Seminários, para a disseminação informal de resultados de investigação, trabalho exploratório de equipas de investigação, actividades de difusão, etc., constituem a forma mais simples de encontros num centro de investigação de matemática.
O CAMGSD regista e publica o calendário dos seus seminários há bastante tempo, servindo páginas como esta não só como um método de anúncio dessas actividades mas também como um registo histórico.
We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with $N $ minima on $S^{1}$. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of $N$-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with 't Hooft anomaly for even $N$ and global inconsistency for odd $N$. By using Delabaere Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.
This talk is based on:
On exact-WKB analysis, resurgent structure, and quantization conditions, N. Sueishi, S. K., T. Misumi, and M. Ünsal, arXiv:2008.00379.
Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$, N. Sueishi, S.K., T. Misumi, and M. Ünsal, arXiv.2103.06586
Topological states of matter have fascinated physicists since a long time. The notion of topology is however ususally associated with ground states of (many-body)-Hamiltonians, which are pure. So what is left of it at finite temperatures and can topological protection be extended to non-equilibrium steady states (NESS) of open systems? Can suitable observables be constructed that preserve the integrity of topological invariants for mixed states and what are measurable consequences of their existence? Can we classify the topology of finite temperature and NESS using generalized symmetries? Motivated by topological charge pumps, first introduced by Thouless, I will first discuss a topological invariant for systems that break time reversal symmetry based on the many-body polarization, called ensemble geometric phase (EGP) [1]. In contrast to charge transport, the EGP can be used to probe topology in one dimensional non-interacting [2] and interacting [3], closed and open systems alike. Furthermore different from other constructions, such as the Uhlmann phase, it can be extended to two dimensions [4]. I will then extend the definition to systems with time-reversal symmetry and finally talk about measurable consequences of mixed-states topological invariants.
[1] C.E. Bardyn, L. Wawer, A. Altland, M. Fleischhauer, S.Diehl, (PRX 2018) [2] D. Linzner, L. Wawer, F. Grusdt, M. Fleischhauer, (PRB 2016) [3] R. Unanyan, M. Kiefer-Emmanouilidis, M. Fleischhauer, (PRL 2020) [4] L. Wawer, M. Fleischhauer, arxiv 2104.12115
The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In a new version of a joint work with Enrica Mazzon and Matthew Stevenson, we prove the full geometric P=W conjecture for elliptic curves: this is the first non-trivial evidence of the conjecture for compact Riemann surfaces. As a byproduct, we show that certain character varieties appear in degenerations of compact hyper-Kähler manifolds. We also explain how the geometric P=W conjecture by Katzarkov-Noll-Pandit-Simpson is related to the cohomological version formulated by de Cataldo-Hausel-Migliorini.
Computational imaging is a rapidly growing area that seeks to enhance the capabilities of imaging instruments by viewing imaging as an inverse problem. There are currently two distinct approaches for designing computational imaging methods: model-based and learning-based. Model-based methods leverage analytical signal properties and often come with theoretical guarantees and insights. Learning-based methods leverage data-driven representations for best empirical performance through training on large datasets. This talk presents Regularization by Artifact Removal (RARE), as a framework for reconciling both viewpoints by providing a learning-based extension to the classical theory. RARE relies on pre-trained “artifact-removing deep neural nets” for infusing learned prior knowledge into an inverse problem, while maintaining a clear separation between the prior and physics-based acquisition model. Our results indicate that RARE can achieve state-of-the-art performance in different computational imaging tasks, while also being amenable to rigorous theoretical analysis. We will focus on the applications of RARE in biomedical imaging, including magnetic resonance and tomographic imaging.
This talk will be based on the following references
J. Liu, Y. Sun, C. Eldeniz, W. Gan, H. An, and U. S. Kamilov, “RARE: Image Reconstruction using Deep Priors Learned without Ground Truth,” IEEE J. Sel. Topics Signal Process., vol. 14, no. 6, pp. 1088-1099, October 2020.
Z. Wu, Y. Sun, A. Matlock, J. Liu, L. Tian, and U. S. Kamilov, “SIMBA: Scalable Inversion in Optical Tomography using Deep Denoising Priors,” IEEE J. Sel. Topics Signal Process., vol. 14, no. 6, pp. 1163-1175, October 2020.
J. Liu, Y. Sun, W. Gan, X. Xu, B. Wohlberg, and U. S. Kamilov, “SGD-Net: Efficient Model-Based Deep Learning with Theoretical Guarantees,” IEEE Trans. Comput. Imag., in press.
Tensor network states provide a comprehensive framework for the analytic and numerical study of strongly correlated many-body systems. In recent years, this framework has been successfully applied to topological phases of matter. In this talk, I will present two dual tensor network representations of the (3+1)d toric code ground state subspace, which are obtained by initially imposing either family of stabilizer constraints. I will discuss topological properties of the model from the point of view of these virtual symmetries, demonstrate that one of these representations is stable to all local tensor perturbations---including those that do not map to local operators on the physical Hilbert space---and explain, both from a physical and category theoretical viewpoint, how the distinguishing properties of these representations are related to the phenomenon of bulk-boundary correspondence.
This work develops a class of relaxations in between the big-M and convex hull formulations of disjunctions, drawing advantages from both. We show that this class leads to mixed-integer formulations for trained ReLU neural networks. The approach balances model size and tightness by partitioning node inputs into a number of groups and forming the convex hull over the partitions via disjunctive programming. At one extreme, one partition per input recovers the convex hull of a node, i.e., the tightest possible formulation for each node. For fewer partitions, we develop smaller relaxations that approximate the convex hull, and show that they outperform existing formulations. Specifically, we propose strategies for partitioning variables based on theoretical motivations and validate these strategies using extensive computational experiments. Furthermore, the proposed scheme complements known algorithmic approaches, e.g., optimization-based bound tightening captures dependencies within a partition.
Each fusion higher category has a "framed S-matrix" which encodes the commutator of operators of complementary dimension. I will explain how to construct and interpret this pairing, and I will emphasize that it may fail to exist if you drop semisimplicity requirements. I will then outline a proof that the framed S-matrix detects (non)degeneracy of the fusion higher category. This is joint work in progress with David Reutter.
