Papers

We investigate the filtration structure of the skein algebra induced by the (quantum) trace map. Utilizing the structure, we prove that relative $\operatorname{SL}(3, \mathbb{C})$ character varieties of punctured surfaces admit log Calabi-Yau compactifications and prove the (weak) geometric P=W conjecture.

We leverage Polishchuk's Poisson blow-up criterion in the context of algebro-geometric integrable systems. In more detail, one may associate an integrable system $\tau:\mathfrak{X}\longrightarrow\mathfrak{B}$ to each affine Poisson scheme $\mathfrak{X}$ over $\mathbb{C}$. We prove that the blow-ups of $\mathfrak{X}$ along fibers of $\tau$ are Poisson schemes occurring in a family $\widetilde{\mathfrak{X}\times\mathfrak{B}}\longrightarrow\mathfrak{B}$, where $\widetilde{\mathfrak{X}\times\mathfrak{B}}$ is itself a Poisson scheme. This result is subsequently specialized to the adjoint quotient $\tau:\mathfrak{g}\longrightarrow\mathfrak{g}/\!/G=:\mathfrak{c}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with integrating algebraic group $G$. We show that the family $\widetilde{\mathfrak{g}\times\mathfrak{c}}\longrightarrow\mathfrak{c}$ is flat, conical, and equipped with a canonical Poisson Hamiltonian $G$-variety structure. We also obtain Poisson-geometric results on the fibers of this family, which are blow-ups of $\mathfrak{g}$ along regular adjoint orbit closures.

We prove the transcendental Morse inequality conjecture of Boucksom-Demailly-Păun-Peternell. Among the corollaries of this result, this proves orthogonality of divisorial Zariski decompositions, shows that the pseudoeffective and movable cones are dual, and establishes the differentiability of the volume function on the big cone, on all compact Kähler manifolds.

The tropical Schottky problem asks which principally polarized tropical abelian varieties are isomorphic to tropical Jacobian varieties equipped with their canonical principal polarizations. We prove the tropical Matsusaka criterion and the tropical Matsusaka–Ran criterion, which are the main results of this paper, and show that their hypotheses are equivalent. Together with the tropical Poincaré formula proved by Gross–Shokrieh, these results give a geometric solution to the tropical Schottky problem. In particular, we give an affirmative answer to a question posed by Brannetti–Melo–Viviani. Several tools are developed for the proofs of the two criteria. In particular, we establish Hodge-type inequalities for ample tropical Cartier divisors on tropical abelian varieties and prove a splitting criterion for principally polarized tropical abelian varieties.

Moustrou, Riener, and Verdure related the dominance order on partitions to Specht ideal and variety inclusion for $S_n$; we previously extended this to direct products of symmetric groups. Here we take a first step toward the imprimitive complex reflection group $G(r,1,n)=(\mathbb{Z}/r\mathbb{Z})\wr S_n$. The correct geometric invariant of a point $x\in K^n$ is a pair $(\mu,z)$: a classical partition $\mu$ on the $r$-th powers of the nonzero coordinates, together with the number $z$ of zero coordinates. We construct a $G(r,1,n)$-invariant ideal $I_{(\mu,z)}$ for each such pair, and show that a dominance order on $(\mu,z)$, with $z$ taking strict priority over $\mu$, exactly characterizes both ideal inclusion and reverse variety inclusion. We settle a monomial criterion in three regimes, recover the classical $B_n$-Specht ideal theory at $r=2$ when $z=0$, and establish radicality of $I_{(\mu,z)}$ at the two extremes of the $z=0$ level. Four computational examples, checked exhaustively for group invariance, illustrate the construction throughout.

The Learning With Errors (LWE) problem, introduced by Regev in 2005, is central to modern cryptography and post-quantum security. The algorithms to solve the search version of the problem, Search-LWE, can be broadly categorised into algebraic, combinatorial and lattice-based. In this work we propose a new algebraic algorithm for the Search-LWE problem. At a high level, the algorithm combines linear-algebraic techniques with S-polynomial-based methods from Groebner basis computation. We provide a direct complexity analysis of our algorithm, avoiding semi-regularity assumptions and complexity bounds derived from the degree of regularity. Our algorithm achieves a polynomial improvement in complexity over prior results that use Groebner basis methods to solve Search-LWE.

