Georgia Institute of Technology College of Sciences

Sampling is a fundamental and widespread algorithmic primitive that lies at the heart of Bayesian inference and scientific computing, among other disciplines. Recent years have seen a flood of works aimed at laying down the theoretical underpinnings of sampling, in analogy to the fruitful and widely used theory of convex optimization. In this talk, I will discuss some of my work in this area, focusing on new convergence guarantees obtained via a proximal algorithm for sampling, as well as a new framework for studying the complexity of non-log-concave sampling.

Mammalian cortical networks are known to process sensory information utilizing feedforward and feedback connections along the cortical hierarchy as well as intra-areal connections between different cortical layers. While feedback and feedforward signals have distinct layer-specific connectivity motifs preserved across species, the computational relevance of these connections is not known. Motivated by predictive coding theory, we study how expected and unexpected visual information is encoded along the cortical hierarchy, and how layer-specific feedforward and feedback connectivity contributes to differential, context-dependent representations of information across cortical layers, by analyzing experimental recordings of neural populations and also by building a recurrent neural network (RNN) model of the cortical microcircuitry. Experimental evidence shows that information about identity of the visual inputs and expectations are encoded in different areas of the mouse visual cortex, and simulations with our RNNs which incorporate biologically plausible connectivity motifs suggest that layer-specific feedforward and feedback connections may be the key contributor to this differential representation of information.
 

Reaction networks are commonly used to model a variety of physical systems ranging from the microscopic world like cell biology and chemistry, to the macroscopic world like epidemiology and evolution biology. A biologically relevant property that reaction networks can have is absolute concentration robustness (ACR), which refers to when a steady-state species concentration is maintained even when initial conditions are changed. Networks with ACR have been observed experimentally, for example, in E. coli EnvZ-OmpR and IDHKP-IDH systems. Another reaction network property that might be desirable is multistationarity-the capacity for two or more steady states, since it is often associated with the capability for cellular signaling and decision-making.

While the two properties seem to be opposite, having both properties might be favorable as a biochemical network may require robustness in its internal operation while maintaining flexibility as a signal-response mechanism. As such, our driving motivation is to explore what network structures can produce ACR and multistationarity. We show that it is highly atypical for both properties to coexist in very small and very large reaction networks without special structures. However, it is possible for them to coexist in certain classes of reaction networks. I will discuss in detail one such class of networks, which consists of multisite phosphorylation-dephosphorylation cycles with a ``paradoxical enzyme".

After finishing the proof of equivalence of the Chebyshev constant of a set and its logarithmic capacity, we will start to discuss classical and recent results on estimates and asymptotics of Chebyshev numbers.