MediMath Science

This blog aims to help mathematicians understand Robert E. Hall and Michael Woodford’s contributions to the microeconomic foundations of macroeconomic stabilization, recognized by the 2026 Nobel Prize in Economic Sciences. I am not an expert in this field; as a mathematician learning about it, I wish to share the understanding I have developed with assistance from ChatGPT. The central mathematical structure is familiar: a household chooses a consumption path subject to constraints, and an interior optimum must be stationary under every admissible variation. The consumption Euler equation expresses this condition for transfers of resources between dates. The equation predates both laureates. Hall developed and empirically tested its implications for consumption, while Woodford integrated forward-looking household behavior with firms’ pricing decisions and economic equilibrium to analyze monetary policy. Begin with a deterministic economy at dates $t=0,\ldots,T$, where $T\geq1$. Let $c_t...

The 2026 Nobel Prize in Physiology or Medicine, awarded to Peter Hegemann, Georg Nagel, and Karl Deisseroth for discoveries concerning light-gated ion channels and optogenetics, brought me back to a research problem I worked on about 20 years ago. Our work was part of a broader effort to detect neural activity through changes in bioimpedance using electrical impedance tomography (EIT). The award reminded me that the opening of an ion channel can be approached both as something to measure and as something to control. Around 25 years ago, a clinical neurophysiologist at University College London explained the physical basis of this approach to me. When a neuron becomes active, ion channels open and its membrane conductance changes. The hope was that the resulting change in tissue impedance would allow researchers to follow the propagation of an action potential. A central challenge was obtaining reliable measurements of small, brief changes within signals influenced by many cells, extrac...

One of the most difficult skills in research is not learning how to develop an idea, but knowing when to abandon one. Scientific training naturally emphasizes persistence. Researchers are taught not to give up when experiments fail, reviewers disagree, or evidence appears inconsistent with an initial hypothesis. This persistence is essential. Most important discoveries would never have been made without it. Yet persistence has a less discussed counterpart. What happens when the idea itself is wrong? The problem becomes particularly difficult when a research program has developed over many years. A theory or methodology gradually accumulates papers, data, techniques, grants, students, collaborations, and professional recognition. What began as a hypothesis becomes an intellectual ecosystem. At that point, abandoning the idea is no longer a simple scientific decision. There are also powerful intellectual mechanisms that favor continuation. When observations conflict with a theory, it is ...

Recent discussions about slowing the development of artificial intelligence are often framed in dramatic terms: artificial general intelligence may become smarter than humans, recursively improve itself, and eventually escape human control. I find this framing somewhat misleading. Modern AI systems are not unconstrained intelligences. Transformer-based models operate within very real physical and computational limits: available compute, memory capacity and bandwidth, electrical power, training time, data quality, and the efficiency of optimization algorithms. Recursive self-improvement therefore does not imply that intelligence can increase without bound. The more interesting problem is different: What happens when an increasingly capable optimizer repeatedly optimizes an imperfect objective function without sufficiently independent external correction? We must consider AI as an optimization system. Consider a simplified learning system with parameters $\theta$ and objective function $...

Before beginning, I want to clarify what I mean by a “generalist.” I do not mean someone with superficial knowledge of many fields, but someone capable of handling a substantial range of work across multiple domains, though not with the depth of a specialist in each. This kind of generalist may become particularly valuable in aging societies, where experienced older people represent a large reservoir of accumulated knowledge and judgment that AI can help put to productive use. Modern science has been built on specialization. This specialization was necessary and enormously productive. As scientific knowledge expanded during the twentieth century, no individual could master everything. Mathematics, physics, engineering, biology, and medicine each divided into increasingly sophisticated subfields. By the latter part of the twentieth century, becoming a serious researcher often meant spending many years mastering a progressively narrower area of knowledge. This system produced extraor...

When I look at recent medical research from leading American institutions, I am often impressed by the intelligence, dedication, and technical sophistication of the researchers. Yet I have also felt increasingly uneasy about the direction in which our research system pushes these talented people. Too much intellectual energy is devoted to making already advanced medicine even more advanced, while comparatively little is devoted to making adequate medicine dramatically cheaper, simpler, and more accessible.  This is not an argument against frontier medical research. We absolutely need high-field MRI, photon-counting CT, advanced PET imaging, molecular diagnostics, precision oncology, and sophisticated medical AI. A small number of outstanding laboratories should continue pushing these technologies to their limits. Breakthrough research is inherently uncertain, and government support is essential because private markets will often underinvest in discoveries whose benefits may lie de...