Spring 2023
Organizers | Gautam Iyer |
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Location | All talks are Tuesdays 2:00PM in WEH 7218, unless otherwise noted. |
Current Schedule | https://www.cmu.edu/math/news-events/cna_working_group_seminars.html |
Mailing List | Please email Gautam Iyer to be added to the list. |
Jan 24 | Alan Frieze: Markov Chains for Counting, and Related Problems (Special time 2:30PM) |
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Jan 31 | Wesley Pegden: Markov Chains for Sampling vs Markov Chains for Outlier detection |
Feb 7 | Wesley Pegden: Markov Chains for Sampling vs Markov Chains for Outlier detection (cont.) |
Feb 14 | Alan Frieze: Volume Computation: A brief overview |
Feb 21 | Martin Larsson: Minimixing asymptotic relative growth |
Feb 28 | Prasad Tetali: Modified Log-Sobolev Inequality (MLSI): theory, examples and consequence |
Mar 7 | (No meeting, Spring Break) |
Mar 14 | Konstantin Tikhomirov: Regularized modified log-Sobolev inequalities, and comparison of Markov chains. |
Mar 21 | Mykhaylo Shkolnikov: Shuffling Cards and Stopping Times |
Mar 28 | Mykhaylo Shkolnikov: Strong Stationary Times, and Intertwining |
Here are a few topics we are considering for the semester in no particular order. If you’d like to give a lecture or two on one of these topics, browse through the reference and send me an email. Talks are all informal so not knowing every last detail about the subject is OK ðŸ˜„.
If you have other things you like to think about, let me know and I can add them to the list.
Rate of entropy decay in finite (discrete space) Markov chains is better captured by a variant of the standard LSI. This brief overview will introduce the MLSI and mention some example applications - such as the transposition chain on permutations, and the matroid basis exchange walk - and a consequence to measure concentration. Time permitting, a general decomposition theorem for LSI, spectral gap and the MLSI will also be outlined, along with some more recent results. References: Bobkov Tetali, Jerrum, Son, Tetali, Vigoda, Herman, Salez Salez, Tikhomirov, Youssef
The ``regularized’‘ version of the Modified Log-Sobolev Inequality (MLSI) for log-Lipschitz functions allows using standard comparison techniques to get strong mixing time bounds for some Markov chains. References: Tikhomirov, Youssef
How many times do you need to shuffle a deck of cards before it’s random? (And applications to random walks.) Reference: Aldous, Diaconis
Intertwining for diffusions.
Fix $N$ and look at the random walk $X_{n+1} = f(X_n) + Îµ_{n+1} \pmod{N}$, where $Îµ_n$’s are i.i.d.\ uniformly distributed on $\{\pm 1, 0\}$, and $f$ is a bijection on $\{0, \dots, N-1\}$. For “almost every” $f$, the mixing time is $O(\log N)$. Reference: Chatterjee, Diaconis
Look at $dX = u(X) \, dt + \sqrt{2\kappa} \, dW$ on the Torus, where the deterministic flow of $u$ is “sufficiently mixing”. Then the mixing time of $X$ is $O(\abs{\ln \kappa}^3)$. References: Feng, Iyer ‘19, Iyer, Zhou ‘22.
Consider an overdamped Langevin system $dX_t = -\nabla U(X) \, dt + \sqrt{2\kappa} dW_t$ on the Torus. When the potential is non-convex, the mixing time is $O(e^{1/\kappa})$. You can add a mixing drift to this to make the mixing time polynomial in $1/\kappa$.
Blow-up suppression in Nonlinear PDEs: If the mixing time of $dX = u(X) \, dt + \sqrt{\kappa} \, dW_t$ is small enough, then you can avoid “blow up” in certain nonlinear PDEs (like the Keller–Segel model for chemotaxis). Reference: Iyer, Xu, ZlatoÅ¡ ‘19.