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1. The Loewner Energy of Loops and Regularity of Driving Functions

   https://doi.org/10.1093/imrn/rnz071  

   Rohde, Steffen; Wang, Yilin (April 2019, International Mathematics Research Notices)

   Abstract Loewner driving functions encode simple curves in 2D simply connected domains by real-valued functions. We prove that the Loewner driving function of a $$C^{1,\beta }$$ curve (differentiable parametrization with $$\beta$$-Hölder continuous derivative) is in the class $$C^{1,\beta -1/2}$$ if $$1/2<\beta \leq 1$$, and in the class $$C^{0,\beta + 1/2}$$ if $$0 \leq \beta \leq 1/2$$. This is the converse of a result of Carto Wong [26] and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint. 

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2. Negative Moments of L –Functions With Small Shifts Over Function Fields

   https://doi.org/10.1093/imrn/rnad118  

   Florea, Alexandra (June 2023, International Mathematics Research Notices)

   Abstract We consider negative moments of quadratic Dirichlet $$L$$–functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $$\mathbb{F}_{q}[x]$$, we obtain an asymptotic formula for the $$k^{\textrm{th}}$$ shifted negative moment of $$L(1/2+\beta ,\chi _{D})$$, in certain ranges of $$\beta $$ (e.g., when roughly $$\beta \gg \log g/g $$ and $k<1$). We also obtain non-trivial upper bounds for the $$k^{\textrm{th}}$$ shifted negative moment when $$\log (1/\beta ) \ll \log g$$. Previously, almost sharp upper bounds were obtained in [ 3] in the range $$\beta \gg g^{-\frac{1}{2k}+\epsilon }$$. 

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3. Frozen pipes: lattice models for Grothendieck polynomials

   https://doi.org/10.5802/alco.277  

   Brubaker, Ben; Frechette, Claire; Hardt, Andrew; Tibor, Emily; Weber, Katherine (June 2023, Algebraic combinatorics)

   We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$$ -- {\it biaxial double} $$(\beta,q)$$-{\it Grothendieck polynomials} -- which specialize at $$q=0$ and $v=1$ to double $$\beta$$-Grothendieck polynomials from torus-equivariant connective K-theory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in $$n$$ pairs of variables is a Drinfeld twist of the $$U_q(\widehat{\mathfrak{sl}}_{n+1})$$ $$R$$-matrix. By leveraging the resulting Yang-Baxter equations of the lattice model, we show that these polynomials simultaneously generalize double $$\beta$$-Grothendieck polynomials and dual double $$\beta$$-Grothendieck polynomials for arbitrary permutations. We then use properties of the model and Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for $$\beta$$-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double $$\beta$$-Grothendieck polynomials, and prove a new branching rule for double $$\beta$$-Grothendieck polynomials. 

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4. Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons

   https://doi.org/10.1007/s10955-020-02500-8  

   Boßmann, Lea; Pavlović, Nataša; Pickl, Peter; Soffer, Avy (March 2020, Journal of Statistical Physics)

   Abstract In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N $$d$$ d -dimensional bosons for large N . The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form $$(N-1)^{-1}N^{d\beta }v(N^\beta \cdot )$$ ( N - 1 ) - 1 N d β v ( N β · ) for $$\beta \in [0,\frac{1}{4d})$$ β ∈ [ 0 , 1 4 d ) . We derive a sequence of N -body functions which approximate the true many-body dynamics in $$L^2({\mathbb {R}}^{dN})$$ L 2 ( R dN ) -norm to arbitrary precision in powers of $$N^{-1}$$ N - 1 . The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution. 

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5. Different origins of running in the 2D CP(1) model

   https://doi.org/10.1103/PhysRevD.110.096007  

   Kiaee, Kasra; Monin, Alexander (November 2024, Physical Review D)

   We revisit the computation of the beta function in the two-dimensional CP(1) sigma model. We show that in different schemes, different diagrams are responsible for the running, such as momentum- independent tadpoles or even UV-finite bubble diagrams. We also comment on the relation between the beta functions and the energy dependence of scattering amplitudes. 

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