James Zhao

Hi, I'm James. I'm a fourth year PhD student in the maths* department at Stanford. My advisor is Persi Diaconis.

My research interests lie at the intersection of Probability and Combinatorics, especially problems with practical applications in Statistics and Computer Science. I am particularly interested in random combinatorial structures (probability measures on families of permutations, partitions, graphs, etc.), and more specifically, limiting behaviour and sampling algorithms for these structures.

I am currently working on a sampling algorithm for graphs with given degree sequence, which is useful for answering questions like "How surprising is it that the mean Erdős number is 4.65 and the mean degree of separation on Twitter is 4.67?"

Contact

Publications

• The Hopfield Model with Superlinearly Many Patterns (Statistics and Probability Letters, 2013)
  This paper proves a conjecture of Dembo and Montanari, showing that the free energy of the Hopfield model in the many-patterns limit is governed by the free energy of the Sherrington-Kirkpatrick model, by using Guerra-Toninelli's interpolation technique and Carmona-Hu's universality method.

• Universality of Asymptotically Ewens Measures on Partitions (Electronic Communications in Probability, 2012)
  This paper generalises the theory of Logarithmic Combinatorial Structures, removing the need for the "conditioning relation", while simultaneously consolidating the main theorems into a single universality theorem with a short elementary proof, by taking advantage of the Feller coupling for the Ewens sampling formula.

• Hidden Markov Models with Multiple Observation Processes (Masters Thesis, 2009)
  My masters thesis studies hidden Markov models where there is more than one way to observe the underlying Markov chain, and attempts to find an optimal strategy for choosing how to make the observation.

Teaching

• MATH51 Linear Algebra and Differential Calculus of Several Variables (Spring 2012)
• MATH113 Linear Algebra and Matrix Theory (Winter 2012)
• MATH51 Linear Algebra and Differential Calculus of Several Variables (Winter 2011)
• MATH51 Linear Algebra and Differential Calculus of Several Variables (Autumn 2009)
• 620-201 Probability (Semester 1, 2008)
• 620-123 Applied Mathematics (Advanced) (Semester 2, 2007)
• 620-121 Mathematics A (Advanced) (Semester 1, 2007)
• 620-123 Applied Mathematics (Advanced) (Semester 2, 2006)
• 620-121 Mathematics A (Advanced) (Semester 1, 2006)

Miscellaneous

• I used to run a movie night for the Stanford maths grad students.
• I was President of the Melbourne University Mathematics and Statistics Society.
• I enjoy playing go, badminton and piano.
• (*) You might notice I say "maths" instead of "math". That's because it's short for "mathematics", which is (etymologically) plural. Sometimes, I almost convince myself that "math" is a reasonable abbreviation, but then I remember that Americans say "math and stats". That's just silly.