arXiv subjects
Publications and source records attributed to Malors Espinosa.
In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.
For every Gorenstein cubic $\mathbb Z$-order, we prove an identity equating Yun's zeta function, defined by counting finite-index submodules of the trace dual, with the explicit overorder zeta function introduced in our previous work on Beyond Endoscopy for $\mathrm{GL}_3$. This is posed as Conjecture A in an early draft of Deng-Espinosa and Lee subsequently proved the functional equation of the overorder zeta function, making the Deng--Espinosa isolation of the trivial representation fully unconditional. His argument computes the local factors explicitly. Our proof is independent of Lee's and does not evaluate the individual cubic overorder factors: it matches natural decompositions of the two sides and concludes by induction. As an application, we give a short, uniform evaluation of the local $\mathrm{GL}_3 $ Kloosterman Dirichlet series in the Poisson-summation argument of Deng--Espinosa. The direct local analysis of the Kloosterman series occupies nearly ninety pages in Deng--Espinosa while our treatment here replaces its case-by-case enumeration with a short uniform proof.
In this paper we define a generating function for buildings of type $\widetilde{A}_1$ (i.e. trees) that are enhanced with a certain filtration structure. We prove that this generating function recovers the zeta function of certain quadratic orders. We do this by studying how the ideals of the orders distribute in the building of $SL(2, K)$.
In this paper we study knots created by galleries in the affine Coxeter complex of type \widewedge{B3}. We bound the stick number by 40 and prove that the smallest length of threefold rotationally symmetric trefoils is 42. We construct explicit galleries that knot as 9_35, 9_40, 9_41 and 9_47 in a way that has threefold rotational symmetry. We explain the construction of these galleries for 9_47 carefully. We conclude with three questions inspired by this work.
We prove that all knots can be embedded into the Menger Sponge fractal. We prove that all Pretzel knots can be embedded into the Sierpinski Tetrahedron. Then we compare the number of iterations of each of these fractals needed to produce a given knot as a mean to compare the complexity of the two fractals.
Langlands has introduced a formula for a specific product of orbital integrals in $\mbox{GL}(2, \mathbb{Q})$. Altuğ employs this formula to manipulate the regular elliptic part of the trace formula, with the aim of eliminating the contribution of the trivial representation from the spectral side. Arthur predicts that this formula coincides with a product of polynomials associated with zeta functions of orders developed by Zhiwei Yun. In a previous paper, the author determined the explicit polynomials for the relevant quadratic orders. This paper demonstrates how these polynomials can effectively generalize Langlands' formula to $GL(2, K)$, for general algebraic number fields $K$. Furthermore, we also use this formula to extend a well-known formula of Zagier to any algebraic number field and explain its applications in the contexts of the strategy of Beyond Endoscopy proposed by Langlands.
Langlands provides a formula for certain product of orbital integrals in $GL(2, \mathbb{Q})$. Its generalization has become an important question for the strategy of Beyond Endoscopy. Arthur predicts this formula should coincide with a product of polynomials associated to zeta functions of orders constructed by Zhiwei Yun. In this paper we compute, for a certain family of orders, explicit formulas for these zeta functions by a recursive method. We use these zeta functions in a further paper to prove that Arthur's prediction is correct.
We define (higher rank) spinorially twisted spin structures and deduce various curvature identites as well as estimates for the eigenvalues of the corresponding twisted Dirac operators.