One-sided exact categories and glider representations.
Donnerstag, 4.7.19, 10:30-11:30, Raum 403, Ernst-Zermelo-Str. 1
Quillen exact categories provide an excellent framework to do homological algebra and algebraic K-theory. A Quillen exact category is an additive category together with a chosen class of kernel-cokernel pairs (called conflations) satisfying 8 axioms. These 8 axioms can be partitioned into two dual sets of axioms referring to either the kernel-part of a conflation (called an inflation) or the cokernel-part of a conflation (called a deflation). However, 2 of the axioms were quickly found to be redundant. These two dual axioms are known as Quillen's obscure axioms.\n\nA one-sided exact category is defined by keeping either the set of axioms referring to inflation or deflations, however, one might wonder whether the obscure axiom needs to be included. In this talk, we will provide several homological interpretations of the obscure axiom. Moreover, any one-sided exact category can naturally be closed under the obscure axiom and is derived equivalent to this obscure closure. As such, we conclude that the obscure axiom may just as well be included into the definition. \n\nWe apply the theory of one-sided exact categories to obtain a categorical framework for glider representations. Glider representations are a type of filtered representations of a filtered ring. We end by concluding that glider representations remember more information on the original ring than ordinary representation theory.
Localizing (one-sided) exact categories
Freitag, 5.7.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
One-sided exact categories are a generalization of Quillen exact categories; they satisfy many desirable homological properties and provide a comparable framework for K-theory. Similar to the exact setting, one can consider the derived category of a one-sided exact category by taking the Verdier quotient of the homotopy category by the subcategory of acyclic complexes.\n\nMimicking the setting of a Serre subcategory of an abelian category, we introduce percolating subcategories of exact categories. One can show that the corresponding localization is, in general, not exact, but merely one-sided exact.\n\nIn this talk, we will discuss these localizations and the corresponding Verdier localizations on the bounded derived categories.\n(Based on joint work with Ruben Henrard.)
Zero cycles on moduli spaces of curves
Freitag, 12.7.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Tautological zero cycles form a one-dimensional subspace of\nthe set of all algebraic zero-cycles on the moduli space of stable curves. The full group of zero cycles can in general be infinite-dimensional, so not all points of the moduli space will represent a tautological class. In the\ntalk, I will present geometric conditions ensuring that a pointed curve does define a tautological point. On the other hand, given any point Q in the moduli space we can find other points P1, ..., Pm such that Q+P1+ ... +\nPm is tautological. The necessary number m is uniformly bounded in terms of g,n, but the question of its minimal value is open. This is joint work with R. Pandharipande.
Wall crossing morphisms for moduli of stable pairs
Freitag, 19.7.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Consider a moduli space M parametrizing stable pairs of the form (X, \bsum ai Di) with ai n positive rational numbers. Consider n positive rational numbers bi with bi \ble ai, and assume that the objects on the interior of M are pairs with KX +\bsum bi Di big. Then on the interior of M one can send a pair (X, \bsum ai Di) to the canonical model of (X, \bsum bi Di). If N is a moduli space of stable pairs with coefficients bi this gives a set theoretic map from an open substack of M to N. We investigate when such a map can be extended to the whole M. Our main result is if the interior of M parameterizes klt pairs we can extend the map, up to replacing M and N with their normalizations. The extension does not exist if above we replace the word normalization with seminormalizaton instead. This is joint with Kenny Ascher, Dori Bejleri and Zsolt Patakfalvi.
Moduli of special cubic 4-folds
Freitag, 26.7.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1