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Creators/Authors contains: "Ramos, Eric"

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  1. ; (, Selecta Mathematica)
    Abstract We formulate a categorification of Robertson’s conjecture analogous to the categorical graph minor conjecture of Miyata–Proudfoot–Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the homology of configuration spaces of graphs—in fixed degree, with a fixed number of particles, under topological embeddings. We explain how the simplest case of our conjecture follows from work of Barter and Proudfoot–Ramos, implying that the category of cographs is Noetherian, a result of potential independent interest. 
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    Free, publicly-accessible full text available November 1, 2025
  2. ; (, Linear Algebra and its Applications)
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  3. ; ; ; ; ; ; (, Inorganic Chemistry)
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  4. ; (, Advances in Mathematics)
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  5. ; (, Séminaire lotharingien de combinatoire)
    One of the most famous results in graph theory is that of Kuratowski’s theorem, which states that a graph $$G$$ is non-planar if and only if it contains one of $$K_{3,3}$$ or $$K_5$$ as a topological minor. That is, if some subdivision of either $$K_{3,3}$$ or $$K_5$$ appears as a subgraph of $$G$$. In this case we say that the question of planarity is determined by a finite set of forbidden (topological) minors. A conjecture of Robertson, whose proof was recently announced by Liu and Thomas, characterizes the kinds of graph theoretic properties that can be determined by finitely many forbidden minors. In this extended abstract we will present a categorical version of Robertson’s conjecture, which we have proven in certain cases. We will then illustrate how this categorification, if proven in all cases, would imply many non-trivial statements in the topology of graph configuration spaces. 
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    Accepted Manuscript Full Text Available
  6. ; ; (, The Electronic Journal of Combinatorics)
    Let $$V_1, V_2, V_3, \dots $$ be a sequence of $$\mathbb {Q}$$-vector spaces where $$V_n$$ carries an action of $$\mathfrak{S}_n$$. Representation stability and multiplicity stability are two related notions of when the sequence $$V_n$$ has a limit. An important source of stability phenomena arises when $$V_n$$ is the $$d^{th}$$ homology group (for fixed $$d$$) of the configuration space of $$n$$ distinct points in some fixed topological space $$X$$. We replace these configuration spaces with moduli spaces of tuples $$(W_1, \dots, W_n)$$ of subspaces of a fixed complex vector space $$\mathbb {C}^N$$ such that $$W_1 + \cdots + W_n = \mathbb {C}^N$$. These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory. 
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  7. (, Notices of the American Mathematical Society)
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  8. ; (, Representation Theory of the American Mathematical Society)
    We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes a graph to a graded piece of the homology of its unordered configuration space and the second takes a graph to an intersection homology group whose dimension is given by a Kazhdan–Lusztig coefficient; in both cases we prove that the module is finitely generated. This allows us to draw conclusions about torsion in the homology groups of graph configuration spaces, and about the growth of Betti numbers of graph configuration spaces and Kazhdan–Lusztig coefficients of graphical matroids. We also explore the relationship between our category and outer space, which is used in the study of outer automorphisms of free groups. 
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  9. (, Mathematische Zeitschrift)
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  10. ; ; ; (, International Mathematics Research Notices)
    Abstract We expand upon the notion of equivariant log concavity and make equivariant log concavity conjectures for Orlik–Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik–Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $$\mathfrak{s}\mathfrak{l}_n$$, we exploit the theory of representation stability to give computer-assisted proofs of these conjectures in low degree. 
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