109 problems

Witten's conjecture. Then $X$ has simple type; the basic classes of $X$ are precisely the Seiberg–Witten basic classes of $X$; and there is a nonzero constant $c(X)$ depending only…

Let $M=H_g^1\cup_{\Sigma_g}H_g^2$ be a Heegaard decomposition of a homology $3$-sphere, and let $I(M)$ denote its instanton Floer homology. For the trivial $SU(2)$ bundles on the h…

Consider the complex projective plane $\mathbb{C}\mathrm{P}^2$ with gauge group $\mathrm{SU}(2)$. Its regularized $u$-plane integral has a contribution from the cusp at…

Let $G$ be a gauge group with gauge coupling $\tau$, and let $G^\vee$ be a group whose Lie algebra satisfies $\operatorname{Lie}(G^\vee)=\mathfrak{g}=\operatorname{Lie}(G)$. Let…

Let $(M,\mathcal P_M)$ be as in Theorem 61, let $\Sigma$ be the cylindrical end of $M$, and let $R(M;\mathcal P_M)$ and $R(\Sigma;\mathcal P_\Sigma)$ denote the corresponding repre…

Let the parameter space of hyperbolic monopoles contain a compact subset and a disjoint subset, and let the monopoles have sufficiently small mass. Small-mass asymptotic separation…

Let $A$ be an anti-self-dual connection on a bundle $E\rightarrow T\times\mathbb{C}$, with curvature satisfying $|F_A|\sim O(|w|^{-2})$. A holomorphic structure…

Physicists' conjecture. Fix a connection $\nabla_A\in\mathcal{C}$. Then for every $\nabla_{A'}\in\mathcal{C}$, there exists a unique $g\in\mathcal{G}$ such that…

Let $X$ be a closed oriented even $4$-manifold, meaning that its intersection form is even, and let $\sigma(X)$ denote its signature. The $\frac{5}{4}$-conjecture. One has … This i…

Hyperkähler Nahm transform conjecture. Then $M$ is diffeomorphic to $N$, and the hyperkähler Nahm transform is invertible.

Local structure conjecture. (1) There exists a proper continuous map

Let ${\cal V}^{DF}$ be Donaldson's TQFT associated with moduli spaces of connections on a non-trivial $SO(3)$ bundle. For $x_1,x_2,\ldots$, define … Let ${\cal V}^{(D)}$ be the pol…

The $10/8$-conjecture. Then

Non-Abelian gauge-fixing conjecture. There exists a gauge transformation $\lambda\in W^{1,p}(\operatorname{Ad}P)$ such that

Let $M$ be a manifold, $G$ a Lie group, $X=G/H$, and let $\varphi$ be smooth. An admissible lift is a map $u$ whose gauge potential $u^{-1}du$ satisfies the admissibility condition…

Charge-existence conjecture. For a given $\mu$ there are monopoles with any collection of nonnegative magnetic charges $(m_1,\ldots,m_s)$. Given a choice of magnetic charges, there…

Let ${\mathcal M}$ be the monopole moduli space, let ${\mathcal M}^{\infty}$ be the moduli space of boundary data at infinity, and let … be the boundary map. For monopoles…

Affine theta-function expansion conjecture. The $A_{N-1}$ theta function in the blow-up formula satisfies

Let $\mathcal{T}$ be the residual symmetry algebra, let $\mathcal{U}$ be the non-confined algebra, and let $\Gamma:\mathcal{T}\to\mathcal{U}$ be the Hopf map. Define the left and r…

Let $(X_1, \Sigma_1)$ and $(X_2,\Sigma_2)$ be permissible 4-manifolds with $b_1(X_i)=0$, and let $\tilde X_i=X_i\#_{\Sigma_i}B_g$. Write … Choose an identification $\phi$, set…

Let $M$ be a closed oriented Riemannian manifold with a non-zero relevant topological invariant, and let $D=\{(x_+,+1),(x_-,-1)\}$ consist of two oppositely labelled points at geod…

Let $\Gamma$ be a finite subgroup of $SU(2)$, and let $X_\zeta$ be the space in the paper's theorem. Let $S_1,\ldots,S_r$ be the components of the $S^1$-fixed locus of $X_\zeta$ ge…

Let $X$ be a closed, smooth, oriented $4$-manifold with $b_2^+(X)\geq 2$. The Bauer–Furuta invariant has BF blowup simple type and $X$ is of BF homogeneous type. This conjecture ex…

Let $k$ be as in the paper, let $x_1$ and $y$ specify the twist, and let $\alpha_{\mathrm{DW}}$ and $\alpha_{\mathrm{GW}}$ be the Dijkgraaf–Witten and Gu–Wen classes in…

Let $\mathfrak{p}_\zeta:\mathbb{P}\to\mathbb{M}$ be the universal bundle with connection $\mathbb{A}$, and let $p_\zeta:P_\zeta\to M_\zeta$ denote its restriction to a fibre. The f…