Quantum Geometry

Speaker: Murad Alim (TUM)

Abstract:

Quantum theory has not only reshaped our understanding of the physical world; it has also become a powerful source of ideas for modern mathematics. In this talk, I will introduce aspects of the emerging field of quantum geometry, where insights from quantum field theory and string theory interact with symplectic, complex, and algebraic geometry. I will explain how dualities in physical theories often reveal that seemingly different mathematical structures share common underlying principles, leading to deep new results and unexpected bridges between diverse areas. A central example is mirror symmetry, a duality relating symplectic and complex geometry with far-reaching consequences for enumerative geometry, representation theory and number theory.

We consider a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices.

After explaining that these Scattering Quantum Walks encompass several known Quantum Walks, we further introduce two classes of Scattering Open Quantum Walks on arbitrary graphs based on that construction, whose asymptotic states we discuss.

Real-life financial time series exhibit heavy tails and clusters of extreme values. In this talk we will address models that exhibit these stylized facts. This is the class of regularly varying time series, introduced by Davis and Hsing (1995, AoP) and further developed by Basrak and Segers (2009, SPA). The marginal distribution of a regularly varying time series has tails of power-law type, and the dynamics caused by an extreme event in this time series is described by the spectral tail process. The perhaps best known financial time series models of this kind are Engle’s (1982) ARCH process, Bollerslev’s (1986) GARCH process and Engle’s and Russell’s (1998) Autoregressive Conditional Duration (ACD) model. The length and magnitude of extremal clusters in such a series can be described by an analog of the autocorrelation function for extreme events: the extremogram. The extremal index is another useful tool for describing expected extremal cluster sizes. Both objects can be expressed in terms of the spectral tail process and allow for statistical estimation. The probabilistic and statistical aspects of regularly varying time series are summarized in the recent monograph by Mikosch and Wintenberger (2024) “Extreme Value Theory for Time Series. Models with Power-Law Tails”. The talk is based on joint work with Olivier Wintenberger (Sorbonne).

The rapid advances in artificial intelligence (AI) have largely been driven by scaling deep neural networks (DNNs) - increasing model size, data, and computational resources. However, performance is ultimately governed by network dynamics. The lack of a principled understanding of DNN dynamics beyond heuristic-based design has contributed to challenges with their robustness, suboptimal performance, high energy consumption and pathologies in continual and AI-generated content learning. In contrast, the human brain does not seem to suffer these problems, and converging evidence suggest that these benefits are achieved by dynamics being poised at a critical phase transition. Inspired by this principle, we propose that criticality provides a unifying framework linking structure, dynamics, and function also in DNNs. First, by analyzing more than 80 state-of-the-art models, we report that a decade of AI progress has implicitly driven successful networks towards criticality – explaining why certain architectures succeeded while others failed. Second, we demonstrate that incorporating criticality explicitly into training improves robustness and accuracy preventing key limitations of current models. Third, we show that catastrophic AI pathologies, including the performance degradation in continual learning and in model collapse - where performance degrades when training on AI-generated data - constitute a loss of critical dynamics. By maintaining networks at criticality, we provide a principled solution to this fundamental AI problem, demonstrating how criticality-based optimization mitigates performance degradation. This work highlights criticality as substrate-independent principle of intelligence, connecting AI advancement with core principles of brain function. It provides theoretical insights along with immediate practical value solving major AI challenges to ensure long-term DNN performance and resilience as models grow in scale and complexity.

Let π∈Π(μ,ν) be a coupling between two probability measures μ and ν on a Polish space. In this talk we propose and study a class of nonparametric measures of association between μ and ν, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between ν and the disintegration πx1 of π with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures μ and ν. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2022]. Our approach applies to probability laws on general Polish spaces.

Consider the regression problem where the response and the covariate are unmatched. Under this scenario, we do not have access to pairs of observations from their joint distribution, but instead we have separate data sets of responses and covariates, possibly collected from different sources. We study this problem assuming that the regression function is linear and the noise distribution is known or can be estimated. We introduce an estimator of the regression vector based on deconvolution (the DLSE) and demonstrate its consistency and asymptotic normality under parametric identifiability. Under non-identifiability of the regression vector but identifiability of the distribution of the predictor, we construct an estimator of the latter based on the DLSE and show that it converges to the true distribution of the predictor at the parametric rate in the Wasserstein distance of order 1. We illustrate the theory with several simulation results. \[ \] This talk is based on my joint work with Mona Azadkia, Antonio di Noia and Cecile Durot

In my talk I will consider the following question: how many random hyperplanes are needed to uniformly tessellate a given subset of Rn with high probability? In my talk I will present an optimal answer to this question for selected distributions for the random hyperplanes and sketch three applications of these results in the mathematical foundations of data science. First, I will show how to create a fast encoding of any given dataset into a minimal number of bits. Second, I will consider performance guarantees for one-bit compressed sensing methods, which aim to reconstruct a signal from a small number of measurements that are each quantized to a single bit using an efficient analog-to-digital converter. Third, I will discuss implications for the robustness of ReLU neural networks. The talk will be a survey-style presentation for a general mathematical audience. Based on joint works with Shahar Mendelson (ANU Canberra), Alexander Stollenwerk (Louvain), Patrick Finke, Nigel Strachan (Utrecht), Paul Geuchen, Dominik Stöger, Felix Voigtlaender (Eichstätt-Ingolstadt) ___________________________ Invited by Prof. Johannes Maly

It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter. These estimates are called local laws and they are the cornerstone in most of the recent results in random matrix theory. In this talk, I will present a novel method of proving single-resolvent and multi-resolvent local laws for random matrices, the Zigzag strategy, which is a recursive tandem of the characteristic flow method and a Green function comparison argument. Novel results, which we obtained via the Zigzag strategy, include the optimal Eigenstate Thermalization Hypothesis (ETH) for Wigner matrices, uniformly in the spectrum, and universality of eigenvalue statistics at cusp singularities for correlated random matrices. Based on joint works with G. Cipolloni, L. Erdös, O. Kolupaiev, and V. Riabov.

TBA