26 problems

Teissier's conjecture. The Łojasiewicz exponent $\mathcal{L}_0(F)$ may be a topological invariant of the singularity $X$.

Let $X$ be an oriented 3-manifold with $b_1(X)>0$, and let $I$ denote the invariant considered in the paper. Let the Seiberg–Witten invariant of $X$ be given in the relevant parame…

Let $H$ be an involutory Hopf category, and let its 2-category of representations be the corresponding representation 2-category. Spherical representation 2-category conjecture. Th…

Let $H$ be an involutory Hopf category whose representations form a spherical 2-category $C_H$. Crane–Frenkel equivalence conjecture. The Crane–Frenkel invariants using $H$ are equ…

Let $\Sigma\subset S^4$ be an oriented knotted surface. Let $\mathcal{G}=(G,E,\partial,\triangleright)$ be a finite crossed module. Write $I_\mathcal{G}(S^4\setminus\nu(\Sigma))$ f…

Let a 2-complex be embedded in a triangulated 4-manifold, and let its Crane-Yetter partition function be the state-sum quantity defined by the model. Normalization conjecture. The…

Let $L(p,q)$ be a three-dimensional lens space with arbitrary $p$ and $q$, and let $I_k$ denote the invariant associated with the integer $k$. The relevant values are indexed by…

Torsion-polynomial conjecture. For every prime $p\ge3$, the map $q\mapsto \tau(p,q)$ is constant precisely on the orbits $\{\pm q^{\pm1}\}$. Equivalently, $\tau(p,q)$ distinguishes…

A zigzag multiplicity is the multiplicity of a zigzag summand in the bicomplex of differential forms of a compact complex manifold, and a Chern number is a characteristic number ob…

Let $Y$ be a negative-definite plumbed manifold, let $\spc{Y}$ denote its set of $spin^c$ structures, and let $\Delta_b$ be the corresponding invariant for $b\in \spc{Y}$. Let…

Yumoto–Misumi conjecture. Under these conditions, the maximal number of fermion species is

Electric-zero bound conjecture. The quantity $Z(K)$ satisfies

Let $M$ be a connected, non-compact, orientable 3-manifold with one toroidal end and a complete hyperbolic structure, and let $\rho:\pi_1(M)\to PSL(2,\mathbb{R})$ be an irreducible…

Let $M$ be a cusped 3-manifold, let $\rho$ be a non-real boundary-parabolic representation of $\pi_1(M)$, and suppose that a suitable shaped ideal triangulation and a solution $z$…

Furuta–Ohta mod-2 reduction conjecture. The Furuta–Ohta invariant satisfies

Seiberg–Witten invariant conjecture. The equivariant geometric genera are topological and are given by the Seiberg–Witten invariant of the link.

A knot type is an ambient isotopy class of tame knots in $S^3$. An optimal coil is a current-carrying knot arranged so that the Lorentz force vanishes on it. The Alexander norm and…

Let $D=2n$ be an even dimension, let $E_{2n}$ denote the corresponding topological invariant, and let \chi^[^^[^u_{2n} be a metric-dependent vector function. Conjecture 2. For ev…

Let $D=2n$ be any even dimension, with $n=1,2,3,4,\ldots$. Let $E_{2n}$ denote the topological invariant in dimension $2n$, and let the auxiliary scalar field $\Phi$ transform unde…

Let a unitary Atiyah–Segal-type $(3+1)$-TQFT be a unitary topological quantum field theory of the Atiyah–Segal type in three spatial dimensions, with partition function assigned to…

The Grassmann-algebraic Pachner-move invariant. The expression

Consider the $2\to4$ Pachner move replacing the 4-simplices $(12345)$ and $(12346)$ by $(12356)$, $(12456)$, $(13456)$, and $(23456)$. Let $\tilde{\mathcal W}_{ijklm}$ denote the c…

Let $\mathcal{D}^{H}$ be the category of modules under consideration, and let $\varpi\in\mathcal{U}^{H}$ be the central element defined by $\varpi=uS(u)^{-1}K_{2\rho}^{2(r-1)}$. Le…

Superspace topological-invariants conjecture. Some topological invariants of $\mathcal{S}_{\mathrm{F}}(\Sigma)$ should bear the same relation to anomalies and sectorial structure a…

Let $\mathcal{C}$ be a spherical fusion category. Its center $\mathcal{Z}(\mathcal{C})$ is modular, and let $\mathrm{TV}_{\mathcal{C}}$ and $\mathrm{RT}_{\mathcal{Z}(\mathcal{C})}$…