I am a lecturer at the University of St Andrews in the School of Mathematics and Statistics. My research is currently supported by an EPSRC Open Fellowship.
I grew up in Cardiff-by-the-Sea, California, before getting my MA at UCLA and my PhD at UC Berkeley under Denis Auroux.
I study the relation between symplectic, algebraic, and tropical geometry through mirror symmetry. Outside of mathematics, my hobbies include piano, partner dancing, board games, and puzzles.
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Tropicalization is a process by which we replace algebraic geometry with the geometry of piecewise linear (tropical) objects. One of the central questions in the field is when this process can be reversed: that is, when can we realize a tropical object with an honest algebraic one. In this talk, I'll discuss some recent work on the tropical to Lagrangian correspondence, and state under what conditions homological mirror symmetry allows us to transfer Lagrangian realizations into algebraic ones.
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Mirror symmetry predicts an action by the fundamental group of a conjectural stringy Kähler moduli space on the derived category of an algebraic variety. For a toric variety, a model for this space is understood, but constructing the action is still an open problem in general. We propose that this action can be studied on the $A$-side via a moduli space of Legendrians isotopic to the FLTZ Legendrian. For the $A_{n-1}$ singularity, we construct an annular braid-group action on the corresponding partially wrapped Fukaya category by exact autoequivalences. The standard braid subgroup recovers the Seidel--Thomas action on the derived category, while the additional annular generator corresponds to tensor product with $\mathcal O(-1)$. We additionally extend the Floer-theoretic approach to homological mirror symmetry for toric varieties to the setting of semiprojective toric Deligne--Mumford stacks over an arbitrary field.
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For a Calabi-Yau manifold $X$, physics predicts that the Kähler moduli space should admit a complexification, the stringy Kähler moduli space (SKMS). While the complex structure of $X$ remains fixed over the SKMS, one expects the derived category of $X$ to assemble into a local system of categories over this space. Mirror symmetry suggests a parallel picture on the symplectic side: a local system of Fukaya categories arising from variation of complex structure on the mirror. One might ask: can we obtain this local system of categories without referencing the Calabi-Yau structure? In this talk, I will illustrate this in the simple example of the $A_n$ singularity and its mirror Landau-Ginzburg model and explain how a local system of Fukaya categories can be constructed over a purely symplectic moduli space, the moduli space of Legendrian stops. I will also show that the braid group embeds into the fundamental group of the moduli space of Legendrian stops. This is based on current joint work with Michela Barbieri and Andrew Hanlon.
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In 1758, Leonhard Euler made the following wonderful observation: if you count the vertices, edges, and faces of a polyhedron with alternating signs, you always get 2. Since then, topologists have been faithfully counting things with plus and minus signs to obtain new invariants. In this talk, we'll discuss how homology is an algebraic upgrade of "counting with signs," and how symplectic geometry can re-encode that data as intersections between curves on surfaces. Finally, we'll run the story in reverse to derive the existence of periodic orbits in Hamiltonian dynamics using the machinery (Floer theory) we developed along the way.
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Given a tropical curve in $\mathbb{R}^n$, the realizability problem asks whether it arises as the tropicalization of an honest algebraic curve in the corresponding algebraic torus. In this talk, I will report on current progress toward solving the symplectic-geometric version of this problem: the unobstructedness of Floer theory on the Lagrangian submanifold lift associated with the tropical curve.
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The polynomial ring in $k$ variables is one of the best understood rings. One of the fundamental results in the area is Hilbert's Syzygy theorem (1890), which in the modern formulation states that a finitely generated module over this ring admits a free resolution of length at most $k$. In the 1990s Bayer and Sturmfels observed that many of these free resolutions had "underlying structure" given by a topological space. From this intuition, they introduced a framework (cellular resolutions) that provided a method for constructing new resolutions of a module from the data of a topological space. Unsurprisingly, many properties of these modules were encoded in the topology of the associated space. If you, for some reason, think of the polynomial ring as the monoid ring of $\mathbb{N}^k$, then you might ask if there is a similar story for non-commutative monoids. A partial answer, which predates Bayer and Sturmfels, is in the foundational work of Eilenberg and Mac Lane (around 1950) on classifying spaces and group cohomology. In this talk, I will mostly give an exposition of Bayer-Sturmfels' work and its relationship to Eilenberg-Mac Lane spaces. Time permitting, I will discuss some work in progress with Lauren Cranton Heller, Mahrud Sayrafi, and Jay Yang on the general case of monoid rings and applications.
