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ReLU networks are a standard example of neural networks. We will discuss the expressivity of neural networks, focusing on their depth. The plan is to start with a general introduction and a survey of basic results, including the duality between ReLU networks and a model for constructing polytopes. We will then discuss polytopes and discuss how their properties can help us understand the depth of neural networks. 

Based on joint works with Bakaev, Brunck, Hertrich and Stade.

X-ray microscopy (XRM) or X-ray micro computed tomography (micro-CT) is a non-destructive and noninvasive imaging technique that uses X-rays to characterize cross-sectional and three-dimensional internal structures. While CT was initially developed for medical imaging, the need for higher resolution and the advancement of technology led to the creation of micro-CT in the early 1980s, specifically for evaluating material structure and other applications requiring higher detail.In this talk, I will introduce XRM, its brief history and basic principles, along with some examples of applications.

At modest temperatures, and especially above 1000 K, most of the entropy of solids comes from atomic vibrations. In 1907, Einstein proposed a quantized harmonic oscillator as a starting point. Today, normal modes of crystal vibrations are quantized, and the quanta are called "phonons." Phonons in crystals were first measured by inelastic neutron scattering in the 1950s. Using inelastic neutron scattering and electronic structure calculations, we have compared the entropy from phonons to the entropy obtained by calorimetry. In short, excellent agreement is found when all known sources of entropy are included, such as from electrons, spins, and interactions between phonons, electrons, and spins. Interactions that cause only small departures from harmonic behavior are treated with many-body perturbation theory. Neutron scattering revealed new anharmonic features in the phonon spectra of NaBr and Cu2O. These anharmonic features, such as phonon frequency doubling and intermodulation sidebands, can be understood with molecular dynamics or methods based on the Heisenberg-Langevin equation or the Schrödinger-Langevin equation. For Cu2O and ZnO, we found diffuse inelastic intensity (DII) at high energies, well above the phonon bands. This DII originates from brief anharmonic interactions between atoms as they vibrate, and is a new probe of anharmonic interatomic potentials.

The macromolecular composition of phytoplankton shapes the nutrition available to marine ecosystems and regulates the interwoven global cycles of carbon and nutrients. Despite these fundamental roles, there are currently no mechanistic, predictive models of the global distribution of phytoplankton macromolecular composition and its variation in response to environmental changes. Here, we simulate the cellular allocation of proteins, carbohydrates, and lipids in a global ocean model in the present day and over the 21st century under a climate change scenario. Our simulations indicate systematic spatial variations in phytoplankton macromolecular composition, which are consistent with available observations. Our model simulations further suggest variable geographic responses to climate change. Specifically, phytoplankton in polar regions are projected to have more carbohydrates and lipids at the expense of proteins, due to warming and relief from light limitation. We compiled and analyzed in situ macromolecular measurements of polar phytoplankton spanning several decades, finding trends consistent with our model predictions. Our findings indicate that changes in the macromolecular composition of phytoplankton can serve as indicators of shifting environmental conditions. Such changes will reshape the nutritional landscape at the base of the marine food web and alter global biogeochemical cycles.

Artificial neural networks have driven remarkable advances in AI in recent years, but their use also raises serious privacy and security concerns. Recent studies have shown that, under certain conditions, parts of the training data can be reconstructed directly from the model parameters, posing significant risks to privacy in sensitive domains such as medicine, and challenging the safe deployment of these models more broadly.

In this talk, I will survey two of my recent works that aim to uncover the theoretical foundations of this phenomenon, shedding light on the extent to which trained models may reveal information about their training data. In particular, we will show - both empirically and theoretically - that reconstructing the training data requires additional prior knowledge, and that trained models do not necessarily leak information.

Based on joint work with Guy Smorodinsky, Gal Vardi, Yehonatan Rafael, and Ofir Lindenbaum.