Geometric and Topological Combinatorics

The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions

Algorithms and Lower Bounds in Circuit Complexity

Computational complexity theory aims to understand what problems can be efficiently solved by computation. This thesis studies computational complexity in the model of Boolean circuits. Boolean circuits provide a basic mathematical model for computation and play a central role in complexity theory, with important applications in separations of complexity classes, algorithm design, and pseudorandom constructions. In this thesis, we investigate various types of circuit models such as threshold circuits, Boolean formulas, and their extensions, focusing on obtaining complexity-theoretic lower bounds and algorithmic upper bounds for these circuits. (1) Algorithms and lower bounds for generalized threshold circuits: We extend the study of linear threshold circuits, circuits with gates computing linear threshold functions, to the more powerful model of polynomial threshold circuits where the gates can compute polynomial threshold functions. We obtain hardness and meta-algorithmic results for this circuit model, including strong average-case lower bounds, satisfiability algorithms, and derandomization algorithms for constant-depth polynomial threshold circuits with super-linear wire complexity. (2) Algorithms and lower bounds for enhanced formulas: We investigate the model of Boolean formulas whose leaf gates can compute complex functions. In particular, we study De Morgan formulas whose leaf gates are functions with "low communication complexity". Such gates can capture a broad class of functions including symmetric functions and polynomial threshold functions. We obtain new and improved results in terms of lower bounds and meta-algorithms (satisfiability, derandomization, and learning) for such enhanced formulas. (3) Circuit lower bounds for MCSP: We study circuit lower bounds for the Minimum Circuit Size Problem (MCSP), the fundamental problem of deciding whether a given function (in the form of a truth table) can be computed by small circuits. We get new and improved lower bounds for MCSP that nearly match the best-known lower bounds against several well-studied circuit models such as Boolean formulas and constant-depth circuits

Anatomy of quantum chaotic eigenstates

The eigenfunctions of quantized chaotic systems cannot be described by explicit formulas, even approximate ones. This survey summarizes (selected) analytical approaches used to describe these eigenstates, in the semiclassical limit. The levels of description are macroscopic (one wants to understand the quantum averages of smooth observables), and microscopic (one wants informations on maxima of eigenfunctions, "scars" of periodic orbits, structure of the nodal sets and domains, local correlations), and often focusses on statistical results. Various models of "random wavefunctions" have been introduced to understand these statistical properties, with usually good agreement with the numerical data. We also discuss some specific systems (like arithmetic ones) which depart from these random models.Comment: Corrected typos, added a few references and updated some result

The limiting spectral law for sparse iid matrices

Let $A$ be an $n\times n$ matrix with iid entries where $A_{ij} \sim \mathrm{Ber}(p)$ is a Bernoulli random variable with parameter $p = d/n$. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distribution as $n \rightarrow \infty$. This essentially resolves a long line of work to determine the spectral laws of iid matrices and is the first known example for non-Hermitian random matrices at this level of sparsity.Comment: 44 page

Zeros of Random Holomorphic Sections of Semipositive Line Bundles on Punctured Riemann Surfaces

With early works dating back to the 1930’s until today, here is a growing interest in the theory of asymptotic distributions of expected zeros of random polynomials when their degree grows indefinitely. A natural geometric generalization of random polynomials are random sections of a holomorphic line bundle over a complex manifold. In 1999, Shiffman and Zelditch proved that on a compact K¨ahler manifold, the zeros of sections of high tensor powers of a holomorphic line bundle asymptotically equidistribute with respect to the normalized curvature of the line bundle. Their result has numerous applications in mathematical physics and was generalized in many different directions. In this thesis we generalize their result to a semipositively curved holomorphic line bundle over a punctured Riemann surface. To achieve this, we discuss many tools that have proven themselves to represent an appropriate framework to study statistical properties of ensembles of zeros on complex manifolds. We start by proving the existence of spectral gap for the Kodaira Laplacian that is associated to the line bundle. We use this result, together with the technique of analytic localization by Ma and Marinescu, to prove a pointwise global on-diagonal asymptotic expansion of the associated Bergman kernel in our setting. Moreover, we show locally uniform estimates on the Bergman kernel and its derivatives. We use these estimates to prove the locally uniform convergence of the induced Fubini-Study metrics and their potentials to the global curvature and its potential, respectively. We conclude by showing that the expected zeros of holomorphic sections equidistribute with respect to the normalized curvature of the line bundle. Moreover, we apply the theory of meromorphic transforms by Dinh and Sibony estimate the speed of convergence in our equidistribution result