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We extend Groman and Solomon's reverse isoperimetric inequality to pseudoholomorphic curves with punctures at the boundary and whose boundary components lie in a collection of Lagrangian submanifolds with intersections locally modelled on $\mathbb{R}^n\cap (\mathbb{R}^{k}\times \sqrt{-1}\mathbb{R}^{n-k})$ inside $\mathbb{C}^n$. Our construction closely follows the methods used by Duval (2016) and Abouzaid (2021) and corrects an error appearing in the latter approach.
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Cellular resolutions give us a way to encode resolutions of modules over multigraded rings into the data of a topological space. These resolutions were first employed in the commutative algebra community in work of Bayer and Sturmfels, but in the last few years several generalizations--inspired by mirror symmetry--have arisen in the literature to address questions in algebraic geometry. A parallel and much older theme links topology to algebra via classifying spaces and group (co)homology: cellular chains on the Eilenberg-MacLane space of a group with appropriate local coefficients yield free resolutions over the group ring. In this talk, I'll discuss how to construct some "cellular resolutions" over noncommutative monoids that generalize both of these stories.
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An embedded tropical curve in $n$-dimensional Euclidean space is an embedded graph whose edges are straight line segments or rays with integral direction vectors that satisfy a "balancing condition" at each vertex (namely, the signed sum of the incident edge direction vectors is zero). These curves were introduced in algebraic geometry, where there is a "tropicalization" process that produces a tropical curve from an embedded algebraic curve. In this talk, we'll discuss how to associate a CAT(0) space with an embedded trivalent tropical curve. Time permitting, we'll discuss how this tropical object relates to the algebraic-geometric tropicalization process and mention connections to noncommutative algebra.
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We prove the `integrality of Taylor coefficients of mirror maps' conjecture for Greene--Plesser mirror pairs as a natural byproduct of an arithmetic refinement of homological mirror symmetry. We also prove homological mirror symmetry for Greene--Plesser mirror pairs in all characteristics such that the B-side family has good reduction, generalizing work of the fifth author and Smith over the complex numbers. A key technical ingredient is a new versality argument which allows us to work throughout over a Novikov-type ring with integer coefficients.
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There are several different measures of the complexity of a triangulated category. Perhaps the best studied of these is the Rouquier dimension of a triangulated category, which roughly measures how long it takes a set of objects to generate a category. For instance, if the category in consideration is the category of sheaves on a smooth affine variety, the Rouquier dimension agrees with the dimension of the affine variety (and is conjectured to agree with the dimension for all smooth varieties). In this talk, we will discuss some work (joint with Oleg Lazarev and Andrew Hanlon) understanding relations between notions of complexity of a topological space (dimension, LS category, or minimal Morse critical number) $M$ and measures of complexity of the triangulated category $W(M)$ --- the category of chains on the based loop space of $M$.
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We say that a tropical subvariety $V\subset \mathbb R^n$ is $B$-realizable if it can be lifted to an analytic subset of $(\Lambda^*)^n$. When $V$ is a smooth curve or hypersurface, there always exists a Lagrangian submanifold lift $L_V\subset (\mathbb C^*)^n$. We prove that whenever $L_V$ has well-defined Floer cohomology, we can find for each point of $V$ a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with $L_V$ is non-vanishing. Assuming an appropriate homological mirror symmetry result holds for toric varieties, it follows that whenever $L_V$ is a Lagrangian submanifold that can be made unobstructed by a bounding cochain, the tropical subvariety $V$ is $B$-realizable. As an application, we show that the Lagrangian lift of a genus zero tropical curve is unobstructed, thereby giving a purely symplectic argument for Nishinou and Siebert's proof that genus-zero tropical curves are $B$-realizable. We also prove that tropical curves inside tropical abelian surfaces are $B$-realizable.
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The Weinstein neighborhood theorem states that every Lagrangian submanifold $L$ of a symplectic manifold $X$ has a symplectic neighborhood modeled on the cotangent bundle of $L$. A consequence of this is that, in some nice cases, we have a Viterbo restriction functor from the Fukaya category of $X$ to the Fukaya category of $T^*L$. In this talk, I'll discuss some parallels between this restriction functor and the restriction to an affine neighborhood of a point in an algebraic variety, and use this to provide a definition of a "noncommutative affine neighborhood" of an object in a triangulated category. We'll then construct a few examples of these noncommutative affine neighborhoods and discuss applications. While we will use some intuition from symplectic geometry in this talk, we won't use any symplectic techniques in our examples or computations.
