Georgia Institute of Technology College of Sciences

Diffusion models have become ubiquitous for image generation and are increasingly being used for scientific applications. To date, many flavors of diffusion models have been developed by varying the stochastic process that noises data, but also the domain on which these processes act. Typically, generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact analysis. We relate the theory to the existing continuous-state Gaussian diffusion in discrete and continuous time. This approach is validated using a simple stochastic decay process, in which the reverse process generates images from a single all-black image, rather than a noisy prior distribution.

In this talk, we'll sketch how one might hope to construct spaces (or spectra) from Floer theories, including framed flow categories and finite-dimensional approximation.  If time allows, we'll talk about some questions Floer spaces (or spectra) can be useful for.

Large-scale parallel-processing infrastructures such as data centers and cloud networks form the cornerstone of the modern digital environment. Central to their efficiency are resource management policies, especially load balancing algorithms (LBAs), which are crucial for meeting stringent delay requirements of tasks. A contemporary challenge in designing LBAs for today's data centers is navigating data locality constraints that dictate which tasks are assigned to which servers. These constraints can be naturally modeled as a bipartite graph between servers and various task types. Most LBA heuristics lean on the mean-field approximation's accuracy. However, the non-exchangeability among servers induced by the data locality invalidates this mean-field framework, causing real-world system behaviors to significantly diverge from theoretical predictions. From a foundational standpoint, advancing our understanding in this domain demands the study of stochastic processes on large graphs, thus needing fundamental advancements in classical analytical tools.

In this presentation, we will delve into recent advancements made in extending the accuracy of mean-field approximation for a broad class of graphs. In particular, we will talk about how to design resource-efficient, asymptotically optimal data locality constraints and how the system behavior changes fundamentally, depending on whether the above bipartite graph is an expander, a spatial graph, or is inhomogeneous in nature.

 A graph is called k-critical if its chromatic number is k but any proper subgraph has chromatic number less than k. There have been extensive reseaches on k-critical graphs over the past decades, yet several basic problems remain widely open. One of such problems is to determine the maximum number of edges in an n-vertex k-critical graph. In this talk, we will discuss some recent results on extremal aspects of k-critical graphs, including improvments on the extremal number of edges/cliques/critical subgraphs in k-critical graphs.  This is based on some joint works with Jun Gao, Cong Luo and Tianchi Yang.