More Like this

1. New lower bounds for matrix multiplication and

   https://doi.org/10.1017/fmp.2023.14  

   Conner, Austin; Harper, Alicia; Landsberg, J.M. (January 2023, Forum of Mathematics, Pi)

   Abstract Let$$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$$denote the matrix multiplication tensor (and write$$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$$), and let$$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$$denote the determinant polynomial considered as a tensor. For a tensorT, let$$\underline {\mathbf {R}}(T)$$denote its border rank. We (i) give the first hand-checkable algebraic proof that$$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$$, (ii) prove$$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$$and$$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$$M_{\langle 2\rangle }$$, (iii) prove$$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$$, (iv) prove$$\underline {\mathbf {R}}(\operatorname {det}_3)=17$$, improving the previous lower bound of$$12$$, (v) prove$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$$for all$$\mathbf {n}\geq 25$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$$was known, as well as lower bounds for$$4\leq \mathbf {n}\leq 25$$, and (vi) prove$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$$for all$$\mathbf {n} \ge 18$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$$was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors. The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, calledborder apolaritydeveloped by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorTand an integerr, in a finite number of steps, either outputs that there is no border rankrdecomposition forTor produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenThas a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory. 

   more » « less
2. Stability in the high-dimensional cohomology of congruence subgroups

   https://doi.org/10.1112/s0010437x20007046  

   Miller, Jeremy; Nagpal, Rohit; Patzt, Peter (April 2020, Compositio Mathematica)

   We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $$\mathbf{SL}_{n}(\mathbb{Z})$$ . This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $$\mathbf{SL}_{n}(K)$$ for $$K$$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $$\mathbf{SL}_{n}(\mathbb{Z})$$ . 

   more » « less
3. ON THE EXISTENCE OF ADMISSIBLE SUPERSINGULAR REPRESENTATIONS OF -ADIC REDUCTIVE GROUPS

   https://doi.org/10.1017/fms.2019.50  

   HERZIG, FLORIAN; KOZIOŁ, KAROL; VIGNÉRAS, MARIE-FRANCE (January 2020, Forum of Mathematics, Sigma)

   Suppose that $$\mathbf{G}$$ is a connected reductive group over a finite extension $$F/\mathbb{Q}_{p}$$ and that $$C$$ is a field of characteristic  $$p$$ . We prove that the group $$\mathbf{G}(F)$$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over  $$C$$ . 

   more » « less
4. Uniform congruence counting for Schottky semigroups in SL2(𝐙)

   https://doi.org/10.1515/crelle-2016-0072  

   Magee, Michael; Oh, Hee; Winter, Dale (August 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})} ,and for {q\in\mathbf{N}} , let {\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}} be its congruence subsemigroupof level q . Let δ denote the Hausdorff dimension of the limit set of Γ.We prove the following uniform congruence counting theoremwith respect to the family of Euclidean norm balls {B_{R}} in {M_{2}(\mathbf{R})} of radius R :for all positive integer q with no small prime factors, \#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(%\mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon}) as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q .Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})} ,which arises in the study of Zaremba’s conjecture on continued fractions. 

   more » « less
5. Kato square root problem with unbounded leading coefficients

   https://doi.org/https://doi.org/10.1090/proc/14224  

   Escauriaza, L.; Hofmann, S. (October 2018, Proceedings of the American Mathematical Society)

   We prove the Kato conjecture for elliptic operators, $$L=-\nabla\cdot\left((\mathbf A+\mathbf D)\nabla\ \right)$$, with $$\mathbf A$$ a complex measurable bounded coercive matrix and $$\mathbf D$$ a measurable real-valued skew-symmetric matrix in $$\re^n$$ with entries in $$BMO(\re^n)$$;\, i.e., the domain of $$\sqrt{L}\,$$ is the Sobolev space $$\dot H^1(\re^n)$$, with the estimate $$\|\sqrt{L}\, f\|_2 \lesssim \| \nabla f\|_2\,.$$ 

   more » « less