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We present an unobstructedness criterion for Lagrangian threefolds $L\subset X^A$ using the $H_1(L)$-class associated with the boundary of a pseudoholomorphic disk. As an application, let $X^A\to Q$ be a Lagrangian torus fibration whose base $Q$ is a tropical abelian threefold. Given $V\subset Q$ a rigid tropical curve with a pair-of-pants decomposition, we prove that the Lagrangian lift $L_V\subset X^A$ is unobstructed. Provided that an appropriate homological mirror symmetry statement holds, this implies the existence of a realization $Y_V$ in the mirror abelian threefold $X^B\to Q$.
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Given a dg category $\mathcal C$, we introduce a new class of objects (weakly product bimodules) in $\mathcal C^{op}\otimes \mathcal C$ generalizing product bimodules. We show that the minimal generation time of the diagonal by weakly product bimodules provides an upper bound for the Rouquier dimension of $\mathcal C$. As an application, we give a purely algebro-geometric proof of a result of Burban and Drozd that the Rouquier dimension of the derived category of coherent sheaves on an $n$-cycle of projective lines is one. Our approach explicitly gives the generator realizing the minimal generation time.
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A fundamental result in symplectic geometry is that every Lagrangian submanifold has a small standard neighborhood. The existence of this Weinstein neighborhood implies that there are no "local" invariants of a Lagrangian submanifold other than the topology of the Lagrangian submanifold. Mirror symmetry is a conjectured dictionary between symplectic and algebraic geometry on a pair of mirror spaces. Under this dictionary, Lagrangian submanifolds in a symplectic manifold are related to sheaves on the mirror algebraic space. In this talk, which will only use intuition, not techniques from symplectic geometry, we give a candidate definition of the "Weinstein neighborhood" of a sheaf on an algebraic space, which extends the definition of an affine neighborhood of a point. We'll prove the existence of such a neighborhood for the structure sheaf of a hyperplane in projective space. Time permitting, we'll draw connections to tropical geometry and spaces of nonpositive curvature.
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Given any toric subvariety $Y$ of a smooth toric variety $X$ of codimension $k$, we construct a length $k$ resolution of $\mathcal O_Y$ by line bundles on $X$. Furthermore, these line bundles can all be chosen to be direct summands of the pushforward of $\mathcal O_X$ under the map of toric Frobenius. The resolutions are built from a stratification of a real torus that was introduced by Bondal and plays a role in homological mirror symmetry. As a corollary, we obtain a virtual analogue of Hilbert's syzygy theorem for smooth projective toric varieties conjectured by Berkesch, Erman, and Smith. Additionally, we prove that the Rouquier dimension of the bounded derived category of coherent sheaves on a toric variety is equal to the dimension of the variety, settling a conjecture of Orlov for these examples. We also prove Bondal's claim that the pushforward of the structure sheaf under toric Frobenius generates the derived category of a smooth toric variety and formulate a refinement of Uehara's conjecture that this remains true for arbitrary line bundles.
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The Strominger-Yau-Zaslow conjecture proposes that "mirror pairs" (pairs of spaces whose symplectic and complex invariants are interchanged) can be constructed by considering dual Lagrangian torus fibrations. In this introductory talk, we will demonstrate how to build dual Lagrangian torus fibrations from the data of an affine manifold. I'll also discuss how to lift affine subspaces to complex and Lagrangian submanifolds and perform some sample computations that motivate Kontsevich's homological mirror symmetry conjecture. Finally, I'll state some theorems illustrating how symplectic geometry can be applied to algebraic geometry. \par To every trivalent tropical curve in an $n$-dimensional affine manifold with metric, we'll build an $n$-dimensional manifold with a locally CAT metric. We'll then switch gears and talk about a relationship between coherent sheaves on projective space and modules over the free group. At the end we'll tie both stories together, giving us an algebraic geometric interpretation of a Weinstein-neighborhood of a tropical Lagrangian via the mirror to Viterbo restriction. \par One version of the realizability problem asks, "Which tropical subvarieties can be lifted to algebraic subvarieties?" In his early work on the subject, Mikhalkin exhibited a superabundant tropical curve that could not be lifted to an algebraic variety. However, in 2014, Cheung, Fantini, Park, and Ulirsch showed that all trivalent tropical curves that are non-superabundant possess algebraic lifts. On the mirror side, every trivalent tropical curve can be lifted to a Lagrangian submanifold! However, these Lagrangian submanifolds may bound holomorphic disks (and therefore be unsuitable for homological mirror symmetry). It is expected that the realizable tropical subvarieties have "unobstructed" Lagrangian submanifold lifts--in the sense that the counts of holomorphic disks cancel out in homology. In this talk, we'll show that the non-superabundance condition implies the unobstructedness of the corresponding tropical Lagrangian lift in dimension three.
