Digital data capture is the backbone of all modern day systems and “Digital Revolution” has been aptly termed as the Third Industrial Revolution. Underpinning the digital representation is the Shannon-Nyquist sampling theorem and more recent developments include compressive sensing approaches. The fact that there is a physical limit to which sensors can measure amplitudes poses a fundamental bottleneck when it comes to leveraging the performance guaranteed by recovery algorithms. In practice, whenever a physical signal exceeds the maximum recordable range, the sensor saturates, resulting in permanent information loss. Examples include (a) dosimeter saturation during the Chernobyl reactor accident, reporting radiation levels far lower than the true value and (b) loss of visual cues in self-driving cars coming out of a tunnel (due to sudden exposure to light). 

To reconcile this gap between theory and practice, we introduce the Unlimited Sensing framework or the USF that is based on a co-design of hardware and algorithms. On the hardware front, our work is based on a radically different analog-to-digital converter (ADC) design, which allows for the ADCs to produce modulo or folded samples. On the algorithms front, we develop new, mathematically guaranteed recovery strategies.  

In the first part of this talk, we prove a sampling theorem akin to the Shannon-Nyquist criterion. We show that, remarkably, despite the non-linearity in sensing pipeline, the sampling rate only depends on the signal’s bandwidth. Our theory is complemented with a stable recovery algorithm. Beyond the theoretical results, we will also present a hardware demo that shows our approach in action.

Moving further, we reinterpret the unlimited sensing framework as a generalized linear model that motivates a new class of inverse problems. We conclude this talk by presenting new results in the context of single-shot high-dynamic-range (HDR) imaging, sensor array processing and HDR tomography based on the modulo Radon transform.

Zilber's Restricted Trichotomy Conjecture predicts that every sufficiently rich strongly minimal structure which can be interpreted from an algebraically closed field K, must itself interpret K. Progress toward this conjecture began in 1993 with the work of Rabinovich, and recently Hasson and Sustretov gave a full proof for structures with universe of dimension 1. In this talk I will discuss a partial result in characteristic zero for universes of dimension greater than 1: namely, the conjecture holds in this case under certain geometric restrictions on definable sets. Time permitting, I will discuss how this result implies the full conjecture for expansions of abelian varieties.

I will connect approaches to classical integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations. In particular, I will consider holomorphic Chern-Simons theory on twistor space, defined using a range of meromorphic (3,0)-forms. On shell these are, in most cases, found to agree with actions for anti-self-dual Yang-Mills theory on space-time. Under symmetry reduction, these space-time actions yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces the holomorphic Chern-Simons action to 4d Chern-Simons theory.

Continuous product systems were introduced and studied by Arveson in the late 1980s. The study of their discrete analogues started with the work of Dinh in the 1990’s and it was formalized by Fowler in 2002. Discrete product systems are semigroup versions of C*-correspondences, that allow for a joint study of many fundamental C*-algebras, including those which come from C*-correspondences, higher rank graphs and elsewhere.
Katsura’s covariant relations have been proven to give the correct Cuntz-type C*-algebra for a single C*-correspondence X. One of the great advantages of Katsura's Cuntz-Pimsner C*-algebra is its co-universality for the class of gauge-compatible injective representations of X. In the late 2000s Carlsen-Larsen-Sims-Vittadello raised the question of existence of such a co-universal object in the context of product systems. In their work, Carlsen-Larsen-Sims-Vittadello provided an affirmative answer for quasi-lattices, with additional injectivity assumptions on X. The general case has remained open and will be addressed in these talk using tools from non-selfadjoint operator algebra theory.

The recent work on nc convex sets of Davidson, Kennedy, and Shamovich show that there is a rich interplay between the category of operator systems and the category of compact nc convex sets, leading to new insights even in the case of C*-algebras. The category of nc convex sets are a generalization of the usual notion of a compact convex set that provides meaningful connections between convex theoretic notions and notions in operator system theory. In this talk, we present a duality theorem for norm closed self-adjoint subspaces of B(H) that do not necessarily contain the unit. Using this duality, we will describe various C*-algebraic and operator system theoretic notions such as simplicity and subkernels in terms of their convex structure. This is joint work with Matthew Kennedy and Nicholas Manor.