More Like this

1. Actions of 𝐴𝑙𝑡(𝑛) on groups of finite Morley rank without involutions

   https://doi.org/10.1090/proc/16530  

   Altınel, Tuna; Wiscons, Joshua (January 2024, Proceedings of the American Mathematical Society)

   We investigate faithful representations of Alt(n) as automorphisms of a connected group 𝐺 of finite Morley rank. We target a lower bound of 𝑛 on the rank of such a nonsolvable 𝐺, and our main result achieves this in the case when 𝐺 is without involutions. In the course of our analysis, we also prove a corresponding bound for solvable 𝐺 by leveraging recent results on the abelian case. We conclude with an application towards establishing natural limits to the degree of generic transitivity for permutation groups of finite Morley rank. 

   more » « less
2. Finite $$F$$-representation type for homogeneous coordinate rings of non-Fano varieties

   https://doi.org/10.46298/epiga.2023.10868  

   Mallory, Devlin (December 2023, Épijournal de Géométrie Algébrique)

   Finite $$F$$-representation type is an important notion in characteristic-$$p$$commutative algebra, but explicit examples of varieties with or without thisproperty are few. We prove that a large class of homogeneous coordinate ringsin positive characteristic will fail to have finite $$F$$-representation type. Todo so, we prove a connection between differential operators on the homogeneouscoordinate ring of $$X$$ and the existence of global sections of a twist of$$(\mathrm{Sym}^m \Omega_X)^\vee$$. By results of Takagi and Takahashi, thisallows us to rule out FFRT for coordinate rings of varieties with$$(\mathrm{Sym}^m \Omega_X)^\vee$$ not ``positive''. By using results positivityand semistability conditions for the (co)tangent sheaves, we show that severalclasses of varieties fail to have finite $$F$$-representation type, includingabelian varieties, most Calabi--Yau varieties, and complete intersections ofgeneral type. Our work also provides examples of the structure of the ring ofdifferential operators for non-$$F$$-pure varieties, which to this point havelargely been unexplored. 

   more » « less
3. Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha}$$ Velocity and Boundary

   https://doi.org/https://doi.org/10.1007/s00220-021-04067-1  

   Chen, Jiajie; Hou, Thomas Y (April 2021, Communications in mathematical physics)

   null (Ed.)

   Inspired by the numerical evidence of a potential 3D Euler singularity by Luo- Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $$C^{1,\alpha}$$ initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with $$C^{1,\alpha}$$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30,31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $$C^{1,\alpha}$$ initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with $$C^{1,\alpha}$$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time. 

   more » « less
4. The spectrum of Artin motives

   https://doi.org/10.1090/tran/9306  

   Balmer, Paul; Gallauer, Martin (March 2025, Transactions of the American Mathematical Society)

   We analyze the tt-geometry of derived Artin motives, via modular representation theory of profinite groups. To illustrate our methods, we discuss Artin motives over a finite field, in which case we also prove stratification. 

   more » « less
5. Slack reactants: A state-space truncation framework to estimate quantitative behavior of the chemical master equation

   https://doi.org/10.1063/5.0013457  

   Kim, Jinsu; Dark, Jason; Enciso, German; Sindi, Suzanne (August 2020, The Journal of Chemical Physics)

   State space truncation methods are widely used to approximate solutions of the chemical master equation. While most methods of this kind focus on truncating the state space directly, in this work, we propose modifying the underlying chemical reaction network by introducing slack reactants that indirectly truncate the state space. More specifically, slack reactants introduce an expanded chemical reaction network and impose a truncation scheme based on desired mass conservation laws. This network structure also allows us to prove inheritance of special properties of the original model, such as irreducibility and complex balancing. We use the network structure imposed by slack reactants to prove the convergence of the stationary distribution and first arrival times. We then provide examples comparing our method with the stationary finite state projection and finite buffer methods. Our slack reactant system appears to be more robust than some competing methods with respect to calculating first arrival times. 

   more » « less