Fall 2024
Monday 4.15  5.15 pm
Room 2143
Scheduled virtual talks will be held on Zoom, Monday 4:155:15 pm.
Zoom Link
Schedule

Sept 9
Jacopo Borga (MIT)
Long increasing subsequences in Browniantype permutations
Abstract: What is the behavior of the longest increasing subsequence in a uniformly random permutation? Its length is of order 2n^{1/2} plus TracyWidom fluctuations of order n^{1/6}. Its scaling limit is the directed geodesic of the directed landscape.
This talk discusses how this behavior changes dramatically when one looks at universal Browniantype permutations, i.e., permutations sampled from the Brownian separable permutons. We show that there are explicit constants 1/2 < alpha< beta < 1 such that the length of the longest increasing subsequence in a random permutation of size n sampled from the Brownian separable permutons is between n^{alpha  o(1)} and n^{beta + o(1)} with high probability. We present numerical simulations which suggest that the lower bound is close to optimal and a very recent conjecture for the exact value of the exponent. Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002).
If time permits, we conclude by discussing some conjectures for permutations sampled from the skew Brownian permutons, a model of universal permutons generalizing the Brownian separable permutons: here, the longest increasing subsequences should be closely related with some models of random directed metrics on planar maps.
Based on joint works with William Da Silva and Ewain Gwynne, and with Arka Adhikari, Thomas Budzinski, William Da Silva, and Delphin Sénizergues. 
Sept 16
Ewain Gwynne (University of Chicago)
Random conformal geometry in dimension $d\geq 3$
Abstract: There has been enormous progress in the last few decades concerning random geometric objects in two dimensions which interact nicely with conformal maps. Such objects include SchrammLoewner evolution (SLE), Liouville quantum gravity (LQG), and discrete analogs thereof. However, much less is known about analogs of these objects in dimension $d\geq 3$. I will give an overview of a few known results and many open problems concerning random geometry in dimension $d\geq 3$. Some of the known results come from recent joint works with Jian Ding and Zijie Zhuang, with Ahmed BouRabee, and with Federico Bertacco. I will not assume any background knowledge about random geometry for $d=2$.

Sept 23
Sahar Diskin
(TelAviv University)Component sizes in percolation on finite regular graphs
Abstract: A classical result by Erdős and Rényi from 1960 shows that the binomial random graph $G(n,p)$ undergoes a fundamental phase transition in its component structure when the expected average degree is around $1$ (i.e., around $p=1/n$). Specifically, for $p = (1\epsilon)/n$, where $\epsilon > 0$ is a constant, all connected components are typically logarithmic in $n$, whereas for $p = (1+\epsilon)/n$ a unique giant component of linear order emerges, and all other components are of order at most logarithmic in $n$.
A similar phenomenon — the typical emergence of a unique giant component surrounded by components of at most logarithmic order — has been observed in random subgraphs $G_p$ of specific host graphs $G$, such as the $d$dimensional binary hypercube, random $d$regular graphs, and pseudorandom $(n,d,\lambda)$graphs, though with quite different proofs.
This naturally leads to the question: What assumptions on a $d$regular $n$vertex graph $G$ suffice for its random subgraph to typically exhibit this phase transition around a critical probability $p=1/(d1)$? Furthermore, is there a unified approach that encompasses these classical cases? In this talk, we demonstrate that it suffices for $G$ to have mild global edge expansion and (almostoptimal) expansion of sets of (poly)logarithmic order in $n$. This result covers many previously considered concrete setups.
We also discuss the tightness of our sufficient conditions.
Joint work with Michael Krivelevich. 
Sept 30
Allan Sly
(Princeton, currently visiting MIT)Rotationally invariant first passage percolation: Concentration and scaling relations
Abstract: For rotationally invariant first passage percolation on the plane, we use a multiscale argument to prove stretched exponential concentration of the passage times at the scale of the standard deviation. Our results are proved for several standard rotationally invariant models of first passage percolation, e.g. Riemannian FPP, Voronoi FPP and the HowardNewman model. As a consequence, we prove a version of the scaling relations between the passage times fluctuation and transversal fluctuations of geodesics. These are the first such unconditional results.

Oct 7
Nytia Mani
(MIT)Fourth moment theorems for monochromatic subgraphs
Abstract: Given a graph sequence $\{G_n\}_{n\ge1}$ and simple, connected subgraph $H$, denote by $T(H,G_n)$ the number of monochromatic copies of $H$ in a uniformly random vertex coloring of $G_n$ with $c \ge 2$ colors. For general $c$, we prove a central limit theorem for $T(H,G_n)$ with explicit error rates that arise from subgraph counts. Based on these counts, we show that for $c \ge 30$, convergence of the 4th moment is sufficient for $T(H, G_n)$ to enjoy a central limit theorem (and that it is always necessary).
In the special case of 2 colors, we distil failures of the 4th moment phenomenon for $T(H, G_n)$ into easytoverify local properties of $\{G_n\}$. Along the way, we extend the fourth moment phenomenon to a broader class of Rademacher and Gaussian polynomials, which do not necessarily belong to a single Wiener chaos.
Based on joint work with Sayan Das, Zoe Himwich, and Dan Mikulincer 
Oct 14
Indigenous people's day

Oct 21
Xin Sun (PKU)

Oct 28
HT Yau (Harvard)

Nov 4

Nov 11
Veteran's day

Nov 18
Jonathan Weitsman
(Northeastern) 
Nov 25
Manan Bhatia
(MIT) 
Dec 2
William Da Silva (University of Vienna)

Dec 9
Mark Sellke (Harvard)