Huy Pham
California Institute of Technology, CA·Mathematics & Statistics·Early-career
When random choices build a large network, such as connecting pairs of points by coin flips, sharp changes often appear. Add a few more connections and the network can suddenly contain a structure it almost never had before.
- Fields
Mathematics & Statistics
Computer Science
- Topics
random graphs
threshold phenomena
first moment method
- Grants
- 1 known grant, through Jun 2031
What they work on
- Develop new toolkits for first moment obstructions, where counting expected copies of a structure rules it out, to close gaps in the theory of threshold phenomena in random discrete systems.
- Extend threshold theory beyond random subsets (the Boolean lattice) to settings such as random subspaces of finite vector spaces, which are useful in cryptography.
- Connect threshold phenomena with suprema of stochastic processes (the largest values of random processes) and large deviations (rare-event probabilities), with relevance to large random networks and wide neural networks.
- Address open questions about random Cayley graphs, networks built from group structure, and use the results to study sets with small doubling in additive combinatorics.
More about the research
Understanding these sudden switches, called threshold phenomena, is a central problem in probability and combinatorics, the mathematics of counting and arrangement. It also connects to computer science, number theory, artificial intelligence and quantum information.
Why it matters
This is basic mathematics. It could strengthen the theory behind large networks and random systems. In the longer run, it could inform areas the grant names: cryptography built on random subspaces, the analysis of very wide neural networks in artificial intelligence, and the design of quantum codes for quantum information.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Jacob Fox
Stanford University, CA·Mathematics & Statistics
Determine how the number of colors and the structure of hypergraphs, networks whose edges can join more than two points, govern the off-diagonal growth of hypergraph Ramsey numbers.
Ryan O'Donnell
Carnegie Mellon University, PA·Computer Science
Find new elementary methods for constructing high-dimensional expanders, highly interconnected mathematical networks, that avoid the advanced algebra and number theory used in current constructions.
Bhargav Narayanan
Rutgers University-New Brunswick, NJ·Mathematics & Statistics·Early-career
Study up-sets and down-sets in the Boolean hypercube from a probabilistic viewpoint, where they encode events of interest in probabilistic combinatorics and statistical physics.
The school
California Institute of Technology: 18 labs on PI Pages, $445M a year on research, 200 research doctorates a year.
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