Let $k$ be an algebraically closed field and let $\Lambda$ be a finite-dimensional associative $k$-algebra. We apply Joyce and Brideland's notion of Ringel–Hall algebra to prove results about picture spaces and picture groups. For instance, we construct a faithful group functor for $\Lambda$, i.e. a faithful functor from the $\tau$-cluster morphism category of $\Lambda$ into a groupoid. Consequently, by results of Hanson–Igusa, the picture space of $\Lambda$ is locally CAT(0) provided that the $\tau$-cluster morphism category of $\Lambda$ admits compatibility of last factors. We also show that green sequences are in bijection with certain positive expressions in the picture group, which generalizes a result of Igusa–Todorov beyond hereditary $k$-algebras.

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain whose boundary is a finite union of pairwise disjoint Jordan curves. We prove the planar Khavinson–Shapiro conjecture in this setting: if every polynomial on $\mathbb{R}^2$ agrees on $\partial\Omega$ with a harmonic polynomial, then $\partial\Omega$ is an ellipse and $\Omega$ is its bounded interior.

Let $\mathcal{F}$ be a canonical foliation of general type on a smooth projective surface $X$, and write $\mathrm{vol}(\mathcal{F}):=\mathrm{vol}(K_{\mathcal{F}})$. Let $G\subseteq\operatorname{Bir}(X,\mathcal{F})$ be a finite subgroup, and let $\mathcal{G}:=\mathcal{F}/G$ be the quotient foliation in the birational sense. For a canonical foliation $\mathcal{H}$, define its $r$-th pluricanonical index by \[ \delta_r(\mathcal{H}) := \min \bigl\{ m\in\mathbb{Z}_{>0} \mid h^0(mK_{\mathcal{H}})\geq r \bigr\}, \] where $\min\varnothing:=\infty$. For an arbitrary foliation, these indices are computed on any canonical birational model. If $\kappa(\mathcal{G})\geq0$, we prove \[ |G| \leq \begin{cases} 4\delta_1(\mathcal{G})\,\mathrm{vol}(\mathcal{F}), &\kappa(\mathcal{G})=0,\\[1mm] \displaystyle \frac{4}{3}\delta_2(\mathcal{G})\,\mathrm{vol}(\mathcal{F}), &\kappa(\mathcal{G})=1,\\[3mm] \delta_2(\mathcal{G})^2 \bigl(1+\delta_2(\mathcal{G})\bigr)\, \mathrm{vol}(\mathcal{F}), &\kappa(\mathcal{G})=2. \end{cases} \] Since $\operatorname{Bir}(X,\mathcal{F})$ is finite, one may in particular take $G=\operatorname{Bir}(X,\mathcal{F})$. When $\kappa(\mathcal{G})=0$ or $1$, the effective bounds $\delta_1(\mathcal{G})\leq12$ and $\delta_2(\mathcal{G})\leq42$ give \[ |G|\leq48\,\mathrm{vol}(\mathcal{F}) \qquad\text{and}\qquad |G|\leq56\,\mathrm{vol}(\mathcal{F}), \] respectively. The main new ingredient is a cluster formula for adjoint volumes, which yields index-dependent lower bounds for tangency-free foliated surface pairs whose underlying foliation has Kodaira dimension zero or one.

The Virasoro conjecture predicts a set of universal relations among all genera Gromov–Witten invariants of any smooth projective variety. The conjecture is well understood for semisimple theories, but remains largely open in the non-semisimple setting. We prove the genus-one Virasoro conjecture on the ambient state space of smooth Fano complete intersections in projective space. For most of these complete intersections, the big quantum cohomology is nowhere semisimple. We also generalize the wall-crossing formula for quasimap invariants with weighted markings to the equivariant twisted setting, allowing descendant insertions at light markings. Together with genus-one quantum Lefschetz for quasimaps with light markings, this wall-crossing formula provides the key bridge from the Gromov–Witten theory of the complete intersection to the semisimple equivariant twisted theory of the projective space.

