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European Mathematical Society Publishing House
2024-03-28 19:35:32
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=PRIMS&vol=52&iss=4&update_since=2024-03-28
Publications of the Research Institute for Mathematical Sciences
Publ. Res. Inst. Math. Sci.
PRIMS
0034-5318
1663-4926
General
10.4171/PRIMS
http://www.ems-ph.org/doi/10.4171/PRIMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Research Institute for Mathematical Sciences, Kyoto University
52
2016
4
Dialogue Categories and Chiralities
Paul-André
Melliès
Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE
Categories and chiralities, cartesian closed categories and chiralities, dialogue categories and chiralities, classical logic, linear logic, tensorial logic, game semantics, logical polarities
In this paper, we consider a two-sided notion of dialogue category which we call dialogue chirality and which we formulate as an adjunction between a monoidal category $\mathcal A$ of proofs and a monoidal category $\mathcal B$ of counter-proofs equivalent to its opposite category $\mathcal A^{\mathrm {op}(0,1)}$. The two-sided formulation of dialogue categories is compared to the original one-sided formulation by exhibiting a 2-dimensional equivalence between a 2-category of dialogue categories and a 2-category of dialogue chiralities. The resulting coherence theorem clari es in what sense every dialogue chirality may be stricti ed to an equivalent dialogue category.
Mathematical logic and foundations
Category theory; homological algebra
359
412
10.4171/PRIMS/185
http://www.ems-ph.org/doi/10.4171/PRIMS/185
Perverse Results on Milnor Fibers inside Parameterized Hypersurfaces
Brian
Hepler
Northeastern University, BOSTON, UNITED STATES
David
Massey
Northeastern University, BOSTON, UNITED STATES
Milnor fiber, hypersurface, stable unfolding, intersection cohomology, perverse sheaf
We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we defi ne a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set influences the topology of the Milnor fiber. We also discuss applications to stable unfoldings of fi nite maps with isolated instabilities.
Several complex variables and analytic spaces
413
433
10.4171/PRIMS/186
http://www.ems-ph.org/doi/10.4171/PRIMS/186
Remarks on Arithmetic Restricted Volumes and Arithmetic Base Loci
Hideaki
Ikoma
Kyoto University, KYOTO, JAPAN
Arakelov theory, linear series, restricted volumes, base loci
We collect some fundamental properties of arithmetic restricted volumes (or arithmetic multiplicities) of adelically metrized line bundles. Every arithmetic restricted volume has the concavity property and characterizes the corresponding arithmetic augmented base locus as the null locus.We also establish a generalized Fujita approximation for arithmetic restricted volumes.
Algebraic geometry
Number theory
435
495
10.4171/PRIMS/187
http://www.ems-ph.org/doi/10.4171/PRIMS/187
New Realization of Cyclotomic $q$-Schur Algebras
Kentaro
Wada
Shinshu University, MATSUMOTO, JAPAN
Cyclotomic $q$-Schur algebras, Lie algebras, quantum groups
We introduce a Lie algebra $\mathfrak{g}_{\mathbf{Q}}(\mathbf{m})$ and an associative algebra $\mathcal{U}_{q,\mathbf{Q}}(\mathbf{m})$ associated with the Cartan data of $\mathfrak{gl}_m$ which is separated into $r$ parts with respect to $\mathbf{m}=(m_1, \dots, m_r)$ such that $m_1+ \dots + m_r =m$. We show that the Lie algebra $\mathfrak{g}_{\mathbf{Q}} (\mathbf{m})$ is a filtered deformation of the current Lie algebra of $\mathfrak{gl}_m$, and we can regard the algebra $\mathcal{U}_{q, \mathbf{Q}}(\mathbf{m})$ as a \lq\lq $q$-analogue" of $U(\mathfrak{g}_{\mathbf{Q}}(\mathbf{m}))$. Then, we realize a cyclotomic $q$-Schur algebra as a quotient algebra of $\mathcal{U}_{q, \mathbf{Q}}(\mathbf{m})$ under a certain mild condition. We also study the representation theory for $\mathfrak g_{\mathbf Q} (\mathbf m)$ and $\mathcal U_{q,\mathbf Q}(\mathbf m)$, and we apply them to the representations of the cyclotomic $q$-Schur algebras.
Group theory and generalizations
Nonassociative rings and algebras
497
555
10.4171/PRIMS/188
http://www.ems-ph.org/doi/10.4171/PRIMS/188