Elizabeth Gasparim, Irena Swanson
We present an algorithm for computing instanton numbers of curve singularities. A comparison is made between these and some other invariants of curve singularities. The algorithm has been implemented in the symbolic computer algebra program Macaulay2, and can be downloaded from http://www.math.nmsu.edu/\~{}iswanson/instanton.m2.}
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This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. In this paper we construct a model for the singularities of some would-be Schubert varieties in the affine Grassmannian for an affine Kac-Moody group. We formulate a conjecture describing the (local) intersection cohomology of these varieties in terms of integrable representations of the Langlands dual affine Kac-Moody group and check this conjecture in a number of cases. Roughly speaking the above singularities are constructed by looking at the Uhlenbeck sp…
The two applications are: 1. sometimes instanton numbers stratify moduli of bundles better than Chern numbers. 2. sometimes instanton numbers distinguish singularities better than the classical numerical invariants.
We use instanton numbers to: (i) stratify moduli of vector bundles, (ii) calculate relative homology of moduli spaces and (iii) distinguish curve singularities.
Instanton slices and their superpolynomials
研究任意秩平面曲线奇点的动机超多项式与瞬子切片超多项式等的联系，通过平面曲线奇点适当族超多项式的归纳极限得瞬子和新公式，有望影响相关领域，还对特定双曲结超多项式有动机解释及相关猜想。
We prove some combinatorial results required for the proof of the following conjecture of Nekrasov: The generating function of closed string invariants in local Calabi-Yau geometries obtained by appropriate fibrations of $A_N$ singularities over $P^1$ reproduce the generating function of equivariant $\hat{A}$-genera of moduli space of instants on $C^2$.
Instanton Floer homology and Milnor Fibers
本文针对Brieskorn-Hamm完全交奇异点，用米尔诺纤维描述其链环的瞬子弗洛尔同调，并借此独立证明Donaldson与Seiberg-Witten理论中Casson不变量相等，同时用类似方法表示环面结的Heegaard Floer d-不变量。
This thesis is an investigation of the moduli spaces of instanton bundles on the Fano threefold $Y_5$ (a linear section of $\mathbb{G}r(2,5)$). It contains new proofs of classical facts about lines, conics and cubics on $Y_5$, and about linear sections of $Y_5$. The main original results are a Grauert-Mülich theorem for the splitting type of instantons on conics, a bound to the splitting type of instantons on lines and an $SL_2$-equivariant description of the moduli space in charge 2 and 3. Using these results we prove the existence of a unique $SL_2$-equivariant instanton of minimal charge an…
Let $G$ be a finite group of order $n$, and $ξ$ an $n$-th primitive root of unity. Consider the affine scheme $C:=\mbox{Spc}({\mathbb Z}[ξ]\otimes_{\mathbb Z} R(G))$ where $R(G)$ is the representation ring of $G$. We study the fibers of the formal tangent sheaf of $C$ by computing their dimension and also finding (and measuring) the singularities of $C$. We present explicit computations for noncommutative groups of small order, and develop practical methods to compute these invariants for an arbitrary finite group.
We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kahler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspond…
In the present paper MAPLE subroutines computing Milnor and Tyurina numbers of an isolated algebraic hypersurface singularity are presented and described in detail. They represents examples, and perhaps the first ones, of a MAPLE implementation of local monomial ordering. As an application, the last section is devoted to writing down equations of algebraic stratifications of Kuranishi spaces of simple Arnol'd singularities: they geometrically represents, by means of inclusions of algebraic subsets, the partial ordering on classes of simple singularities induced by the adjacency relation.
Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over C, this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in quasi-linear time with respect to the discriminant valuation of a Weierstrass equation.
Let $G\subseteq GL(n)$ be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution $X\rightarrow \mathbb{C}^n/G$, which is based just on the geometry of the singularity $\mathbb{C}^n/G$, without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular. As an application, we determine the Cox rings of resolutions $X\rightarrow \mathbb{C}^3/G$ for all $G\subseteq GL(3)$ with the aforementioned property and of order $|G|\leq 12$. We also provide examples in dimension 4.
Given a rational projective parametrization $\cP(\ttt,\sss,\vvv)$ of a rational projective surface $\cS$ we present an algorithm such that, with the exception of a finite set (maybe empty) $\cB$ of projective base points of $\cP$, decomposes the projective parameter plane as $\projdos\setminus \cB=\cup_{k=1}^{\ell} \cSm_k$ such that if $(\ttt_0:\sss_0:\vvv_0)\in \cSm_k$ then $\cP(\ttt_0,\sss_0,\vvv_0)$ is a point of $\cS$ of multiplicity $k$.
Given a hypersurface, $X$, prime $p$, the zeta function is a generating function for the number of $\mathbb{F}_{p}$ rational points of $X$. Until now, there is no algorithm for computing hypersurfaces with ADE singularities. Scott Stetson and Vladimir Baranovsky provided an algorithm with Mathematica for the ordinary double point case. In this paper, I go over a Sage algorithm for computing the zeta function of a hypersurface with ADE singularities over 3-dimensional projective space. To make the algorithm more efficient, I established an equivalence between a polynomial belonging to the Jacob…
There is presented a method for computing the algebraic number of cross-cap singularities for mapping from m dimensional compact manifold with boundary M into R^(2m-1), m is odd. As an application, the intersection number of an immersion g from (m-1)-dimensional sphere to R^(2m-2) is described as the algebraic number of cross-caps of a mapping naturally associated with g.
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