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Biran and Cornea showed that monotone Lagrangian cobordisms give an equivalence of objects in the Fukaya category. However, there are currently no known non-trivial examples of monotone Lagrangian cobordisms with two ends. We look at an extension of their theory to the pearly model of Lagrangian Floer cohomology and unobstructed Lagrangian cobordisms. In particular, we examine the suspension cobordism of a Hamiltonian isotopy and the Haug mutation cobordism between mutant Lagrangian surfaces. In both cases, we show that these Lagrangian cobordisms can be unobstructed by bounding cochain and additionally induce an $A_\infty$ homomorphism between the Floer cohomology of the ends. This gives a first example of a two-ended Lagrangian cobordism giving a non-trivial equivalence of Lagrangian Floer cohomology. A brief computation is also included which shows that the incorporation of bounding cochain from this equivalence accounts for the "instanton-corrections" considered by Auroux, Ekholm, Pascaleff, Rizell, and Tonkonog for the wall-crossing formula between Chekanov and product tori in $(\mathbb C^2)\setminus \{z_1z_2=1\}$. We additionally prove some auxiliary results that may be of independent interest. These include a weakly filtered version of the Whitehead theorem for $A_\infty$ algebras and an extension of Charest-Woodward's stabilizing divisor model of Lagrangian Floer cohomology to Lagrangian cobordisms.
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Given a tropical curve that admits a pair-of-pants decomposition in the base of a Lagrangian torus fibration, Mikhalkin constructed a Lagrangian submanifold whose projection under the Lagrangian fibration approximates the original tropical curve. Under appropriate conditions, this Lagrangian submanifold is mirror to the algebraic-geometric realization of the underlying tropical curve. In this talk, we will employ tools from geometric topology to demonstrate that these Lagrangian submanifolds admit a metric with non-positive curvature in dimension three. Additionally, I will explore the algebraic-geometric implications of this structure as viewed through the mirror. Time permitting, I will also discuss connections to non-superabundance and Floer-theoretic unobstructedness.
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We prove Kontsevich's homological mirror symmetry conjecture for a large class of mirror pairs of Calabi--Yau hypersurfaces in toric varieties. These mirror pairs were constructed by Batyrev from dual reflexive polytopes. The theorem holds in characteristic zero and in all but finitely many positive characteristics.
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One way to obtain a nice set of generators for the derived category of coherent sheaves is to exhibit a resolution of the diagonal bimodule by product objects. For example, the Beilinson resolution of the diagonal for projective space proves that the line bundles $\mathcal{O}$, $\mathcal{O}(1)$, \ldots, $\mathcal{O}(n)$ generate any coherent sheaf in time $n$. In the first part of the talk, I'll discuss a generalization of this approach. In the second part of the talk, we'll focus on the specific example of a cycle of projective lines. Using some intuition from mirror symmetry, we'll exhibit a set of generators for the derived category of this space with generation time one. As a corollary, we obtain a short proof of a result of Burban and Drozd that the Rouquier dimension of this category is one. There will be lots of pictures and we won't use any symplectic geometry to prove our results.
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There are several different notions of "complexity" for a topological space $M$. For instance, when $M$ is a manifold one can study: the Lebesgue covering dimension; sum of Betti numbers; minimal number of Morse critical values; or the Lusternik-Schnirelmann category. Similarly, given a triangulated category $C$, one can measure its complexity using invariants such as the Rouquier dimension; diagonal dimension; or minimal length of presentation as a homotopy colimit. In this talk, I will discuss some of the relations between these categorical invariants and topological invariants when the category $C$ is the Fukaya category of the cotangent bundle of $M$ (equivalently, the category of modules over chains on the based loop space on $M$). I will mostly focus on introducing the above invariants of topological spaces and categories and discussing how Lagrangian cobordisms play a role in bounding the diagonal dimension of $C$ in terms of the minimal number of critical values of a Morse function on $M$.
