@phdthesis{9022,
abstract = {In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime.
In the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time
(up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions.
In the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result
improves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices.},
author = {Cipolloni, Giorgio},
issn = {2663-337X},
pages = {380},
publisher = {IST Austria},
title = {{Fluctuations in the spectrum of random matrices}},
doi = {10.15479/AT:ISTA:9022},
year = {2021},
}
@phdthesis{9056,
abstract = {In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations.
The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density,
and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics.
We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration
function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss
the implications on persistence for periodic data sets.},
author = {Osang, Georg F},
issn = {2663-337X},
pages = {134},
publisher = {IST Austria},
title = {{Multi-cover persistence and Delaunay mosaics}},
doi = {10.15479/AT:ISTA:9056},
year = {2021},
}
@phdthesis{8934,
abstract = {In this thesis, we consider several of the most classical and fundamental problems in static analysis and formal verification, including invariant generation, reachability analysis, termination analysis of probabilistic programs, data-flow analysis, quantitative analysis of Markov chains and Markov decision processes, and the problem of data packing in cache management.
We use techniques from parameterized complexity theory, polyhedral geometry, and real algebraic geometry to significantly improve the state-of-the-art, in terms of both scalability and completeness guarantees, for the mentioned problems. In some cases, our results are the first theoretical improvements for the respective problems in two or three decades.},
author = {Goharshady, Amir Kafshdar},
issn = {2663-337X},
pages = {278},
publisher = {IST Austria},
title = {{Parameterized and algebro-geometric advances in static program analysis}},
doi = {10.15479/AT:ISTA:8934},
year = {2021},
}
@phdthesis{7996,
abstract = {Quantum computation enables the execution of algorithms that have exponential complexity. This might open the path towards the synthesis of new materials or medical drugs, optimization of transport or financial strategies etc., intractable on even the fastest classical computers. A quantum computer consists of interconnected two level quantum systems, called qubits, that satisfy DiVincezo’s criteria. Worldwide, there are ongoing efforts to find the qubit architecture which will unite quantum error correction compatible single and two qubit fidelities, long distance qubit to qubit coupling and
calability. Superconducting qubits have gone the furthest in this race, demonstrating an algorithm running on 53 coupled qubits, but still the fidelities are not even close to those required for realizing a single logical qubit. emiconductor qubits offer extremely good characteristics, but they are currently investigated across different platforms. Uniting those good characteristics into a single platform might be a big step towards the quantum computer realization.
Here we describe the implementation of a hole spin qubit hosted in a Ge hut wire double quantum dot. The high and tunable spin-orbit coupling together with a heavy hole state character is expected to allow fast spin manipulation and long coherence times. Furthermore large lever arms, for hut wire devices, should allow good coupling to superconducting resonators enabling efficient long distance spin to spin coupling and a sensitive gate reflectometry spin readout. The developed cryogenic setup (printed circuit board sample holders, filtering, high-frequency wiring) enabled us to perform low temperature spin dynamics experiments. Indeed, we measured the fastest single spin qubit Rabi frequencies reported so far, reaching 140 MHz, while the dephasing times of 130 ns oppose the long decoherence predictions. In order to further investigate this, a double quantum dot gate was connected directly to a lumped element
resonator which enabled gate reflectometry readout. The vanishing inter-dot transition signal, for increasing external magnetic field, revealed the spin nature of the measured quantity.},
author = {Kukucka, Josip},
issn = {2663-337X},
pages = {178},
publisher = {IST Austria},
title = {{Implementation of a hole spin qubit in Ge hut wires and dispersive spin sensing}},
doi = {10.15479/AT:ISTA:7996},
year = {2020},
}
@phdthesis{8032,
abstract = {Algorithms in computational 3-manifold topology typically take a triangulation as an input and return topological information about the underlying 3-manifold. However, extracting the desired information from a triangulation (e.g., evaluating an invariant) is often computationally very expensive. In recent years this complexity barrier has been successfully tackled in some cases by importing ideas from the theory of parameterized algorithms into the realm of 3-manifolds. Various computationally hard problems were shown to be efficiently solvable for input triangulations that are sufficiently “tree-like.”
In this thesis we focus on the key combinatorial parameter in the above context: we consider the treewidth of a compact, orientable 3-manifold, i.e., the smallest treewidth of the dual graph of any triangulation thereof. By building on the work of Scharlemann–Thompson and Scharlemann–Schultens–Saito on generalized Heegaard splittings, and on the work of Jaco–Rubinstein on layered triangulations, we establish quantitative relations between the treewidth and classical topological invariants of a 3-manifold. In particular, among other results, we show that the treewidth of a closed, orientable, irreducible, non-Haken 3-manifold is always within a constant factor of its Heegaard genus.},
author = {Huszár, Kristóf},
isbn = {978-3-99078-006-0},
issn = {2663-337X},
pages = {xviii+120},
publisher = {IST Austria},
title = {{Combinatorial width parameters for 3-dimensional manifolds}},
doi = {10.15479/AT:ISTA:8032},
year = {2020},
}