In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit a $3$-isogeny. Our methods are Galois-theoretic, and lead us to a thorough analysis of the Galois closure of $f:\mathbb{P}^1\to\mathbb{P}^1$. We exploit its rich geometry to prove a Selmer companionship theorem for the family $E_f$, allowing us to exhibit elements in certain Tate-Shafarevich groups which are visible in an abelian surface. We also give some dynamical and Diophantine applications of our constructions, as well as new examples of Jacobians isogenous to a power of an elliptic curve.

We establish an analogue of André Weil's classical theorem for irreducible nodal curves. Let \(X_0\) be an irreducible projective nodal curve. We prove that an indecomposable vector bundle or torsion-free coherent sheaf \(E\) on \(X_0\) admits a holomorphic logarithmic connection \(\nabla\colon E\to E\otimes\omega_{X_0}\) with respect to the dualizing sheaf if and only if \(\deg E=0\). Moreover, when \(\deg E=0\), such a connection can be chosen so that the induced logarithmic connection on the normalization has scalar residues \(\lambda\cdot I\) at one preimage of the node and \(-\lambda\cdot I\) at the other, for some \(\lambda\in\mathbb{C}\). Explicit one-parameter families of flat connections are constructed on the irreducible rational nodal cubic curve as an illustration.

We develop the infinitesimal deformation theory of generalized parabolic Hitchin pairs (GPHs) on irreducible nodal curves. For any GPH $(E,\phi,F(E))$ we associate an explicit three-term deformation complex $\mathcal{C}_{(E_\bullet,\phi)}$ whose first hypercohomology $\mathbb{H}^1(\mathcal{C}_{(E_\bullet,\phi)})$ parameterizes first-order deformations. As the main application, we prove that the coarse moduli space $\mathcal{M}_{\mathrm{GPH}}$ of $1$-stable generalized parabolic Hitchin pairs of rank $n$ and degree $d$ with $\gcd(n,d)=1$ is a local complete intersection, smooth in codimension one, and hence a normal variety. We further show that this moduli space carries a natural Poisson structure.

We completely characterize the $n$-fold connected sums of $S^3\times S^4$ that admit Sasaki-Einstein structures by proving that such a structure exists if and only if $n$ is even.

We study generalized Hamming weights along the one-point code flag of an AJ-Gorenstein curve. We organize these weights in a graded array, the zero diagram, whose entries are generalized coweights: the largest numbers of evaluation points on which subcodes of prescribed dimensions vanish simultaneously. Twisted and Wei duality show that each row of the zero diagram determines both the generalized Hamming weights of a lower block of short codes and the missing weights of a reflected upper block of long codes in the GHW diagram of the complete flag. Our main quantitative result is a uniform coverage theorem. For an AJ-Gorenstein curve of genus $g$ and evaluation length $n>2g$, the proportion of generalized-weight positions determined exactly throughout the complete flag satisfies $\operatorname{Cov}_{\mathrm{full}} \ge \frac{n(n-1)+4g}{n(n+2g-1)}>\frac12$. Thus more than half of all generalized-weight positions in the complete flag are determined uniformly. For the smallest Suzuki curve, the general and Castle-specific mechanisms together determine $2{,}280$ of the $2{,}912$ positions, giving an exact coverage of $78.30\%$.

In this paper, we construct the $(\infty,\infty)$-category $\mathsf{Span}_\infty(\mathcal{C})$ of spans, also known as correspondences, in any given $(\infty,1)$-category $\mathcal{C}$ with finite limits. This yields new models for the span $(\infty,n)$-categories for $n \in \mathbb{N} \cup \{\infty\}$. We characterize the mapping $(\infty, n-1)$-categories in these $(\infty,n)$-categories, and thereby verify that our model agrees with other models for spans. Finally, and most importantly, we prove a new universal property, characterizing functors into span $(\infty,n)$-categories, which specializes to the well-known relation with the twisted arrow categories in dimension $1$. These results will be used in the sequels to construct higher analogs of the classical Hall algebra construction, where "higher" refers to both higher categorical and "higher monoidal" structures, i.e., $\mathsf{E}_k$-algebras in $(\infty, n)$-categories for $n, k>1$.

