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The title is a bit of a mouthful, so let us unpack it together:


A C*-correspondence is a certain kind of bimodule over a C*-algebra B that has a B-valued inner product.
Sub and super-product systems are families of C*-correspondences that enjoy certain semigroup-like properties under the tensor product. 
A CP-semigroup is a family of completely positive maps that form a semigroup under composition

Successful goal-directed navigation requires estimating one’s current position in the environment, representing the future goal location, and maintaining a map that preserves the topological relationship between positions. In addition, we often need to implement similar navigational strategies in a continuously changing environment, thereby necessitating certain invariance in the underlying spatial maps. Previous research has identified neurons in the hippocampus and parahippocampal cortices that fire specifically when an animal visits a particular location, implying the presence of a spatial map in the brain. However, this map largely encodes the current position of an animal and is context-dependent, whereby changing the room or shape of the arena results in a new map orthogonal to the previous one. These observations raise the question, are there other spatial maps that fulfill the cognitive requirements necessary for goal-directed navigation?
Using a goal-directed navigation task with multiple reward locations, we observed that neurons in the orbitofrontal cortex (OFC) exhibit distinct firing patterns depending on the goal location, and this goal-specific OFC activity originates even before the onset of the journey. Further, the difference in the ensemble firing patterns representing two target locations is proportional to the physical distance between these locations, implying the preservation of spatial topology. Finally, carrying out the task across different spatial contexts revealed that the mapping of target locations in the OFC is largely preserved and that the maps formed in two different contexts occupy similar neural subspaces and could be aligned by a linear transformation. Taken together, the OFC forms a topology-preserved schema of spatial locations that is used to represent the future spatial goal, making it a potentially crucial brain region for planning context-invariant goal-directed navigational strategies.

In this talk, we delve into several fundamental questions in deep learning. We start by addressing the question, "What are good representations of data?" Recent studies have shown that the representations learned by a single classifier over multiple classes can be easily adapted to new classes with very few samples. We offer a compelling explanation for this behavior by drawing a relationship between transferability and an emergent property known as neural collapse. Later, we explore why certain architectures, such as convolutional networks, outperform fully-connected networks, providing theoretical support for how their inherent sparsity aids learning with fewer samples. Lastly, I present recent findings on how training hyperparameters implicitly control the ranks of weight matrices, consequently affecting the model's compressibility and the dimensionality of the learned features.

Additionally, I will describe how this research integrates into a broader research program where I aim to develop realistic models of contemporary learning settings to guide practices in deep learning and artificial intelligence. Utilizing both theory and experiments, I study fundamental questions in the field of deep learning, including why certain architectural choices improve performance or convergence rates, when transfer learning and self-supervised learning work, and what kinds of data representations are learned in practical settings.
 

 

Due to their intrinsic nonequilibrium and disordered nature, glasses feature low-frequency, nonphononic vibrations, in addition to phonons. These excess modes generate a peak —the boson peak— in the ratio of the vibrational density of state (VDoS) and Debye’s VDoS of phonons. Yet, the excess vibrations and the boson peak are not fully understood. After presenting the experimental evidence of the boson peak, we will discuss additional universal characteristics of glassy low frequency VDoS obtained through numerical simulations. We will then examine a recently analyzed mean-field model capturing the universal low-frequency glassy VDoS characteristics. Combining reanalyzed experimental data and computer simulations, we will observe that the same mean-field model also captures the origin, nature and properties of the boson peak, yielding a unified physical picture of the low-frequency VDoS spectra of glasses.

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Periodic Jacobi matrices on the line have a very rich spectral theory, one of whose ingredients is the Floquet theory of eigenfunctions. In this talk we will discuss ongoing work that attempts to generalize this theory to more general trees. We will describe some results obtained in joint works with Jess Banks, Jorge Garza Vargas, Eyal Seelig and Barry Simon.

Statistical physics successfully accounts for phenomena involving a large number of components using a probabilistic approach with predictions for collective properties of the system. While biological cells contain a very large number of interacting components, (proteins, RNA molecules, metabolites, etc.), the cellular network is understood as a particular, highly specific, choice of interactions shaped by evolution, and therefore difficultly amenable to a statistical physics description. Here we show that when a cell encounters an acute but non-lethal stress, its perturbed state can be modelled as random network dynamics. Strong perturbations may therefore reveal the dynamics of the underlying network that are amenable to a statistical physics description. We show that our experimental measurements of the recovery dynamics of bacteria from a strong perturbation can be described in the framework of physical aging in disordered systems (Kaplan Y. et al, Nature 2021). Further experiments on gene expression confirm predictions of the model. The predictive description of cells under and after strong perturbations should lead to new ways to fight bacterial infections, as well as the relapse of cancer after treatment.