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1. THE COHOMOLOGY OF UNRAMIFIED RAPOPORT–ZINK SPACES OF EL-TYPE AND HARRIS'S CONJECTURE

   https://doi.org/10.1017/S1474748020000535  

   Meli, Alexander Bertoloni ( January 2021 , Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Abstract We study the l -adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm {GL_n}$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $\mathrm {Mant}_{b, \mu }$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ an admissible representation and prove it when $\rho $ is essentially square-integrable. Our proof works for general $\rho $ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup. 

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2. Regular Supercuspidal Representations

   Tasho Kaletha ( July 2019 , Journal of the American Mathematical Society)

   We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups. 

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3. On the construction of tame supercuspidal representations

   https://doi.org/10.1112/S0010437X21007636  

   Fintzen, Jessica ( December 2021 , Compositio Mathematica)

   Let $F$ be a non-archimedean local field of residual characteristic $p \neq 2$ . Let $G$ be a (connected) reductive group over $F$ that splits over a tamely ramified field extension of $F$ . We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [ Construction of tame supercuspidal representations , J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference. 

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4. Cohomological representations of parahoric subgroups

   https://doi.org/10.1090/ert/557  

   Chan, Charlotte ; Ivanov, Alexander ( January 2021 , Representation Theory of the American Mathematical Society)

   We give a geometric construction of representations of parahoric subgroups P P of a reductive group G G over a local field which splits over an unramified extension. These representations correspond to characters θ \theta of unramified maximal tori and, when the torus is elliptic, are expected to give rise to supercuspidal representations of G G . We calculate the character of these P P -representations on a special class of regular semisimple elements of G G . Under a certain regularity condition on θ \theta , we prove that the associated P P -representations are irreducible. This generalizes a construction of Lusztig from the hyperspecial case to the setting of an arbitrary parahoric. 

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5. How does a Neural Network's Architecture Impact its Robustness to Noisy Labels?

   Li, Jingling ; Zhang, Mozhi ; Xu, Keyulu ; Dickerson, John P ; Ba, Jimmy ( January 2021 , Advances in Neural Information Processing Systems 34)

   Noisy labels are inevitable in large real-world datasets. In this work, we explore an area understudied by previous works --- how the network's architecture impacts its robustness to noisy labels. We provide a formal framework connecting the robustness of a network to the alignments between its architecture and target/noise functions. Our framework measures a network's robustness via the predictive power in its representations --- the test performance of a linear model trained on the learned representations using a small set of clean labels. We hypothesize that a network is more robust to noisy labels if its architecture is more aligned with the target function than the noise. To support our hypothesis, we provide both theoretical and empirical evidence across various neural network architectures and different domains. We also find that when the network is well-aligned with the target function, its predictive power in representations could improve upon state-of-the-art (SOTA) noisy-label-training methods in terms of test accuracy and even outperform sophisticated methods that use clean labels. 

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