Let $S$ be a smooth complex projective surface endowed with a fibration $f \colon S \to C$ onto a smooth projective curve $C$. We prove that, if $S$ is a Mori dream space (or, more generally, if its pseudo-effective cone is polyhedral) then the Picard number $\rho(S)$ can be effectively computed by counting the irreducible components of the reducible fibres of $f$. A first simple consequence is that, given an elliptic fibration $f \colon S \to \mathbb{P}^1$ with a section and such that $S$ is a Mori dream space, the Mordell-Weil group of the general fibre of $f$ is finite. The main application is a simple criterion for proving that a surface fibred over a curve is not a Mori dream space. We show that certain Horikawa surfaces, Fermat surfaces in $\mathbb{P}^{3}$ of every degree $\ge 4$, particular product-quotient surfaces, and the minimal simply connected numerical Godeaux surface constructed by Craighero and Gattazzo are not Mori dream spaces.

The boundary of a plane amoeba is always contained in its contour, and equality is a characteristic feature of simple Harnack curves. We show that the converse fails, even under strong smoothness and nondegeneracy assumptions. For every two-dimensional lattice polygon, except unimodular triangles, we construct a smooth Newton-nondegenerate curve with smooth logarithmic critical locus and smooth embedded contour satisfying $\mathcal C(\mathscr A_f)=\partial\mathscr A_f$, although the curve is not Harnack. We also provide explicit primitive and nonprimitive families that are not torus-equivalent to simple Harnack curves. A concrete primitive example is certified by exact elimination and Sturm root counting. These results disprove contour–boundary rigidity and show that the contour as a set does not detect the real structure or the multiplicity of coincident critical sheets, thereby refining the compensation problem proposed by Lang, Shapiro, and Shustin.

The contour of an amoeba records the critical values of the logarithmic map and forms a fundamental interface between complex algebraic geometry and real-analytic geometry. Building on the known semianalyticity of amoeba contours, we prove the stronger statement that the componentwise exponential image of the contour of any algebraic hypersurface in $(\mathbb C^*)^n$ is semialgebraic. For a smooth hypersurface, we establish a natural criterion guaranteeing that the contour is a closed real-analytic hypersurface: the logarithmic Gauss map is transverse to $\mathbb RP^{n-1}$ and the logarithmic map has maximal rank along its critical locus. We then show that neither hypothesis is necessary, since singular or ramified critical parametrizations may still have complete real-analytic images. Finally, we derive intrinsic restrictions imposed by analyticity. In particular, every irreducible contour component of dimension $n-1$ is generated, over a dense open subset of its regular locus, by a real-analytic critical stratum on which the logarithmic map has maximal rank. These results distinguish the analytic geometry of the contour from the singularities of its critical parametrization and answer the question of Lang, Shapiro, and Shustin concerning effective sufficient conditions for analyticity.

We compute the arithmetic rank of nullcone ideals arising from natural actions of the special linear, orthogonal, and symplectic groups on direct sums of copies of their standard and dual representations. Over an infinite field of characteristic different from two, we show that the arithmetic rank equals the Krull dimension of the invariant ring, and compute this dimension. In positive characteristic, the required lower bounds are not detected by local cohomology, and are obtained using étale cohomology.

This article presents a systematic study of the critical cohomology of $\QM^{\xi}(\bbP^1, X)$, the moduli space of based quasimaps from $\bbP^1$ to a Nakajima quiver variety $X$. We develop two derived equivalent presentations of this moduli space as a global critical locus. Accordingly, we obtain two equivalent explicit realizations of the canonical DT sheaf of the moduli space. Using each critical presentation, we describe canonical actions of shifted Yangians and quantized Coulomb branches on the quasimap critical cohomologies. We produce, by geometric means, a canonical morphism from an appropriate Hall algebra to the Coulomb branch, and show that the two actions we describe are compatible via this morphism under an additional hypothesis. We conjecture the compatibility holds generally, and reduce the claim to a sheaf-theoretic statement that may be approached via Joyce conjecture.

