Mathematical Foundations of Artificial Intelligence (MFAI)
NSF support for interdisciplinary collaborations developing mathematical and theoretical foundations for explainable, reliable, sustainable and trustworthy artificial intelligence.
Simons Fellows in Mathematics
Sabbatical fellowship providing salary replacement (up to 50% of academic-year salary, capped at $125,000) plus up to $10,000 in leave-related expenses so mathematicians can extend a sabbatical term to a full academic year; maximum total budget $162,000 including 20% overhead.
Targeted Grants to Institutes
Supports established institutes and research centers in mathematics and the physical sciences to strengthen contacts within the international scientific community; up to $750,000 (inclusive of 20% indirect costs) over a three-year award period, with up to ten grants each year.
Simons Symposia 2026 call for proposals
The Simons Foundation invites proposals to organize new symposium series in mathematics and theoretical physics, including theoretical computer science. Selected series may comprise up to three weeklong meetings with travel, lodging, and meals covered for participants, creating sustained time for collaboration on timely research problems.
Briefing Chat: Sweet! Elusive sugar molecules found in space
Nature staff discuss the discovery of a four-carbon sugar molecule in interstellar space and a newly decoded formula that identifies a Maya mathematician.
Computational modelling of ocular pharmacokinetics
Pharmacokinetics in the eye is an important factor for the success of ocular drug delivery and treatment. Pharmacokinetic features determine the feasible routes of drug administration, dosing levels and intervals, and it has impact on eventual drug responses. Several physical, biochemical, and flow-related barriers limit drug exposure of anterior and posterior ocular target tissues during treatment during local (topical, subconjunctival, intravitreal) and systemic administration (intravenous, per oral). Mathematical models integrate joint impact of various barriers on ocular pharmacokinetics (PKs) thereby helping drug development. The models are useful in describing (top-down) and predicting (bottom-up) pharmacokinetics of ocular drugs. This is useful also in the design and development of new drug molecules and drug delivery systems. Furthermore, the models can be used for interspecies translation and probing of disease effects on pharmacokinetics. In this lecture, ocular pharmacokinetics and current modelling methods (noncompartmental analyses, compartmental, physiologically based, and finite element models) are introduced. Future challenges are also highlighted (e.g. intra-tissue distribution, prediction of drug responses, active transport).
Why age-related macular degeneration is a mathematically tractable disease
Among all prevalent diseases with a central neurodegeneration, AMD can be considered the most promising in terms of prevention and early intervention, due to several factors surrounding the neural geometry of the foveal singularity. • Steep gradients of cell density, deployed in a radially symmetric fashion, can be modeled with a difference of Gaussian curves. • These steep gradients give rise to huge, spatially aligned biologic effects, summarized as the Center of Cone Resilience, Surround of Rod Vulnerability. • Widely used clinical imaging technology provides cellular and subcellular level information. • Data are now available at all timelines: clinical, lifespan, evolutionary • Snapshots are available from tissues (histology, analytic chemistry, gene expression) • A viable biogenesis model exists for drusen, the largest population-level intraocular risk factor for progression. • The biogenesis model shares molecular commonality with atherosclerotic cardiovascular disease, for which there has been decades of public health success. • Animal and cell model systems are emerging to test these ideas.
Mathematical and computational modelling of ocular hemodynamics: from theory to applications
Changes in ocular hemodynamics may be indicative of pathological conditions in the eye (e.g. glaucoma, age-related macular degeneration), but also elsewhere in the body (e.g. systemic hypertension, diabetes, neurodegenerative disorders). Thanks to its transparent fluids and structures that allow the light to go through, the eye offers a unique window on the circulation from large to small vessels, and from arteries to veins. Deciphering the causes that lead to changes in ocular hemodynamics in a specific individual could help prevent vision loss as well as aid in the diagnosis and management of diseases beyond the eye. In this talk, we will discuss how mathematical and computational modelling can help in this regard. We will focus on two main factors, namely blood pressure (BP), which drives the blood flow through the vessels, and intraocular pressure (IOP), which compresses the vessels and may impede the flow. Mechanism-driven models translates fundamental principles of physics and physiology into computable equations that allow for identification of cause-to-effect relationships among interplaying factors (e.g. BP, IOP, blood flow). While invaluable for causality, mechanism-driven models are often based on simplifying assumptions to make them tractable for analysis and simulation; however, this often brings into question their relevance beyond theoretical explorations. Data-driven models offer a natural remedy to address these short-comings. Data-driven methods may be supervised (based on labelled training data) or unsupervised (clustering and other data analytics) and they include models based on statistics, machine learning, deep learning and neural networks. Data-driven models naturally thrive on large datasets, making them scalable to a plethora of applications. While invaluable for scalability, data-driven models are often perceived as black- boxes, as their outcomes are difficult to explain in terms of fundamental principles of physics and physiology and this limits the delivery of actionable insights. The combination of mechanism-driven and data-driven models allows us to harness the advantages of both, as mechanism-driven models excel at interpretability but suffer from a lack of scalability, while data-driven models are excellent at scale but suffer in terms of generalizability and insights for hypothesis generation. This combined, integrative approach represents the pillar of the interdisciplinary approach to data science that will be discussed in this talk, with application to ocular hemodynamics and specific examples in glaucoma research.