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In this talk, I will discuss some recent work constructing resolutions of a toric variety $X$. Given a toric subvariety $Y$ of codimension $k$, we will exhibit a resolution of the direct image of the structure sheaf of $Y$ of length $k$; furthermore every term in this resolution will be a direct sum of line bundles. The resolution is constructed by computing the homology of the (real!) torus with coefficients in the category of coherent sheaves on $X$. I will mostly work through a specific example (resolving the skyscraper sheaf of a point on the projective plane) and describe how the proof that this provides a resolution arises from invariance of Morse homology under variation of Morse function. All work is joint with Andrew Hanlon and Oleg Lazarev.
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Lagrangian $k$-surgery modifies an immersed Lagrangian submanifold by topological $k$-surgery while removing a self-intersection. Associated to a $k$-surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism is exactly homotopic to a concatenation of suspension cobordisms and Lagrangian surgery traces. This exact homotopy can be chosen with as small Hofer norm as desired. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated with the handle attachment to the Floer cochain generated by the self-intersection. We give a sample computation for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds. In an appendix, we describe a 2-ended embedded monotone Lagrangian cobordism which is not the suspension of a Hamiltonian isotopy following a suggestion of Abouzaid and Auroux.
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We study several notions of dimension for (pre-)triangulated categories naturally arising from topology and symplectic geometry. We prove new bounds on these dimensions and raise several questions for further investigation. For instance, we relate the Rouquier dimension of the wrapped Fukaya category of either the cotangent bundle of a smooth manifold $M$ or more generally a Weinstein domain $X$ to quantities of geometric interest. These quantities include the minimum number of critical values of a Morse function on $M$, the Lusternik-Schnirelmann category of $M$, the number of distinct action values of a Hamiltonian diffeomorphism of $X$, and the smallest $n$ such that $X$ admits a Weinstein embedding into $\mathbb{R}^{2n+1}$. Along the way, we introduce a notion of the Lusternik-Schnirelmann category for dg-categories and construct exact Lagrangian cobordisms for restriction to a Liouville subdomain.
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Given a toric subvariety $Y$ of a smooth toric variety $X$, we construct a resolution of $\mathcal{O}_Y$ by line bundles on $X$. The resolution has length the codimension of $Y$, and the line bundles are chosen from a finite collection of line bundles on $X$ which are direct summands of the pushforward of $\mathcal{O}_X$ under the map of toric Frobenius. As a consequence, we show that the Rouquier dimension of the derived category of a toric variety is equal to its Krull dimension. Joint work with Andrew Hanlon and Oleg Lazarev.
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A Lagrangian cobordism, as introduced by Arnold, is a Lagrangian submanifold with a notion of "ends" in the stabilization of a symplectic manifold $X$. The ends are Lagrangian submanifolds of $X$. The Floer theory of Lagrangian cobordisms was first studied by Biran and Cornea, who showed that the ends of a monotone Lagrangian cobordism are related by an iterated mapping cone in the Fukaya category. Many known HMS correspondences rely on matching generators of categories. Provided we can meet the monotonicity requirement, Lagrangian cobordisms allow us to bootstrap to more general objects of the Fukaya category by exhibiting a Lagrangian cobordism from the generating Lagrangians to the desired object. In this talk, I will discuss a few applications in HMS where this monotonicity requirement is met, what one can say beyond the monotone setting (unobstructedness), and how the geometry of Lagrangian cobordisms determines unobstructedness.
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The realizability problem asks if a given tropical subvariety is the tropicalization of some algebraic subvariety. Realizability is already an interesting question for curves in $\mathbb{R}^3$, where Mikhalkin exhibited a tropical curve of genus 1 which is non-realizable. In recent independent work, Mak-Ruddat, Matessi, Mikhalkin, and I show that for many examples of tropical subvarieties in $\mathbb{R}^n$ there exists a Lagrangian lift. This is a Lagrangian submanifold of $(\mathbb{C}^*)^n$ whose image under the moment map approximates a given tropical subvariety. In particular, every smooth tropical curve in $\mathbb{R}^n$ can be lifted to a Lagrangian submanifold (in contrast to the algebraic setting!). In this talk, I'll discuss what it means to be a Lagrangian lift of a tropical curve. We will then look at what symplectic conditions on the resulting Lagrangian detect realizability of the underlying tropical curve. As an application, we will prove that every tropical curve in a tropical abelian surface has a rigid-analytic realization.
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We provide some constructions using Lagrangian cobordisms which improve known examples for some symplectic squeezing problems. Additionally, we prove a flexibility result that Lagrangian submanifolds which are Lagrangian isotopic are also Lagrangian cobordant.
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We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we develop techniques for constructing Lagrangian cobordisms and Lagrangian correspondences in Liouville domains, which may be of independent interest.
