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1. Quantum circuits for toric code and X-cube fracton model

   https://doi.org/10.22331/q-2024-03-13-1276  

   Chen, Penghua; Yan, Bowen; Cui, Shawn X (March 2024, Quantum)

   We propose a systematic and efficient quantum circuit composed solely of Clifford gates for simulating the ground state of the surface code model. This approach yields the ground state of the toric code in $\lceil 2L+2+lo{g}_2\left(d\right)+\frac{L}{2d}\rceil$time steps, where $L$refers to the system size and $d$represents the maximum distance to constrain the application of the CNOT gates. Our algorithm reformulates the problem into a purely geometric one, facilitating its extension to attain the ground state of certain 3D topological phases, such as the 3D toric model in $3L+8$steps and the X-cube fracton model in $12L+11$steps. Furthermore, we introduce a gluing method involving measurements, enabling our technique to attain the ground state of the 2D toric code on an arbitrary planar lattice and paving the way to more intricate 3D topological phases. 

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2. The Vlasov–Poisson–Landau system in the weakly collisional regime

   https://doi.org/10.1090/jams/1014  

   Chaturvedi, Sanchit; Luk, Jonathan; Nguyen, Toan (January 2023, Journal of the American Mathematical Society)

   Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a $3$-torus, i.e. $\begin{array}{c}{\mathrm{\partial <\#comment/>}}_{t}F(t,x,v)+v_i{\mathrm{\partial <\#comment/>}}_{x_i}F(t,x,v)+E_i(t,x){\mathrm{\partial <\#comment/>}}_{v_i}F(t,x,v)=\mathrm{\nu <\#comment/>}Q(F,F)(t,x,v),\\ E(t,x)=\mathrm{\nabla <\#comment/>}{\mathrm{\Delta <\#comment/>}}^{-<\#comment/>1}\left({\int <\#comment/>}_{{\mathbb{R}}^3}F\right(t,x,v)\mathrm{d}v-<\#comment/>{\int <\#comment/>-<\#comment/>}_{{\mathbb{T}}^3}{\int <\#comment/>}_{{\mathbb{R}}^3}F(t,x,v\left)\mathrm{d}v\mathrm{d}x\right),\end{array}$with $\mathrm{\nu <\#comment/>}\ll <\#comment/>1$. We prove that for $\mathrm{ϵ<\#comment/>}>0$sufficiently small (but independent of $\mathrm{\nu <\#comment/>}$), initial data which are $O\left(\mathrm{ϵ<\#comment/>}{\mathrm{\nu <\#comment/>}}^{1/3}\right)$-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as $t\to <\#comment/>\mathrm{\infty <\#comment/>}$. The solutions exhibit uniform-in- $\mathrm{\nu <\#comment/>}$Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo’s weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation. 

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3. Blown-up toric surfaces with non-polyhedral effective cone

   https://doi.org/10.1515/crelle-2023-0022  

   Castravet, Ana-Maria; Laface, Antonio; Tevelev, Jenia; Ugaglia, Luca (May 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We construct projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone.As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space ${\overline{M}}_{0,n}$\overline{M}_{0,n}of stable rational curves is not polyhedral for $n\ge 10$n\geq 10.These results hold both in characteristic 0 and in characteristic 𝑝, for all primes 𝑝.Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order.Our analysis relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang–Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic 0 and in characteristic 𝑝, for an infinite set of primes 𝑝 of positive density. 

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4. The unbounded denominators conjecture

   https://doi.org/10.1090/jams/1053  

   Calegari, Frank; Dimitrov, Vesselin; Tang, Yunqing (February 2025, Journal of the American Mathematical Society)

   We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of ${\mathrm{SL}}_2<\#comment/>\left(\mathbf{Z}\right)$. Our result includes also Mason’s generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna second main theorem, the congruence subgroup property of ${\mathrm{SL}}_2<\#comment/>\left(\mathbf{Z}\right[1/p\left]\right)$, and a close description of the Fuchsian uniformization $D(0,1)/{\mathrm{\Gamma <\#comment/>}}_N$of the Riemann surface $\mathbf{C}\setminus <\#comment/>{\mathrm{\mu <\#comment/>}}_N$. 

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5. Convolution identities for divisor sums and modular forms

   https://doi.org/10.1073/pnas.2322320121  

   Fedosova, Ksenia; Klinger-Logan, Kim; Radchenko, Danylo (October 2024, Proceedings of the National Academy of Sciences)

   Ribet, Kenneth (Ed.)

   We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums $\sum _{n_1\in \mathbb{Z}\setminus \{0,n\}}\phi (n_1,n-n_1){\sigma }_{2m_1}\left(n_1\right){\sigma }_{2m_2}(n-n_1),$ where $\phi (n_1,n_2)$is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to $n_1\in \{0,n\}$, do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms. 

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