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Joshua Mundinger

Publications and source records attributed to Joshua Mundinger.

17 recordsLinked to original sources
arXiv2026

Tannakian reconstruction in derived algebraic geometry

We prove analogues of Tannakian reconstruction theorems of Lurie and Bhatt--Halpern-Leistner in \emph{derived} algebraic geometry, where the basic geometric objects are spectra of animated rings rather than $\mathbb{E}_\infty$-rings. In this setting, symmetric monoidal $\infty$-categories are replaced by the $\Theta$-categories of Nuiten--To\"en. These are enhancements of symmetric monoidal $\infty$-categories which capture the strict commutativity structure on animated rings.

math.AG↗
arXiv2025

Cartier duality via Mittag-Leffler modules

We construct the Cartier duality equivalence for affine commutative group schemes $G$ whose coordinate ring is a flat Mittag-Leffler module over an arbitrary base ring $R$. The dual $G^\vee$ of $G$ turns out to be an ind-finite ind-scheme over $R$. When $R$ is Noetherian and admits a dualizing complex, we construct a Fourier-Mukai transform between quasicoherent derived categories of $G$ and of $BG^\vee$ and also between those of $G^\vee$ and $BG$.

math.AG↗
arXiv2025

Poisson Manifolds of Compact Types with an Appendix by Joshua Mundinger

We develop the theory of Poisson and Dirac manifolds of compact types, a broad generalization in Poisson and Dirac geometry of compact Lie algebras and Lie groups. We establish key structural results, including local normal forms, canonical stratifications, and a Weyl type resolution, which provides a way to resolve the singularities of the original structure. These tools allow us to show that the leaf space of such manifolds is an integral affine orbifold and to define their Weyl group. This group is a Coxeter group acting on the orbifold universal cover of the leaf space by integral affine transformations, and one can associate to it Weyl chambers, reflection hyperplanes, etc. We further develop a Duistermaat-Heckman theory for Poisson manifolds of s-proper type, proving the linear variation of cohomology of leafwise symplectic form and establishing a Weyl integration formula. As an application, we show that every Poisson manifold of compact type is necessarily regular. We conclude the paper with a list of open problems.

math.DG↗
arXiv2024

On the differentials of the Hochschild-Kostant-Rosenberg spectral sequence

The Hochschild-Kostant-Rosenberg theorem implies the existence of a spectral sequence computing the Hochschild homology of a variety in terms of the cohomology of differential forms. When the base field $k$ has characteristic $p>0$, we show that the differentials in this spectral sequence are zero before page $p$; when the variety admits a lift to $W_2(k)$, we give a formula for the differential on page $p$. The formula involves the Bockstein associated to the lift and a $p$th power operation for the Atiyah class. Along the way, we also discuss rudiments of Tannakian reconstruction for derived stacks using the $\Theta$-categories of Nuiten and To\"en.

math.AG↗
arXiv2024

Dieudonn\'e theory for $n$-smooth group schemes

For all $n \geq 1$, there is a notion of an $n$-smooth group scheme over any $\mathbb{F}_p$-algebra $R$, which may be thought of as a ``Frobenius analogue" of an $n$-truncated Barsotti--Tate group over $R$. We prove that the category of $n$-smooth commutative group schemes over $R$ is equivalent to a certain full subcategory of Dieudonn\'e modules over $R$. As a consequence, we show that the moduli stack $\mathrm{Sm}_n$ of $n$-smooth commutative group schemes is smooth over $\mathbb{F}_p$ and that the natural truncation morphism $\mathrm{Sm}_{n+1} \to \mathrm{Sm}_n$ is smooth and surjective. These results affirmatively answer conjectures of Drinfeld.

math.AG↗
arXiv2024

Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each $n\geq 2$ and $d\geq 1$ our construction yields a non-realizable degree $d$ hypersurface scheme in $\mathbb{P}^n$. Maclagan-Rinc\'on produced a non-realizable line in $\mathbb{P}^n$ for each $n$, and for $(d,n)=(1,2)$ the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.

math.AG↗
arXiv2024

Higher Congruences in Character Tables

Motivated by recent work of Peluse and Soundararajan on divisibility properties of the entries of the character tables of symmetric groups, we investigate the question: For a finite group G, when are two columns of the character table of G congruent to one another modulo a power of a prime?