We establish a Grothendieck six-functor formalism for Kummer étale cohomology including Poincaré duality for every separated vertical exact log smooth morphism of noetherian fs log schemes $f\colon X\rightarrow S$ when the coefficient ring $\Lambda$ is killed by an integer invertible on $S$. This is done via log étale rigidity \[\mathrm{D}_{\mathrm{l\acute{e}t}}(S,\Lambda)\simeq \mathrm{DA}_{\mathrm{l\acute{e}t}}(S,\Lambda).\] To achieve this, we also prove that Kummer étale cohomology satisfies $\mathbb{A}^1$-invariance, invariance under virtual isomorphisms, log cdh-descent, and invariance under verticalization.

We first give a complete classification of bi-affine representations of mapping class groups of surfaces with finitely many boundary components or punctures. We also show that every linear representation of the mapping class group of a genus-$g$ surface with two boundary components of dimension at most $3g-3$ is bi-affine. We then classify low-dimensional symplectic representations of the mapping class group associated to triple covers. Let $[\beta]\in H_1(S_g;\mathbb{Z}/3\mathbb{Z})^*$, and let $\widetilde{S}\to S_g$ be the corresponding triple cover with deck transformation $\sigma$. For $h\le g$, every non-abelian homomorphism from either $\mathrm{Mod}(S_g,[\beta])$, the stabilizer of $[\beta]$ in $\mathrm{Mod}(S_g)$, or $\mathrm{Mod}(\widetilde{S},\sigma)$, the centralizer of $\sigma$ in $\mathrm{Mod}(\widetilde{S})$, to $\mathrm{Sp}_{2h}(\mathbb{Z})$ is, up to conjugation, the standard symplectic representation on $H_1(S_g;\mathbb{Z})$. As an application, we obtain a rigidity theorem for holomorphic maps from the moduli space $R_g^{(3)}$ of genus-$g$ curves equipped with a $3$-sheeted (unbranched) normal covering to the moduli space $\mathcal{A}_h$ of $h$-dimensional principally polarized abelian varieties. We prove that, for $g\ge 6$ and $h\le g$, the unique nonconstant holomorphic map from $R_g^{(3)}$, equipped with either of its two natural complex-orbifold structures, to $\mathcal{A}_h$ is the period map sending a cover $Y\to X$ to the Jacobian of the base curve $X$.

We solve two open problems on ample vector bundles posed by Hartshorne in 1966. We prove that tensor products of ample vector bundles on schemes of finite type over an algebraically closed field are ample in arbitrary characteristic, extending Hartshorne's characteristic-zero result and Barton's projective positive-characteristic theorem. More generally, let f: X -> S be a morphism of schemes. Then we prove that tensor products of f-ample vector bundles are f-ample. Moreover, if E is an f-ample vector bundle of rank r>0 and W is a finite locally free polynomial GL(r,S)-module of positive rank with W_0 = 0, then E(W) is f-ample. In particular, Gamma^n E is f-ample for every n>0. Finally, adapting a construction of Ejiri-Fujino-Iwai, we show that in every characteristic a smooth quasi-projective surface carries a non-ample extension of an ample line bundle by itself.

In this paper we give a geometric description of the behavior of analytic intertwining operators between parabolically induced representations of $\mathrm{GL}_n(\mathrm{F})$, where $\mathrm{F}$ is a local non-archimedean field, in terms of hyperbolic localization functors of Braden. As a consequence, we can show that the image of an intertwining operator is always semi-simple and give a lower bound on the order of its pole, which is conjectured to be an equality as well as an upper bound in terms of the singular support of certain perverse sheaves. Moreover, we are able to reduce the conjecture of Lapid and Mínguez on the shape of irreducible subrepresentations of induced representations to a computation of characteristic cycles. The theory of mixed Hodge modules plays a crucial role in the proofs.

