and the Whitney hurdle; Whitney-Thom stratification, the PaPa stratification; Lagrange, Peano, and the Curve Selection Lemma; etc..

### 11 Gus Lehrer (Sydney University, Australia): Geometry and invariant theory

Abstract: I shall show how fundamental questions in the invariant theory of isometry group schemes (orthogonal, symplectic, orthosymplectic) may be reduced to questions about $GL$, using algebraic geometric methods. In this way one can treat the super (${Z}_{2}$-graded) cases, and avoid the mysterious Capelli identities. This is joint work with Ruibin Zhang, and partly with Pierre Deligne.

### 12 Yutaka Matsui (Kindai University, Japan): Topological Radon transforms onGrassmann manifolds

Abstract: In this talk, we will study Radon transforms of constructible functions, whose integrals are based on the Euler characteristics. In particular, we will show the injectivity and a characterization of the images of them on compact Grassmann manifolds. We would also discuss them on affine Grassmann manifolds.

### 13 Scott McCallum (Macquarie University): Singularity theory and computationallogic

Abstract: We describe two applications of parts of singularity theory in computational logic over the real numbers. In 1975, Oscar Zariski published a short paper with the title, ”On equimultiple subvarieties of algebroid hypersurfaces.” This paper extended part of Zariski’s comprehensive theory of equisingularity of the 1960s. The same year (1975), George Collins presented his seminal work on computational logic, ”Quantifier elimination for real closed fields by cylindrical algebraic decomposition (CAD).” These two papers, each probably unknown to the other author at the time, turned out to have an important connection: Zariski’s paper points the way to a significant improvement of Collins’ work with respect to algorithmic efficiency. The key concept involved is that of equimultiplicity, also known as multiplicity-invariance. In 1990, Daniel Lazard proposed an alternative concept, that of valuation-invariance, and he sketched its application to CAD. Lazard’s approach was seen to have some advantages relative to the Zariski inspired one. However some serious flaws in Lazard’s associated theory were noticed quite quickly. This year 2017, Parusinski, Paunescu and I are pleased to announce that the Lazard approach using valuation-invariance can be firmly grounded using different mathematics. This has much potential benefit for the future development of CAD.

### 14 Katsusuke Nabeshima (Tokushima University, Japan): Computing Bruce-RobertsMilnor numbers

Abstract: Bruce-Roberts Milnor number is a generalization of the Milnor number, a multiplicity of an isolated critical point of a holomorphic function germ. This number is defined for a critical point of a holomorphic function on a singular variety. In the hypersurface isolated singularity case, Bruce-Roberts Milnor numbers can be considered in the context of symbolic computation. This talk has mainly two parts. In the first part of the talk, we consider how to compute Bruce-Roberts Milnor numbers, namely, we give an algorithm for computing Bruce-Roberts Milnor numbers. In the second, we report on some computational experiments for Bruce-Roberts Milnor numbers of polynomial functions on semi-quasi homogenous singular varieties. This is a joint work with Prof. Shinichi Tajima and Prof. Takeshi Izawa.

### 15 Osamu Saeki ( Kyushu University, Japan): A Vassiliev type invariant of order one forstable maps of 3-manifolds into surfaces

Abstract: Based on the signature formula for stable maps of closed oriented 4-manifolds into 3-manifolds, the author defined a Vassiliev type invariant of order one for stable maps of closed oriented 3-manifolds into surfaces. In this talk, we give an explicit formula for the invariant in terms of a certain linking form for a framed link in the 3-manifold consisting of regular fibers of a stable map. As a corollary, we get a signature formula for 4-manifolds with boundary in terms of their singular fibers of stable maps. An enlightening explicit example is also given.

### 16 Jiro Sekiguti (Tokyo University of Agriculture and Technology, Japan)): PotentialVector Fields related with Algebraic Solutions to Painleve VI Equation

Abstract: This is based on the joint work with Mitsuo Kato, Toshiyuki Mano. A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painleve VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this talk is to explan the idea how to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions.

### 17 Geordie Williamson (Sydney University, Australia): On torsion in the intersectioncohomology of Schubert varieties

Abstract: A basic invariant of a singularity of a complex algebraic variety is its (local) intersection cohomology. This group makes sense with coefficients in any ring, but is very difficult to compute in general. For an isolated singularity it amounts to computing the cohomology of the link of the singularity. I will explain what is known and not known about this question for singularities of Schubert varieties. This is an interesting question in itself, and also has deep connections to Lie theory (Kazhdan-Lusztig conjeccture, Lusztig conjecture etc.)

### 18 Oded Yacobi (Sydney University, Australia): On equations defining the affine Grassmannianof $S{L}_{n}$

Abstract: The affine Grassmannian $Gr$ of a semisimple group $G$ is an important infinite dimensional variety that appears in representation theory and related areas. This talk concerns the projective geometry of the affine Grassmannian when $G=S{L}_{n}$. More precisely, in this case $Gr$ naturally embeds into the Sato Grassmannian $SGr$, which is a limit of finite dimensional Grassmannians $Gr\left(n,2n\right),n\to \infty$. We are interested in the equations defining the embedding $Gr\subset SGr$. Kreiman, Lakshmibai, Magyar and Weyman constructed linear equations on $SGr$ which vanish on $Gr$ and conjectured that these equations suffice to cut out the affine Grassmannian. We recently proved this conjecture by reducing it to a question about finite dimensional Grassmannians. I’ll describe our method of proof and mention some conjectures that arise from our work. In particular I’ll discuss the relation of our work to the problem of describing the equations of an interesting class of singular varieties: the nilpotent orbit closures in positive characteristic. This is joint work with Dinakar Muthiah and Alex Weekes.

Abstract: Let $R=\mathbf{C}\left[{x}_{1},{x}_{2},...,{x}_{n}\right]∕\left({f}_{1},....,{f}_{m}\right)$ be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, commutative algebra and rational homotopy theory is the non-existence of negative weight derivations on R. Alexsandrov conjectured that there is no negative weight derivation when R is a complete intersection algebra and Yau conjectured there is no negative weight derivation on R when R is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in differential geometry. On the other hand, Wahl conjectured that non-existence of negative weight derivations is still true for positive dimensional positively graded R. In this talk we present our recent progress on these problems. Joint work with Bingyi Chen, Hao Chen, and Huaiqing Zuo.