math.RT↗
arXiv2023

Twisting the Infinitesimal Site

We classify twistings of Grothendieck's differential operators on a smooth variety $X$ in prime characteristic $p$. We prove isomorphism classes of twistings are in bijection with $H^2(X,\mathbb{Z}_p(1))$, the degree 2, weight 1 syntomic cohomology of $X$. We also discuss the relationship between twistings of crystalline and Grothendieck differential operators. Twistings in mixed characteristic are also analyzed.

math.AG↗
arXiv2021

Quantization of restricted Lagrangian subvarieties in positive characteristic

Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties X in positive characteristic which endow the Poisson bracket on X with the structure of a restricted Lie algebra. We consider deformation quantization of line bundles on Lagrangian subvarieties Y of X to modules over such quantizations. If the ideal sheaf of Y is a restricted Lie subalgebra of the structure sheaf of X, we show that there is a certain cohomology class which vanishes if and only if a line bundle on Y admits a quantization.

math.AG↗
arXiv2018

The Manickam-Miklós-Singhi Parameter of Graphs and Degree Sequences

Let $G$ be a simple graph. Consider all weightings of the vertices of $G$ with real numbers whose total sum is nonnegative. How many edges of $G$ have endpoints with a nonnegative sum? We consider the minimum number of such edges over all such weightings as a graph parameter. Computing this parameter has been shown to be NP-hard but we give a polynomial algorithm to compute the minimum of this parameter over realizations of a given degree sequence. We also completely determine the minimum and maximum value of this parameter for regular graphs.

math.CO↗
arXiv2018

The image of a tropical linear space

Given a tropical linear space $L \subseteq \mathbb{T}^n$ and a matrix $A \in \mathbb{T}^{m \times n}$, the image $AL$ of $L$ under $A$ is typically not a tropical linear space. We introduce a tropical linear space $\mathrm{tropim}_A(L)$, the tropical image, containing $AL$. We show under mild hypotheses that $\mathrm{tropim}_A(L)$ is realizable if $L$ is and apply the tropical image to construct the stable sum of two tropical linear spaces without a disjoint pair of bases.

math.AG↗
arXiv2017

A module-theoretic approach to matroids

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form.

math.AG↗
arXiv2016

Quantum State Transfer on Coronas

We study state transfer in quantum walk on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs $G$ and $H$, the Frucht-Harary corona product $G \circ H$ is obtained by taking $|G|$ copies of the cone $K_{1} + H$ and by connecting the conical vertices according to $G$. Our work explores conditions under which the corona $G \circ H$ exhibits state transfer. We also describe new families of graphs with state transfer based on the corona product. Some of these constructions provide a generalization of related known results.

math.CO↗
arXiv2015

Laplacian State Transfer in Coronas

We prove that the corona product of two graphs has no Laplacian perfect state transfer whenever the first graph has at least two vertices. This complements a result of Coutinho and Liu who showed that no tree of size greater than two has Laplacian perfect state transfer. In contrast, we prove that the corona product of two graphs exhibits Laplacian pretty good state transfer, under some mild conditions. This provides the first known examples of families of graphs with Laplacian pretty good state transfer. Our result extends of the work of Fan and Godsil on double stars to the Laplacian setting. Moreover, we also show that the corona product of any cocktail party graph with a single vertex graph has Laplacian pretty good state transfer, even though odd cocktail party graphs have no perfect state transfer.

quant-ph↗
arXiv2015

Demagnetization dynamics of non-interacting trapped fermions

Motivated by several experimental efforts to understand spin diffusion and transport in ultracold fermionic gases, we study the spin dynamics of initially spin-polarized ensembles of harmonically trapped non-interacting spin-1/2 fermionic atoms, subjected to a magnetic field gradient. We obtain simple analytic expressions for spin observables in the presence of both constant and linear magnetic field gradients, with and without a spin-echo pulse, and at zero and finite temperatures. The analysis shows the relevance of spin-motional coupling in the non-interacting regime where the demagnetization decay rate at short times can be faster than the experimentally measured rates in the strongly interacting regime under similar trapping conditions. Our calculations also show that particle motion limits the ability of a spin-echo pulse to remove the effect of magnetic field inhomogeneity, and that a spin-echo pulse can instead lead to an increased decay of magnetization at times comparable to the trapping period.

cond-mat.quant-gas↗