Computational and mathematical approaches to myopigenesis
Myopia is predicted to affect 50% of all people worldwide by 2050, and is a risk factor for significant, potentially blinding ocular pathologies, such as retinal detachment and glaucoma. Thus, there is significant motivation to better understand the process of myopigenesis and to develop effective anti-myopigenic treatments. In nearly all cases of human myopia, scleral remodeling is an obligate step in the axial elongation that characterizes the condition. Here I will describe the development of a biomechanical assay based on transient unconfined compression of scleral samples. By treating the scleral as a poroelastic material, one can determine scleral biomechanical properties from extremely small samples, such as obtained from the mouse eye. These properties provide proxy measures of scleral remodeling, and have allowed us to identify all-trans retinoic acid (atRA) as a myopigenic stimulus in mice. I will also describe nascent collaborative work on modeling the transport of atRA in the eye.
Computational models and experimental methods for the human cornea
The eye is a multi-component biological system, where mechanics, optics, transport phenomena and chemical reactions are strictly interlaced, characterized by the typical bio-variability in sizes and material properties. The eye’s response to external action is patient-specific and it can be predicted only by a customized approach, that accounts for the multiple physics and for the intrinsic microstructure of the tissues, developed with the aid of forefront means of computational biomechanics. Our activity in the last years has been devoted to the development of a comprehensive model of the cornea that aims at being entirely patient-specific. While the geometrical aspects are fully under control, given the sophisticated diagnostic machinery able to provide a fully three-dimensional images of the eye, the major difficulties are related to the characterization of the tissues, which require the setup of in-vivo tests to complement the well documented results of in-vitro tests. The interpretation of in-vivo tests is very complex, since the entire structure of the eye is involved and the characterization of the single tissue is not trivial. The availability of micromechanical models constructed from detailed images of the eye represents an important support for the characterization of the corneal tissues, especially in the case of pathologic conditions. In this presentation I will provide an overview of the research developed in our group in terms of computational models and experimental approaches developed for the human cornea.
Cognitive supports for analogical reasoning in rational number understanding
In cognitive development, learning more than the input provides is a central challenge. This challenge is especially evident in learning the meaning of numbers. Integers – and the quantities they denote – are potentially infinite, as are the fractional values between every integer. Yet children’s experiences of numbers are necessarily finite. Analogy is a powerful learning mechanism for children to learn novel, abstract concepts from only limited input. However, retrieving proper analogy requires cognitive supports. In this talk, I seek to propose and examine number lines as a mathematical schema of the number system to facilitate both the development of rational number understanding and analogical reasoning. To examine these hypotheses, I will present a series of educational intervention studies with third-to-fifth graders. Results showed that a short, unsupervised intervention of spatial alignment between integers and fractions on number lines produced broad and durable gains in fractional magnitudes. Additionally, training on conceptual knowledge of fractions – that fractions denote magnitude and can be placed on number lines – facilitates explicit analogical reasoning. Together, these studies indicate that analogies can play an important role in rational number learning with the help of number lines as schemas. These studies shed light on helpful practices in STEM education curricula and instructions.