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The tropicalization map associates to each curve in the algebraic $n$-torus a piecewise linear object (tropical curve) in real $n$-dimensional space. Given a tropical curve, a natural question is if it can arise as the tropicalization of some algebraic curve. If this is the case we say that the tropical curve is realizable. Determining good realizability criteria for tropical curves remains an important part of tropical geometry since Mikhalkin provided examples of non-realizable tropical curves. We explore the following strategy for realizing tropical curves: (1) Produce a Lagrangian submanifold of the cotangent bundle of the torus whose moment map projection approximates the tropical curve; (2) Use homological mirror symmetry to obtain a mirror algebraic sheaf; (3) Show that the tropicalization of the support of this sheaf is the original tropical curve. We will give full answers to (1) and (3), and explain why (2) is fairly subtle. As applications, we will obtain some new and known realizability statements for tropical curves.
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Given a tropical hypersurface in $n$-dimensional real space, there exists a Lagrangian submanifold in the algebraic $n$-torus whose projection under the log map approximates the given tropical hypersurface. We call this the Lagrangian realization of the tropical hypersurface. The support of a Lagrangian submanifold $L$ is the set of Lagrangian torus fibers whose Lagrangian intersection Floer cohomology with $L$ is non-vanishing. Under mirror symmetry, this corresponds to the support of the sheaf which is mirror to $L$. We discuss how to construct some Lagrangian realizations of tropical hypersurfaces, and when those realizations are exact we will compute their support. Time permitting, we'll make some connections with dimers and mutations.
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Biran and Cornea proved that when $K$ is a monotone Lagrangian cobordism with Lagrangian ends, the Floer cohomology of $K$ is a mapping cocylinder yielding an equivalence between the Floer cohomology of the ends. In this talk, we will discuss how to decompose Lagrangian cobordisms with Lagrangian ends into Lagrangian surgery handles. We then speculate how one would use such a decomposition to provide criteria for when the Lagrangian $K$ is an unobstructed object (in the sense of admitting a bounding cochain). Time permitting, we'll discuss how the same techniques might recover Biran and Cornea's result for unobstructed Lagrangian cobordisms with Lagrangian ends.
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Given an algebraic curve in the $n$-dimensional algebraic torus, there exists a piecewise linear object in real $n$-dimensional space (called its tropicalization) which is supposed to capture the "leading-order" behavior of the curve. One of the goals of tropical geometry is to understand when this process can be reversed; that is, when does a tropical curve "lift" to an honest algebraic one? In this talk, I will say something about the corresponding problem in symplectic geometry--when a tropical curve "lifts" to a Lagrangian submanifold--and how these two problems are related through the mirror correspondence.
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In this survey, we look at some expectations for Lagrangian submanifolds which are built as the lifts of tropical curves from the base of a Lagrangian torus fibration. In particular, we perform a first computation showing that holomorphic triangles can appear with boundary on the Lagrangian submanifold. We speculate how these holomorphic triangles can contribute to the count of holomorphic strips in the Lagrangian intersection Floer cohomology between a tropical Lagrangian submanifold and a fiber of the SYZ fibration.
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We look at how one can construct from the data of a dimer model a Lagrangian submanifold in $(\mathbb{C}^*)^n$ whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori $L_{T^2}$ in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair $(\mathbb{CP}^2\setminus E, W), \check X_{9111}$. We find a symplectomorphism of $\mathbb{CP}^2\setminus E$ interchanging $L_{T^2}$ and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier-Mukai transform on $\check X_{9111}$.
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We produce for each tropical hypersurface $V(\phi)\subset Q=\mathbb{R}^n$ a Lagrangian $L(\phi)\subset (\mathbb{C}^*)^n$ whose moment map projection is a tropical amoeba of $V(\phi)$. When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are unobstructed objects of the Fukaya category, and mirror to sheaves supported on complex hypersurfaces in a toric mirror.
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We produce for each tropical hypersurface $V(\phi)\subset Q = \mathbb R^n$ a Lagrangian $L(\phi)\subset (\mathbb C^*)^n$ whose moment map projection is a tropical amoeba of $V(\phi)$. When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are mirror to hypersurfaces in a toric mirror. These constructions are extended to tropical varieties given by locally planar intersections, and to symplectic 4-manifolds with almost toric fibrations. We also explore relations to wall-crossing, dimer models, and Lefschetz fibrations. A complete example is worked out for the mirror pair $(\mathbb{CP}^2\setminus E, W), X_{9111}$.
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