倒计时2天 | 院士领衔大会报告

An alternative view on AI: Collaborative learning, incentives, and social welfare

Artificial intelligence (Al) has focused on a paradigm in which intelligence inheres in a single, autonomous agent. Social issues are entirely secondary in this paradigm. When Al systems are deployed in social contexts, however, the overall design of such systems is often naive--a centralized entity provides services to passive agents and reaps the rewards. Such a paradigm need not be the dominant paradigm for information technology. In a broader framing, agents are active, they are cooperative, and they wish to obtain value from their participation in learning-based systems. Agents may supply data and other resources to the system, only if it is in their interest to do so. Critically, intelligence inheres as much in the overall system as it does in individual agents, be they humans or computers. This is a perspective familiar in the social sciences and a first goal in this line of work is to bring economics into contact with the computing and data sciences. The long-term goal is two-fold--to provide a broader conceptual foundation for emerging real-world Al systems, and to upend received wisdom in the computational, economic, and inferential disciplines.

The AAA algorithm for rational approximation

With this new algorithm, approximation by rational functions has become fast and practical in a way it was not before. We will discuss many examples and applications including the solution of partial differential equations.

Deep Approximation via Deep Learning

The primary objective of many applications is to approximate or estimate a function using samples obtained from a probability distribution on the input space. Deep approximation involves approximating a function by composing numerous layers of simple functions, which can be seen as a sequence of nested feature extractors. The fundamental concept of deep learning networks is to convert these layers of compositions into layers of adjustable parameters that can be fine-tuned through a learning process, ultimately achieving a high-quality approximation based on the input data. In this presentation, we will delve into the mathematical theory behind this innovative approach and explore the approximation rate of deep networks. Additionally, we will highlight the distinctions between this new approach and traditional approximation theory, and demonstrate how this novel theory can be leveraged to comprehend and design deep learning network.

An infinity category and Floer homology

Floer homology is a half infinite dimensional homology theory invented by A Floer 40 years ago. The further study of the structure in Floer theory shows a kind of higher category theory is useful. I will survey this topic including its application to Symlectic Geometry and Gauge theory.

How to build a random surface

The theory of "random surfaces" has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, combinatorics, analysis and mathematical physics. Just as Brownian motion is a special kind of random path, there is a similarly special kind of random surface, which is characterized by special symmetries, and which arises in many different contexts. Random surfaces are often motivated by physics: statistical physics, string theory, quantum field theory, and so forth. They have also been independently studied by mathematicians working in random matrix theory and enumerative graph theory. But even without that motivation, one may be drawn to wonder what a "typical" two-dimensional manifold look likes, or how one can make sense of that question. I will give a broad overview of what this theory is about, including many computer simulations and illustrations. ln particular, I will highlight some recent works with Jason Miller in which we proved the equivalence of Liouville quantum gravity and the Brownian sphere -- two random surface models that were historically defined in completely different ways.

Towards a geometric theory of cohomology functors: the case of degree 0

We will explain how Olivia Caramello's proposed approach for constructing a Galois-type theory of cohomology functors can be implemented and fully verified in the case of cohomology of degree 0. The models of this theory-or equivalently the points of the associated Classifying Topos-are exactly cohomology functors of degree 0. As its Classifying Topos is Galois, this theory is complete, which means that all its models share the same geometric-logic properties. In particular, their components at all different geometric objects all have the same dimensions and the same algebraic structures. These results are already non-trivial and can be considered as toy-models for cohomology in higher degrees, which is the objective of Caramello's proposed approach.

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