Jack Smith

I am a College Lecturer, and the Director of Studies in Pure Mathematics, at St John's College, Cambridge. I am also an Affiliated Lecturer in the Department of Pure Mathematics and Mathematical Statistics.

Previously I was a visiting researcher at the Fields Institute, Toronto, a postdoc at University College London, and a PhD student of Ivan Smith.

Here you can find my thesis, me on arXiv, and my department profile page.

I can be contacted by email at j.smith AT dpmms.cam.ac.uk.

Research

My main interests are in mirror symmetry and in interactions between algebra and Floer theory.

Research papers:

• Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving
  arXiv, preprint
• Hochschild cohomology of the Fukaya category via Floer cohomology with coefficients
  arXiv, preprint
• Homological Lagrangian monodromy for some monotone tori
  Joint with Marcin Augustynowicz and Jakub Wornbard
  arXiv, preprint
• Superfiltered \(A_\infty\)-deformations of the exterior algebra, and local mirror symmetry
  J. London Math. Soc. Volume 105 (2022), 2203-2248
  arXiv
  Corrections
  • In Section 5.1, the expressions given for the differential \(\mathrm{d}\) and cup product \(\smile\) on the Hochschild cochain complex are really expressions for the \(\mu^1\) and \(\mu^2\) operations. In order to call them the differential and cup product (for example, in order for the product to be associative after passing to cohomology), they should be twisted by the usual signs as given in (3).
  • In the second paragraph of the proof of Lemma 6.10, the \(\gamma_-\) should be \(\gamma_+\).
• A monotone Lagrangian casebook
  Algebr. Geom. Topol. 21 (2021), no. 5, 2273-2312
  arXiv
• Homological Berglund–Hübsch mirror symmetry for curve singularities
  Joint with Matthew Habermann
  J. Symplectic Geom. Volume 18 (2020), no. 6, 1515-1574
  arXiv
  Corrections (thanks to Matt for these)
  • Just after Lemma 2.7, the morphisms from \(K_w\) to \(K_x(l)\) and \(K_y(l)\) should be in \(L\)-degree \((m+1)\vec{c}+l-\vec{x}-\vec{y}\), rather than \((m+1)\vec{c}+l-\vec{y}\) and \((m+1)\vec{c}+l-\vec{x}\) respectively.
  • In the proof of Proposition 2.14, just before the paragraph beginning "So far...", the range should be \(2 \leq j \leq q-1\) instead of \(2 \leq j \leq p-1\).
  • In the proof of Lemma 4.7, in the paragraph beginning "We have therefore...", it should say \((a,b) = (1,1), \dots, (p-1,1)\) instead of \((a,b) = (1,1), \dots, (1,p-1)\).
  • At the beginning of Section 6.1, it is \(L/\mathbb{Z}\vec{c}\) that's isomorphic to \(\mathbb{Z}/p \oplus \mathbb{Z}/q\), not \(L\) itself. \(L\) is an extension of \(\mathbb{Z}\) by \(\mathbb{Z}/d\), where \(d = \gcd(p, q)\).
  • Just above Remark 6.2, the Lagrangian \(V_{\check{x}\check{y}}\) should be shifted by \([-1]\) when the Lagrangians are reordered, so that the morphisms remain in degree zero.
  • Potentially helpful clarification: all of our matrix factorisation morphism spaces \(\hom^\bullet(E, F)\) are computed and described by treating \(E\) as a complex and \(F\) as a module. This means that each morphism we write down only gives half of the corresponding chain map (either the odd or even terms) that would be obtained by viewing both \(E\) and \(F\) as complexes. At times one may need to translate between the two halves, especially if \(F\) receives an odd degree shift.
• Monotone Lagrangians in \(\mathbb{CP}^n\) of minimal Maslov number \(n+1\)
  Joint with Momchil Konstantinov
  Math. Proc. Cambridge Philos. Soc. Volume 171, Issue 1, July 2021, 1–21
  arXiv
• Quantum cohomology and closed-string mirror symmetry for toric varieties
  Q. J. Math. Volume 71, Issue 2, June 2020, 395–438
  Accepted version (incorporating the corrections below)
  arXiv
  Corrections
  • In Corollary 1.3/Corollary 2 (depending on whether you look in the published version or in the arXiv or accepted version) the map should be a \(\Lambda_0\)-algebra homomorphism, not just a \(\Lambda_0\)-module homomorphism. This ensures that the situation reduces to the classical Stanley–Reisner presentation modulo \(\Lambda_+\).
  • In Corollaries 4.17 and 4.18/Corollaries 4.16 and 4.17 (published version/arXiv or accepted version) the rings \(\Lambda_0\) and \(R[T]\) should be \(\Lambda\) and \(R[T^{\pm 1}]\) respectively. It's explained in the surrounding text that you need to invert the Novikov variable \(T\) to make the toric divisors invertible, but this was missed from the statements of the Corollaries.
• Discrete and continuous symmetries in monotone Floer theory
  Selecta Math. (N.S.) 26 (2020), no. 3, Paper No. 47, 65 pp.
  arXiv
• Floer cohomology of Platonic Lagrangians
  J. Symplectic Geom. Volume 17 (2019), no. 2, 477–601
  arXiv (including Mathematica notebook)

Talk slides and videos:

• From Floer to Hochschild via matrix factorisations slides/video
• Homological Lagrangian monodromy for monotone tori slides
• Towards Berglund–Hübsch mirror symmetry slides/video
• \(\mathbb{RP}^n\)-like Lagrangians in \(\mathbb{CP}^n\) slides
• Exterior algebras and local mirror symmetry slides

Teaching

• Part IB Geometry, Cambridge, Lent 2024
• Part III Differential Geometry, Cambridge, Michaelmas 2020, 2021, and 2022.
• Graduate course on K-theory, London, June 2018. Expanded notes here.