Related papers: Hidden sign in Langlands' correspondence
This is a journey through integrals of involutions and surprising consequences of the Lagrange inversion theorem. On the way, we meet unexpected logarithmic identities, hypergeometric functions with a linear regime and other mysterious…
We demonstrate that LLMs may learn indicators of document usefulness and modulate their updates accordingly. We introduce random strings ("tags") as indicators of usefulness in a synthetic fine-tuning dataset. Fine-tuning on this dataset…
The first part of this article is a review of the properties expected of any local Langlands correspondence that aims to be considered "canonical," and of known results that establish some or all of these properties for specific groups. In…
In this brief essay a construction of the $2$-variable L-function of Langlands is sketched in terms of monomial resolutions of admissible representations of reductive locally $p$-adic Lie groups.
Various threshold effects are investigated on a discrete quasi-1D scattering system. In particular, one of these effects is to add corrections to Levinson's theorem. We explain how these corrections are due to the opening or to the closing…
We train multi-task autoencoders on linguistic tasks and analyze the learned hidden sentence representations. The representations change significantly when translation and part-of-speech decoders are added. The more decoders a model…
We explore potential uses of physics formulated in Kleinian (i.e., $2+2$) signature spacetimes as a tool for understanding properties of physics in Lorentzian (i.e., $3+1$) signature. Much as Euclidean (i.e., $4+0$) signature quantities can…
Eq.(19) is added and related issues are further clarified. Also some typos and signs a re corrected.
This expository note outlines why it is sometimes useful to consider the bigraded type A link homology theories as associated with the Lie algebras gl(N) instead of sl(N).
Kaletha extended local Langlands conjectures to a certain class of disconnected groups and proved them for disconnected tori. Our first main result is a reinterpretation of the local Langlands correspondence for disconnected tori. Our…
We show how the sign of a permutation can be deduced from the tableaux induced by the permutation under the Robinson-Schensted-Knuth correspondence. The result yields a simple proof of a conjecture on the squares of imbalances raised by…
We study spectral theory of sign-changing Laplace operators using semi-classical Dirichlet-to-Neumann maps. We prove the existence of modesconcentrated on the interface and describe an effective semi-classical equation for them.
A multi-layered structure consisting of alternate right-handed material (RHM) and left-handed material (LHM)is considered and the unusual narrow transmission bands are explained as the competitive results of the Bragg condition and the…
Minor modifications are given to prove the Main Theorem under the Blaschke (instead of Carleson) condition as well as a small historical comment.
The first part of this paper is a survey of mathematical results on mirror symmetry phenomena between Hitchin systems for Langlands dual groups. The second part introduces and discusses multiplicity algebras of the Hitchin system on…
Let $f$ be a holomorphic or Maass Hecke cusp form for the full modular group and write $\lambda_f(n)$ for the corresponding Hecke eigenvalues. We are interested in the signs of those eigenvalues. In the holomorphic case, we show that for…
We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the…
This is an informal paper presenting historical results around the recent paper of the author about Lang's Conjecture and torsion of elliptic curves. This paper also discusses a few aspects of the proof.
The spectral theory of the normalized Laplacian for chemical hypergraphs is further investigated. The signless normalized Laplacian is introduced and it is shown that its spectrum for classical hypergraphs coincides with the spectrum of the…
The motivation of the note is to obtain a H\"{o}rmander-type $L^2$ estimate for $\bar\partial$ equation. The feature of the new estimate is that the constant is independent of the weight function. Moreover, our estimate can be used for…