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1. Gromov norm and Turaev-Viro invariants of 3-manifolds

   Detcherry, Renaud; Kalfagianni, Efstratia (December 2020, Annales scientifiques de lÉcole normale supérieure)

   null (Ed.)

   We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants $$TV_r $$and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold $$M$$, with (possibly empty) toroidal boundary, $log|TVr(M)|$ is bounded above by a function linear in $$r$$ and whose slope is a positive universal constant times the Gromov norm of $$M$$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of $6j$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $$log|TVr(M)|\geq B r$$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose $SO(3)$- Turaev-Viro invariants grow exponentially. These constructions are essential for the results of article [3] where we make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations. We also study the behavior of the Turaev-Viro invariants under cutting and gluing of 3-manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question of the extent to which relations between the invariants TVr and hyperbolic volume are preserved under Dehn filling. Finally we give constructions of 3-manifolds, both with zero and non-zero Gromov norm, for which the Turaev-Viro invariants determine the Gromov norm. 

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2. Ginzburg-Landau description and emergent supersymmetry of the (3, 8) minimal model

   https://doi.org/10.1007/JHEP02(2023)066  

   Klebanov, Igor R.; Narovlansky, Vladimir; Sun, Zimo; Tarnopolsky, Grigory (February 2023, Journal of High Energy Physics)

   A bstract A pair of the 2D non-unitary minimal models M (2 , 5) is known to be equivalent to a variant of the M (3 , 10) minimal model. We discuss the RG flow from this model to another non-unitary minimal model, M (3 , 8). This provides new evidence for its previously proposed Ginzburg-Landau description, which is a ℤ 2 symmetric theory of two scalar fields with cubic interactions. We also point out that M (3 , 8) is equivalent to the (2 , 8) superconformal minimal model with the diagonal modular invariant. Using the 5-loop results for theories of scalar fields with cubic interactions, we exhibit the 6 − ϵ expansions of the dimensions of various operators. Their extrapolations are in quite good agreement with the exact results in 2D. We also use them to approximate the scaling dimensions in d = 3 , 4 , 5 for the theories in the M (3 , 8) universality class. 

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3. A tutorial on fitting joint models of M/EEG and behavior to understand cognition

   https://doi.org/10.3758/s13428-023-02331-x  

   Nunez, Michael D.; Fernandez, Kianté; Srinivasan, Ramesh; Vandekerckhove, Joachim (February 2024, Behavior Research Methods)

   Abstract We present motivation and practical steps necessary to find parameter estimates of joint models of behavior and neural electrophysiological data. This tutorial is written for researchers wishing to build joint models of human behavior and scalp and intracranial electroencephalographic (EEG) or magnetoencephalographic (MEG) data, and more specifically those researchers who seek to understand human cognition. Although these techniques could easily be applied to animal models, the focus of this tutorial is on human participants. Joint modeling of M/EEG and behavior requires some knowledge of existing computational and cognitive theories, M/EEG artifact correction, M/EEG analysis techniques, cognitive modeling, and programming for statistical modeling implementation. This paper seeks to give an introduction to these techniques as they apply to estimating parameters from neurocognitive models of M/EEG and human behavior, and to evaluate model results and compare models. Due to our research and knowledge on the subject matter, our examples in this paper will focus on testing specific hypotheses in human decision-making theory. However, most of the motivation and discussion of this paper applies across many modeling procedures and applications. We provide Python (and linked R) code examples in the tutorial and appendix. Readers are encouraged to try the exercises at the end of the document. 

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4. Stability conditions in families

   https://doi.org/10.1007/s10240-021-00124-6  

   Bayer, Arend; Lahoz, Martí; Macrì, Emanuele; Nuer, Howard; Perry, Alexander; Stellari, Paolo (June 2021, Publications mathématiques de l'IHÉS)

   Abstract We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers. Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type. Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration. 

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5. KNOTS, THREE-MANIFOLDS AND INSTANTONS

   https://doi.org/10.1142/9789813272880_0024  

   KRONHEIMER, PETER B.; MROWKA, TOMASZ S. (May 2019, Proceedings of the International Congress of Mathematicians (ICM 2018))

   Low-dimensional topology is the study of manifolds and cell complexes in dimensions four and below. Input from geometry and analysis has been central to progress in this field over the past four decades, and this article will focus on one aspect of these developments in particular, namely the use of Yang–Mills theory, or gauge theory. These techniques were pioneered by Simon Donaldson in his work on 4-manifolds, but the past ten years have seen new applications of gauge theory, and new interactions with more recent threads in the subject, particularly in 3-dimensional topology. This is a field where many mathematical techniques have found applications, and sometimes a theorem has two or more independent proofs, drawing on more than one of these techniques. We will focus primarily on some questions and results where gauge theory plays a special role. 

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