45 problems

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be reduced holomorphic function-germs, and let $V(f)$ and $V(g)$ denote their zero-set germs at the origin. Suppose there is a home…

Let $n,d$ be positive integers, let $T,\mathbf{t}\in\mathbb{C}^n\otimes((\mathbb{C}^n)^*)^{\otimes d}$, and assume that $\mathbf{t}$ is non-singular. For $\lambda_0\in\mathbb{C}$,…

Let $X\subset\mathbb{C}^n$ and $Y\subset\mathbb{C}^m$ be complex analytic sets with $\dim X=\dim Y=d$. Their germs at zero are bi-Lipschitz homeomorphic if there is a bi-Lipschitz…

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring with its standard grading, let $I\subset S$ be a graded ideal, and let $R=S/I$ be a standard graded $K$-algebra. Write…

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring, let $\widehat{R}$ denote its $\operatorname{\mathfrak{m}}$-adic completion, and consider … where $I$ ranges over t…

Teissier's conjecture. Any $\mu$-constant family of isolated hypersurface singularities is equimultiple.

Let $V$ be a finite set, let $R=\Bbbk[V]$ be a standard graded polynomial ring, and let $I$ be a homogeneous ideal of height $c$. Let $e(R/I)$ be the Hilbert-Samuel multiplicity, a…

Nonexistence conjecture. There is no Buchsbaum homogeneous integral domain over $k$ with minimal multiplicity of degree $q \ge 2$.

Huneke–Srinivasan–Herzog–Srinivasan conjecture. If $R/I$ is Cohen–Macaulay, then

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, and let $N$ be a finitely generated graded torsion $R$-module. Set $c=n-\dim N$ and, for a minimal grade…

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, and let $I\subset R$ be a graded ideal. For a minimal graded free resolution of $R/I$, let $M_i$ denote…

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, let $I\subset R$ be a graded ideal, and set $c=\operatorname{codim} R/I$. For a minimal graded free reso…

Let $K$ be a field, and let $R$ be a homogeneous $K$-algebra of codimension $s$. For a graded minimal free resolution of $R$ over a polynomial ring, let $M_i$ denote the largest de…

Let $n,d$ be positive integers, let $T,\mathbf{t}\in\mathbb{C}^n\otimes((\mathbb{C}^n)^*)^{\otimes d}$, and assume that $\mathbf{t}$ is non-singular. For $\lambda_0\in\mathbb{C}$,…

Let $x\in X$ be a point on a variety with log canonical singularities. Denote $e:=\dim m_x/m_x^2$ and $d:=\dim X$ at $x$. The minimal log discrepancy at $x$ is denoted by…

Metric Zariski multiplicity conjecture. If the germs $(X,0)$ and $(Y,0)$ are ambient bi-Lipschitz homeomorphic, then

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be reduced holomorphic function-germs. Suppose there is a homeomorphism $\varphi\colon(\mathbb{C}^n,0)\to(\mathbb{C}^n,0)$ such tha…

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be reduced holomorphic function-germs, and let $f_*$ and $g_*$ be their initial forms. Suppose there is a homeomorphism…

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be holomorphic function-germs with isolated singularities at the origin. Suppose there is a homeomorphism…

Let $\mathcal{A}$ be a $k$-th order, $n$-dimensional tensor, let $lambda$ be an eigenvalue, and let … be its eigenvariety. Suppose that this variety has irreducible components…

Let $d\geq 2$ and $N\geq d$ be positive integers. Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring of dimension $d$, and let $I$ be an $\operatorname{\mathfrak{m}}$…

Let Theorem Hanes denote the theorem in the paper concerning the relevant uniform improvement of Lech's inequality under its stated characteristic assumption. Characteristic-free H…

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring of dimension $d\geq 1$. Write $\widehat{R}$ for its completion, $\operatorname{Spec}\widehat{R}$ for its spectrum,…

Let $X,Y\subset\mathbb{R}^n$ be two germs at the origin of irreducible real analytic subsets. Let $h\colon(\mathbb{R}^n,0)\to(\mathbb{R}^n,0)$ be a germ of a subanalytic, arc-analy…

Let $n\geq 3$, and let $\widetilde{\pi}$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$. Write $\mathcal M(\mathrm{L}(\widetilde{\pi}))$ for the number o…