More Like this

1. SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE

   https://doi.org/10.1017/fmp.2020.1  

   LE, DANIEL; LE HUNG, BAO V.; LEVIN, BRANDON; MORRA, STEFANO (January 2020, Forum of Mathematics, Pi)

   We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $$p$$ . This is a generalization to $$\text{GL}_{3}$$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $$n$$ -dimensional Galois representations’, Duke Math. J.   149 (1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math.   212 (1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $$\text{GL}_{3}(\mathbb{F}_{q})$$ . 

   more » « less
2. Stratification and the comparison between homological and tensor triangular support

   https://doi.org/10.1093/qmath/haac040  

   Barthel, Tobias; Heard, Drew; Sanders, Beren (December 2022, The Quarterly Journal of Mathematics)

   Abstract We compare the homological support and tensor triangular support for ‘big’ objects in a rigidly-compactly generated tensor triangulated category. We prove that the comparison map from the homological spectrum to the tensor triangular spectrum is a bijection and that the two notions of support coincide whenever the category is stratified, extending the work of Balmer. Moreover, we clarify the relations between salient properties of support functions and exhibit counter-examples highlighting the differences between homological and tensor triangular support. 

   more » « less
3. Bernstein–Sato functional equations, V-filtrations, and multiplier ideals of direct summands

   https://doi.org/10.1142/S0219199721500838  

   Àlvarez Montaner, Josep; Hernández, Daniel J.; Jeffries, Jack; Núñez-Betancourt, Luis; Teixeira, Pedro; Witt, Emily E. (December 2022, Communications in Contemporary Mathematics)

   This paper investigates the existence and properties of a Bernstein–Sato functional equation in nonregular settings. In particular, we construct [Formula: see text]-modules in which such formal equations can be studied. The existence of the Bernstein–Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of [Formula: see text]-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings. 

   more » « less
4. Search problems in algebraic complexity, GCT, and hardness of generator for invariant rings

   Garg, A; Ikenmeyer, C; Makam, V; Oliveira, R; Walter, M; Wigderson, A (January 2020, Proceedings of the Computational Complexity Conference (CCC) 2020)

   We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SLn(C)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions). We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings. 

   more » « less
5. Search problems in algebraic complexity, GCT, and hardness of generator for invariant rings

   Garg, A; Ikenmeyer, C; Makam, V; Oliveira, R; Walter, M; Wigderson, A. (July 2020, Computational Complexity Conference (CCC) 2020)

   We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SL_n(C)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions). We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings. 

   more » « less