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A survey of general results on the singularities of inverses to meromorphic functions is given, with applications to holomorphic dynamics. This is a lecture delivered at the workshop "The role of complex analysis in complex dynamics" in…
Developing robust representations of chemical structures that enable models to learn topological inductive biases is challenging. In this manuscript, we present a representation of atomistic systems. We begin by proving that our…
By complexifying a Hamiltonian system one obtains dynamics on a holomorphic symplectic manifold. To invert this construction we present a theory of real forms which not only recovers the original system but also yields different real…
We present an explicit and computationally actionable blueprint for constructing vector-valued Siegel modular forms associated to real multiplication (RM) abelian surfaces, leveraging the theta correspondence for the unitary dual pair…
Transformers trained on huge text corpora exhibit a remarkable set of capabilities, e.g., performing basic arithmetic. Given the inherent compositional nature of language, one can expect the model to learn to compose these capabilities,…
Representation learning on text-attributed graphs (TAGs) integrates structural connectivity with rich textual semantics, enabling applications in diverse domains. Current methods largely rely on contrastive learning to maximize cross-modal…
Creating representations of shapes that are invari-ant to isometric or almost-isometric transforma-tions has long been an area of interest in shape anal-ysis, since enforcing invariance allows the learningof more effective and robust shape…
These are lecture notes for a course in Winter 2022/23, updated and completed in October 2025. The goal of the lectures is to present some recent developments around six-functor formalisms, in particular: the abstract theory of 6-functor…
This communication is an introduction to the Langlands Program and to ($G$-) shtukas (over algebraic curves) over function fields. Modular curves and Drinfeld (elliptic) modules and shtukas are used in coding theory. From this point of view…
We give an introduction to the structure theory of extended affine Lie algebras, which provide a common framework for finite-dimensional semisimple, affine and toroidal Lie algebras. The notes are based on a lecture series given during the…
Most recent work on interpretability of complex machine learning models has focused on estimating $\textit{a posteriori}$ explanations for previously trained models around specific predictions. $\textit{Self-explaining}$ models where…
This is an expository article based on my lectures on eigenfunctions of the Laplacian for the 2013 IAS/Park City Mathematics Institute (PCMI) summer school in geometric analysis. Many of the results are based on joint work with H.…
Learning expressive representations for high-dimensional yet sparse features has been a longstanding problem in information retrieval. Though recent deep learning methods can partially solve the problem, they often fail to handle the…
These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to…
The advent of representation learning methods enabled large performance gains on various language tasks, alleviating the need for manual feature engineering. While engineered representations are usually based on some linguistic…
Deploying machine learning models in safety-related do-mains (e.g. autonomous driving, medical diagnosis) demands for approaches that are explainable, robust against adversarial attacks and aware of the model uncertainty. Recent deep…
These are notes based on a series of lectures given at the KITP workshop "Quantum Criticality and the AdS/CFT Correspondence" in July, 2009. The goal of the lectures was to introduce condensed matter physicists to the AdS/CFT…
These notes were prepared for the introductory lectures on Higher Spin Holography presented at the Twelfth Modave Summer School in Mathematical Physics 2016, aimed at Ph.D. students and researchers new to this topic. Contents: 1. Higher…
A study of assisted problem solving formalized via decompositions of deterministic finite automata is initiated. The landscape of new types of decompositions of finite automata this study uncovered is presented. Languages with various…
Despite tremendous progress over the past decade, deep learning methods generally fall short of human-level systematic generalization. It has been argued that explicitly capturing the underlying structure of data should allow connectionist…