Unique features of oxygen delivery to the mammalian retina
Like all neural tissue, the retina has a high metabolic demand, and requires a constant supply of oxygen. Second and third order neurons are supplied by the retinal circulation, whose characteristics are similar to brain circulation. However, the photoreceptor region, which occupies half of the retinal thickness, is avascular, and relies on diffusion of oxygen from the choroidal circulation, whose properties are very different, as well as the retinal circulation. By fitting diffusion models to oxygen measurements made with oxygen microelectrodes, it is possible to understand the relative roles of the two circulations under normal conditions of light and darkness, and what happens if the retina is detached or the retinal circulation is occluded. Most of this work has been done in vivo in rat, cat, and monkey, but recent work in the isolated mouse retina will also be discussed.
Learning by Analogy in Mathematics
Analogies between old and new concepts are common during classroom instruction. While previous studies of transfer focus on how features of initial learning guide later transfer to new problem solving, less is known about how to best support analogical transfer from previous learning while children are engaged in new learning episodes. Such research may have important implications for teaching and learning in mathematics, which often includes analogies between old and new information. Some existing research promotes supporting learners' explicit connections across old and new information within an analogy. In this talk, I will present evidence that instructors can invite implicit analogical reasoning through warm-up activities designed to activate relevant prior knowledge. Warm-up activities "close the transfer space" between old and new learning without additional direct instruction.
Comparing Multiple Strategies to Improve Mathematics Learning and Teaching
Comparison is a powerful learning process that improves learning in many domains. For over 10 years, my colleagues and I have researched how we can use comparison to support better learning of school mathematics within classroom settings. In 5 short-term experimental, classroom-based studies, we evaluated comparison of solution methods for supporting mathematics knowledge and tested whether prior knowledge impacted effectiveness. We next developed supplemental Algebra I curriculum and professional development for teachers to integrate Comparison and Explanation of Multiple Strategies (CEMS) in their classrooms and tested the promise of the approach when implemented by teachers in two studies. Benefits and challenges emerged in these studies. I will conclude with evidence-based guidelines for effectively supporting comparison and explanation in the classroom. Overall, this program of research illustrates how cognitive science research can guide the design of effective educational materials as well as challenges that occur when bridging from cognitive science research to classroom instruction.
Dr Lindsay reads from "Models of the Mind : How Physics, Engineering and Mathematics Shaped Our Understanding of the Brain" 📖
Though the term has many definitions, computational neuroscience is mainly about applying mathematics to the study of the brain. The brain—a jumble of all different kinds of neurons interconnected in countless ways that somehow produce consciousness—has been described as “the most complex object in the known universe”. Physicists for centuries have turned to mathematics to properly explain some of the most seemingly simple processes in the universe—how objects fall, how water flows, how the planets move. Equations have proved crucial in these endeavors because they capture relationships and make precise predictions possible. How could we expect to understand the most complex object in the universe without turning to mathematics? — The answer is we can’t, and that is why I wrote this book. While I’ve been studying and working in the field for over a decade, most people I encounter have no idea what “computational neuroscience” is or that it even exists. Yet a desire to understand how the brain works is a common and very human interest. I wrote this book to let people in on the ways in which the brain will ultimately be understood: through mathematical and computational theories. — At the same time, I know that both mathematics and brain science are on their own intimidating topics to the average reader and may seem downright prohibitory when put together. That is why I’ve avoided (many) equations in the book and focused instead on the driving reasons why scientists have turned to mathematical modeling, what these models have taught us about the brain, and how some surprising interactions between biologists, physicists, mathematicians, and engineers over centuries have laid the groundwork for the future of neuroscience. — Each chapter of Models of the Mind covers a separate topic in neuroscience, starting from individual neurons themselves and building up to the different populations of neurons and brain regions that support memory, vision, movement and more. These chapters document the history of how mathematics has woven its way into biology and the exciting advances this collaboration has in store.
MidsummerBrains - computational neuroscience from my point of view
Computational neuroscience is a highly interdisciplinary field ranging from mathematics, physics and engineering to biology, medicine and psychology. Interdisciplinary collaborations have resulted in many groundbreaking innovations both in the research and application. The basis for successful collaborations is the ability to communicate across disciplines: What projects are the others working on? Which techniques and methods are they using? How is data collected, used and stored? In this webinar series, several experts describe their view on computational neuroscience in theory and application, and share experiences they had with interdisciplinary projects. This webinar is open for all interested students and researchers. If you are interested to participate live, please send a short message to smartstart@fz-juelich.de Please note, these lectures will be recorded for subsequent publishing as online lecture material.
