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1. Uniqueness of some cylindrical tangent cones to special Lagrangians

   https://doi.org/10.1007/s00039-023-00634-x  

   Collins, Tristan C.; Li, Yang (April 2023, Geometric and Functional Analysis)

   Abstract We show that if an exact special Lagrangian $$N\subset {\mathbb {C}}^n$$ N ⊂ C n has a multiplicity one, cylindrical tangent cone of the form $${\mathbb {R}}^{k}\times {\textbf{C}}$$ R k × C where $${\textbf{C}}$$ C is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided $${\textbf{C}}$$ C satisfies an integrability condition. This applies, for example, when $${\textbf{C}}= {\textbf{C}}_{HL}^{m}$$ C = C HL m is the Harvey-Lawson $$T^{m-1}$$ T m - 1 cone for $$m\ne 8,9$$ m ≠ 8 , 9 . 

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2. Compactifications of Moduli of Points and Lines in the Projective Plane

   https://doi.org/10.1093/imrn/rnab200  

   Schaffler, Luca; Tevelev, Jenia (August 2021, International Mathematics Research Notices)

   Abstract Projective duality identifies the moduli spaces $$\textbf{B}_n$$ and $$\textbf{X}(3,n)$$ parametrizing linearly general configurations of $$n$$ points in $$\mathbb{P}^2$$ and $$n$$ lines in the dual $$\mathbb{P}^2$$, respectively. The space $$\textbf{X}(3,n)$$ admits Kapranov’s Chow quotient compactification $$\overline{\textbf{X}}(3,n)$$, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ “broken lines”. Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003. 

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3. Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

   https://doi.org/10.1093/qmath/haz022  

   Basu, Saugata; Lerario, Antonio; Natarajan, Abhiram (October 2019, The Quarterly Journal of Mathematics)

   Abstract Given a sequence $$\{Z_d\}_{d\in \mathbb{N}}$$ of smooth and compact hypersurfaces in $${\mathbb{R}}^{n-1}$$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$ such that each manifold $$Z_d$$ is diffeomorphic to a component of the zero set on $$\Gamma$$ of some polynomial of degree $$d$$. (This is in sharp contrast with the case when $$\Gamma$$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $$p$$ on $$\Gamma$$ is bounded by a polynomial in $$\deg (p)$$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$$ containing a subset $$D$$ homeomorphic to a disk, and a family of polynomials $$\{p_m\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that $$(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$$ i.e. the zero set of $$p_m$$ in $$D$$ is isotopic to $$Z_{d_m}$$ in $${\mathbb{R}}^{n-1}$$. This says that, up to extracting subsequences, the intersection of $$\Gamma$$ with a hypersurface of degree $$d$$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $$0 \leq k \leq n-2$$ and every sequence of natural numbers $$a=\{a_d\}_{d\in \mathbb{N}}$$ there is a regular, compact semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$, a subsequence $$\{a_{d_m}\}_{m\in \mathbb{N}}$$ and homogeneous polynomials $$\{p_{m}\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $$b_k$$ denotes the $$k$$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $$\Gamma$$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $$d$$, of the set $$\Sigma _{d_m,a, \Gamma }$$ of polynomials verifying (0.1) is positive, but there exists a constant $$c_\Gamma$$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $$a$$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $$\Gamma$$, for most polynomials a Bézout-type bound holds for the intersection $$\Gamma \cap Z(p)$$: for every $$0\leq k\leq n-2$$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$ 

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4. Rational Homotopy Type and Computability

   https://doi.org/10.1007/s10208-022-09582-8  

   Manin, Fedor (October 2023, Foundations of Computational Mathematics)

   Given a simplicial pair (X, A), a simplicial complex Y, and a map f:A -> Y, does f have an extension to X? We show that for a fixed Y, this question is algorithmically decidable for all X, A, and f if Y has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable. Conversely, for all other Y, the question is at least as hard as certain special cases of Hilbert's tenth problem which are known or suspected to be undecidable. 

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5. Witnessability of Undecidable Problems

   https://doi.org/10.1145/3571227  

   Ding, Shuo; Zhang, Qirun (January 2023, Proceedings of the ACM on Programming Languages)

   Many problems in programming language theory and formal methods are undecidable, so they cannot be solved precisely. Practical techniques for dealing with undecidable problems are often based on decidable approximations. Undecidability implies that those approximations are always imprecise. Typically, practitioners use heuristics andad hocreasoning to identify imprecision issues and improve approximations, but there is a lack of computability-theoretic foundations about whether those efforts can succeed. This paper shows a surprising interplay between undecidability and decidable approximations: there exists a class of undecidable problems, such that it is computable to transform any decidable approximation to a witness input demonstrating its imprecision. We call those undecidable problemswitnessable problems. For example, if a program propertyPis witnessable, then there exists a computable functionf_P, such thatf_Ptakes as input the code of any program analyzer targetingPand produces an input programwon which the program analyzer is imprecise. An even more surprising fact is that the class of witnessable problems includes almost all undecidable problems in programming language theory and formal methods. Specifically, we prove the diagonal halting problemKis witnessable, and the class of witnessable problems is closed under complements and many-one reductions. In particular, all “non-trivial semantic properties of programs” mentioned in Rice’s theorem are witnessable. We also explicitly construct a problem in the non-witnessable (and undecidable) class and show that both classes have cardinality 2^ℵ₀. Our results offer a new perspective on the understanding of undecidability: for witnessable problems, although it is impossible to solve them precisely, it is always possible to improve any decidable approximation to make it closer to the precise solution. This fact formally demonstrates that research efforts on such approximations are promising and shows there exist universal ways to identify precision issues of program analyzers, program verifiers, SMT solvers, etc., because their essences are decidable approximations of witnessable problems. 

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