@article{14499, abstract = {An n-vertex graph is called C-Ramsey if it has no clique or independent set of size Clog2n (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.}, author = {Kwan, Matthew Alan and Sah, Ashwin and Sauermann, Lisa and Sawhney, Mehtaab}, issn = {2050-5086}, journal = {Forum of Mathematics, Pi}, keywords = {Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Analysis}, publisher = {Cambridge University Press}, title = {{Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture}}, doi = {10.1017/fmp.2023.17}, volume = {11}, year = {2023}, } @article{11916, abstract = {A domain is called Kac regular for a quadratic form on L2 if every functions vanishing almost everywhere outside the domain can be approximated in form norm by functions with compact support in the domain. It is shown that this notion is stable under domination of quadratic forms. As applications measure perturbations of quasi-regular Dirichlet forms, Cheeger energies on metric measure spaces and Schrödinger operators on manifolds are studied. Along the way a characterization of the Sobolev space with Dirichlet boundary conditions on domains in infinitesimally Riemannian metric measure spaces is obtained.}, author = {Wirth, Melchior}, issn = {2538-225X}, journal = {Advances in Operator Theory}, keywords = {Algebra and Number Theory, Analysis}, number = {3}, publisher = {Springer Nature}, title = {{Kac regularity and domination of quadratic forms}}, doi = {10.1007/s43036-022-00199-w}, volume = {7}, year = {2022}, } @article{10643, abstract = {We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap. }, author = {Henheik, Sven Joscha and Teufel, Stefan}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, keywords = {computational mathematics, discrete mathematics and combinatorics, geometry and topology, mathematical physics, statistics and probability, algebra and number theory, theoretical computer science, analysis}, publisher = {Cambridge University Press}, title = {{Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk}}, doi = {10.1017/fms.2021.80}, volume = {10}, year = {2022}, } @article{11135, abstract = {We consider a correlated NxN Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one.}, author = {Reker, Jana}, issn = {2010-3271}, journal = {Random Matrices: Theory and Applications}, keywords = {Discrete Mathematics and Combinatorics, Statistics, Probability and Uncertainty, Statistics and Probability, Algebra and Number Theory}, number = {4}, publisher = {World Scientific}, title = {{On the operator norm of a Hermitian random matrix with correlated entries}}, doi = {10.1142/s2010326322500368}, volume = {11}, year = {2022}, } @article{11545, abstract = {We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category O to the Miličić–Soergel category N . We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings. We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard Whittaker modules. We show that costandard modules give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the associated flag variety, which are defined using holonomic duality of D-modules. We prove that with these costandard modules, blocks of category N have the structure of highest weight categories and we establish a BGG reciprocity theorem for N .}, author = {Brown, Adam and Romanov, Anna}, issn = {0021-8693}, journal = {Journal of Algebra}, keywords = {Algebra and Number Theory}, number = {11}, pages = {145--179}, publisher = {Elsevier}, title = {{Contravariant pairings between standard Whittaker modules and Verma modules}}, doi = {10.1016/j.jalgebra.2022.06.017}, volume = {609}, year = {2022}, } @article{12148, abstract = {We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, keywords = {Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Theoretical Computer Science, Analysis}, publisher = {Cambridge University Press}, title = {{Rank-uniform local law for Wigner matrices}}, doi = {10.1017/fms.2022.86}, volume = {10}, year = {2022}, } @article{12216, abstract = {Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.}, author = {Carlen, Eric A. and Zhang, Haonan}, issn = {0024-3795}, journal = {Linear Algebra and its Applications}, keywords = {Discrete Mathematics and Combinatorics, Geometry and Topology, Numerical Analysis, Algebra and Number Theory}, pages = {289--310}, publisher = {Elsevier}, title = {{Monotonicity versions of Epstein's concavity theorem and related inequalities}}, doi = {10.1016/j.laa.2022.09.001}, volume = {654}, year = {2022}, } @article{12309, abstract = {Take a rational elliptic curve defined by the equation y2=x3+ax in minimal form and consider the sequence Bn of the denominators of the abscissas of the iterate of a non-torsion point. We show that B5m has a primitive divisor for every m. Then, we show how to generalize this method to the terms of the form Bmp with p a prime congruent to 1 modulo 4.}, author = {Verzobio, Matteo}, issn = {0065-1036}, journal = {Acta Arithmetica}, keywords = {Algebra and Number Theory}, number = {2}, pages = {129--168}, publisher = {Institute of Mathematics, Polish Academy of Sciences}, title = {{Primitive divisors of elliptic divisibility sequences for elliptic curves with j=1728}}, doi = {10.4064/aa191016-30-7}, volume = {198}, year = {2021}, } @article{12308, abstract = {Let P and Q be two points on an elliptic curve defined over a number field K. For α∈End(E), define Bα to be the OK-integral ideal generated by the denominator of x(α(P)+Q). Let O be a subring of End(E), that is a Dedekind domain. We will study the sequence {Bα}α∈O. We will show that, for all but finitely many α∈O, the ideal Bα has a primitive divisor when P is a non-torsion point and there exist two endomorphisms g≠0 and f so that f(P)=g(Q). This is a generalization of previous results on elliptic divisibility sequences.}, author = {Verzobio, Matteo}, issn = {2522-0160}, journal = {Research in Number Theory}, keywords = {Algebra and Number Theory}, number = {2}, publisher = {Springer Nature}, title = {{Primitive divisors of sequences associated to elliptic curves with complex multiplication}}, doi = {10.1007/s40993-021-00267-9}, volume = {7}, year = {2021}, } @article{10711, abstract = {In this paper, we investigate the distribution of the maximum of partial sums of families of m -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of ℓ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of m -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.}, author = {Autissier, Pascal and Bonolis, Dante and Lamzouri, Youness}, issn = {1570-5846}, journal = {Compositio Mathematica}, keywords = {Algebra and Number Theory}, number = {7}, pages = {1610--1651}, publisher = {Cambridge University Press}, title = {{The distribution of the maximum of partial sums of Kloosterman sums and other trace functions}}, doi = {10.1112/s0010437x21007351}, volume = {157}, year = {2021}, } @article{12310, abstract = {Let be a sequence of points on an elliptic curve defined over a number field K. In this paper, we study the denominators of the x-coordinates of this sequence. We prove that, if Q is a torsion point of prime order, then for n large enough there always exists a primitive divisor. Later on, we show the link between the study of the primitive divisors and a Lang-Trotter conjecture. Indeed, given two points P and Q on the elliptic curve, we prove a lower bound for the number of primes p such that P is in the orbit of Q modulo p.}, author = {Verzobio, Matteo}, issn = {0022-314X}, journal = {Journal of Number Theory}, keywords = {Algebra and Number Theory}, number = {4}, pages = {378--390}, publisher = {Elsevier}, title = {{Primitive divisors of sequences associated to elliptic curves}}, doi = {10.1016/j.jnt.2019.09.003}, volume = {209}, year = {2020}, } @article{10874, abstract = {In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.}, author = {Ionica, Sorina and Kılıçer, Pınar and Lauter, Kristin and Lorenzo García, Elisa and Manzateanu, Maria-Adelina and Massierer, Maike and Vincent, Christelle}, issn = {2363-9555}, journal = {Research in Number Theory}, keywords = {Algebra and Number Theory}, publisher = {Springer Nature}, title = {{Modular invariants for genus 3 hyperelliptic curves}}, doi = {10.1007/s40993-018-0146-6}, volume = {5}, year = {2019}, }