MidsummerBrains - computational neuroscience from my point of view
Computational neuroscience is a highly interdisciplinary field ranging from mathematics, physics and engineering to biology, medicine and psychology. Interdisciplinary collaborations have resulted in many groundbreaking innovations both in the research and application. The basis for successful collaborations is the ability to communicate across disciplines: What projects are the others working on? Which techniques and methods are they using? How is data collected, used and stored? In this webinar series, several experts describe their view on computational neuroscience in theory and application, and share experiences they had with interdisciplinary projects. This webinar is open for all interested students and researchers. If you are interested to participate live, please send a short message to smartstart@fz-juelich.de Please note, these lectures will be recorded for subsequent publishing as online lecture material.
MidsummerBrains - computational neuroscience from my point of view
Computational neuroscience is a highly interdisciplinary field ranging from mathematics, physics and engineering to biology, medicine and psychology. Interdisciplinary collaborations have resulted in many groundbreaking innovations both in the research and application. The basis for successful collaborations is the ability to communicate across disciplines: What projects are the others working on? Which techniques and methods are they using? How is data collected, used and stored? In this webinar series, several experts describe their view on computational neuroscience in theory and application, and share experiences they had with interdisciplinary projects. This webinar is open for all interested students and researchers. If you are interested to participate live, please send a short message to smartstart@fz-juelich.de Please note, these lectures will be recorded for subsequent publishing as online lecture material.
MidsummerBrains - computational neuroscience from my point of view
Computational neuroscience is a highly interdisciplinary field ranging from mathematics, physics and engineering to biology, medicine and psychology. Interdisciplinary collaborations have resulted in many groundbreaking innovations both in the research and application. The basis for successful collaborations is the ability to communicate across disciplines: What projects are the others working on? Which techniques and methods are they using? How is data collected, used and stored? In this webinar series, several experts describe their view on computational neuroscience in theory and application, and share experiences they had with interdisciplinary projects. This webinar is open for all interested students and researchers. If you are interested to participate live, please send a short message to smartstart@fz-juelich.de Please note, these lectures will be recorded for subsequent publishing as online lecture material.
MidsummerBrains - computational neuroscience from my point of view
Computational neuroscience is a highly interdisciplinary field ranging from mathematics, physics and engineering to biology, medicine and psychology. Interdisciplinary collaborations have resulted in many groundbreaking innovations both in the research and application. The basis for successful collaborations is the ability to communicate across disciplines: What projects are the others working on? Which techniques and methods are they using? How is data collected, used and stored? In this webinar series, several experts describe their view on computational neuroscience in theory and application, and share experiences they had with interdisciplinary projects. This webinar is open for all interested students and researchers. If you are interested to participate live, please send a short message to smartstart@fz-juelich.de Please note, these lectures will be recorded for subsequent publishing as online lecture material.
MidsummerBrains - computational neuroscience from my point of view
Computational neuroscience is a highly interdisciplinary field ranging from mathematics, physics and engineering to biology, medicine and psychology. Interdisciplinary collaborations have resulted in many groundbreaking innovations both in the research and application. The basis for successful collaborations is the ability to communicate across disciplines: What projects are the others working on? Which techniques and methods are they using? How is data collected, used and stored? In this webinar series, several experts describe their view on computational neuroscience in theory and application, and share experiences they had with interdisciplinary projects. This webinar is open for all interested students and researchers. If you are interested to participate live, please send a short message to smartstart@fz-juelich.de Please note, these lectures will be recorded for subsequent publishing as online lecture material.
Relational Reasoning in Curricular Knowledge Components
It is a truth universally acknowledged that relational reasoning is important for learning in Science, Technology, Engineering, and Mathematics (STEM) disciplines. However, much research on relational reasoning uses examples unrelated to STEM concepts (understandably, to control for prior knowledge in many cases). In this talk I will discuss how real STEM concepts can be profitably used in relational reasoning research, using fraction concepts in mathematics as an example.
Targeted Grants in MPS
Supports high-risk projects of exceptional promise and scientific importance in mathematics and the physical sciences, with funding for up to five years; letters of intent are accepted on a rolling basis and there is no recommended or assumed funding level.