The flat DHS connection $\mathcal J_{\mathrm{DHS}}$ constructed in arXiv:2602.01461 is smooth on the configuration space of $n$ points on a fixed compact Riemann surface $\Sigma$ of arbitrary genus $h$, takes values in an infinite-dimensional Lie algebra $\hat{\mathfrak t}_{h,n}$ and is invariant under the modular group $\mathrm{Sp}(2h,\mathbb Z)$. This paper is the first in a series for a program whose goal is to extend the connection $\mathcal J_{\mathrm{DHS}}$ to a global flat connection on the Teichmüller space $\mathcal T_{h,n}$ valued in the Lie algebra of derivations of $\hat{\mathfrak t}_{h,n}$. Upon the choice of local coordinates adapted to the map $\mathcal T_{h,n}\to \mathcal T_h$, such a connection splits into three pieces: $\mathcal J_{\mathrm{DHS}}$, a piece $\mathcal L$ corresponding to holomorphic directions of $\mathcal T_h$ and a third piece corresponding to anti-holomorphic directions in $\mathcal T_h$. In this paper, we isolate the system of equations satisfied by $\mathcal L$ and obtain its solution locally and explicitly. The construction of a global extension of $\mathcal J_{\mathrm{DHS}}$ to $\mathcal T_{h,n}$ and the extension of the meromorphic connection of arXiv:1112.0864 to $\mathcal T_{h,n}$, are relegated to future publications in this series.

We prove an all genus full descendant open mirror symmetry for footballs. The B-model is given by the Chekhov-Eynard-Orantin topological recursion on the mirror curve.

We consider varieties whose singularities are complete intersections within toric singularities, which we call well-formed varieties. Using the construction of equivariant standard resolutions, we can extend several classical formulas in the derived category of bounded coherent sheaves to this singular setting. Finally, we also discuss the Bondal–Orlov localization for certain morphisms between terminal complex threefolds and their stacks.

We prove that a positive-definite quadratic form has nonnegative conorms if and only if it is matroidal, resolving a conjecture of Dutour Sikirić and Kummer. We show that the positive conorm support determines a regular matroid, and the positive conorms are the coefficients of a unimodular integral decomposition of the form. As a result, we show that a positive-definite form lies in the tropical Schottky locus if and only if its conorms are nonnegative and its positive conorm support matroid is cographic. This answers a question of Chua, Kummer, and Sturmfels and provides algorithms for the tropical Schottky decision and recovery problems in every genus.

Positroid varieties $\Pi_f$ provide a decomposition of the Grassmannian $Gr(k,n)$; they can be enumerated using bounded affine permutations ($f$) which have bijections with a number of interesting combinatorial objects. Furthermore, (the nonnegative part of) positroid varieties parameterize the space that is integrated over when calculating amplitudes in planar N=4 supersymmetric Yang-Mills theory. The main question we answer in this thesis is whether a positroid variety $\Pi_f$ has any geometric singularities. We show that it is sufficient to check singularity at the $T$-fixed points ($\lambda$) and we can obtain the multiplicity at these points $\lambda$ by calculating the equivariant cohomology of $\Pi_f$ restricted to the point. We give 2 ways of doing this: (1) A diagrammatic way using affine pipe dreams and (2) A computational method. For (2), we have written code that does the computation and outputs the multiplicity (files at josephflueg.github.io). We have included tables listing the multiplicities of all the points on positroid varieties up to $n=6$. We also describe an ordering on pairs $(\Pi_f,\lambda)$ given by deletion/contraction that interacts nicely with smoothness, and describe the relationship between affine pipe dreams and finite pipe dreams. In Part II of this thesis connects with physics of planar N=4 SYM. We briefly introduce quantum field theory, leading singularities, and positroids in N=4 SYM. We then explain Britto-Cachazo-Feng-Witten (BCFW) recursion and the BCFW bridge decomposition of an on-shell diagram. We describe how to build an on-shell diagram on a pipe dream using BCFW bridges. Finally, we work out some combinatorics related to inverse soft factors for on-shell diagrams and explore whether singularities in positroid varieties have relevance to amplitudes. This is a revised version of my thesis originally written in August 2024.