Talks

Abstract For a given graph H we are interested in the critical value of p so that a sample of an Erdos-Renyi graph contains a copy of H with high probability,  Kahn and Kalai in 2006 conjectured that it should be given (up to a logarithm) by the minimum p so that in expectation all subgraphs H' of H appear in G(n,p). In this work, we will present a proof of a modified version of this conjecture. Our proof is based on a powerful "spread lemma", which played a key role in the recent breakthroughs on the sunflower conjecture and the proof of the fractional Kahn-Kali conjecture. Time permitting, we will discuss a new proof of the spread lemma using Bayesian inference tools. This is joint work with Elchanan Mossel, Jonathan Niles-Weed and Nike Sun

Abstract It is well known that sparse random graphs (where the number of edges is of the order of the number of vertices) contain only a small number of triangles. On the other hand, many networks observed in real life are sparse but contain a large number of triangles. Motivated by this discrepancy we ask the following two questions: How (un)likely is it that a sparse random graph contains a large number of triangles? What does the graph look like when it contains a large number of triangles? We also ask a related question: What is the probability that in a sparse random graph a large number of vertices are part of some triangle? We discuss these questions, as well as some applications to exponential random graph models.

Abstract   In many domains, we are interested in estimating the total causal treatment effect in the presence of network interference, where the outcome of one individual or unit is affected by the treatment assignment of those in its local network. Additional challenges arise when complex cluster randomized designs are not feasible to implement, or the network is unknown and costly to estimate. We propose a new measure of model complexity that characterizes the difficulty of estimating the total treatment effect under the standard A/B testing setup. We provide a class of unbiased estimators whose variance is optimal with respect to the population size and the treatment budget. Furthermore, we show that when the network is completely unknown, we can still estimate the total treatment effect under a richer yet simple staggered rollout experimental design. The proposed design principles, and related estimator, work with a broad class of outcome models. Our solution and statistical guarantees do not rely on restrictive network properties, allowing for highly connected networks that may not be easily clustered. This is joint work with Edoardo Airoldi, Christian Borgs, Jennifer Chayes, Mayleen Cortez, and Matthew Eichhorn

Abstract A random structure exhibits symmetry if its law remains invariant under a group of transformations. Exchangeability (of graphs, sequences, etc) and stationarity are examples. Under suitable conditions, the transformation group can be used to define an estimator that averages over an instance of the structure, and such estimators turn out to satisfy a law of large numbers, a central limit theorem, and further convergence results. Loosely speaking: The large-sample theory of i.i.d averages still holds if the i.i.d assumption is substituted by a suitable symmetry assumption.  Joint work with Morgane Austern.

Within the AdS/CFT correspondence, the entanglement properties of the CFT are related to wormholes in the dual gravity theory. This gives rise to questions about the factorisation properties of the Hilbert spaces on both sides of the correspondence.  We  show how the Berry phase, a geometrical phase encoding information about topology, may be used to reveal the Hilbert space structure. Wormholes are characterized by a non-exact symplectic form that gives rise to the Berry phase. For wormholes connecting two spacelike regions in AdS3 spacetimes, we find that the non-exactness is linked to a variable appearing in the phase space of the boundary CFTs. Mathematical concepts such as coadjoint orbits and geometric actions play an important role in this analysis.  We classify Berry phases according to the type of dual bulk diffeomorphism involved, distinguishing between Virasoro, gauge and modular Berry phases.

In addition to its relevance for quantum gravity, the approach presented also suggests how to experimentally realize the Berry phase and its relation to entanglement in table-top experiments involving photons or electrons. This provides a new example for relations between very different branches of physics that follow from the AdS/CFT correspondence and its generalizations. Based on 2202.11717 and 2109.06190.

Zoom link: https://pitp.zoom.us/j/96113910200?pwd=YXJnSWxiMHRIb21xdGFnNFM0cFFvUT09

Abstract Dense graph limit theory is mainly concerned with law-of large-number type of results. We propose a corresponding central limit theorem - or rather fluctuation theory - based on Janson's theory of Gaussian Hilbert Spaces and generalised U-statistics from the 1990s. Our approach provides rates and allows for proper statistical inference based on subgraph counts.

Abstract   The problem of completing a large low rank matrix using a subset of revealed entries has received much attention in the last ten years. I will give a necessary and sufficient condition, stated in the language of graph limit theory, for a sequence of matrix completion problems with arbitrary missing patterns to be asymptotically solvable. I will also present an algorithm that is able to approximately recover the matrix whenever recovery is possible.

Abstract Ramsey's Theorem states that for every graph H, there is an integer R(H) such that every 2-edge-coloring of R(H)-vertex complete graph contains a monochromatic copy of H. In this talk, we focus on a natural quantitative extension: how many monochromatic copies of H can we find in every 2-edge-coloring of K_N, and what graphs H are so-called common, i.e., the number of monochromatic copies of H is asymptotically minimized by a random 2-edge-coloring. A classical result of Goodman from 1959 states that the triangle is a common graph. On the other hand, Thomason proved in 1989 that no clique of order at least four is common, and the existence of a common graph with chromatic number larger than three was open until 2012, when Hatami, Hladky, Kral, Norin and Razborov proved that the 5-wheel is common. In this talk, we show that for every k>4 there exists a common graph with chromatic number k. This is a joint work with D. Kral and F. Wei

Information about the late-time Universe is imprinted on the small scale CMB as photons travel to us from the surface of last scattering. Several processes are at play and small scale fluctuations are very rich and non-Gaussian in nature. I will review some of the most important effects and I will focus on the Sunyaev-Zel'dovich (SZ) effect and gravitational lensing. I will discuss how a combination of measurements can probe velocity fields at cosmological distances and inform us on cluster energetics. I will also show recent measurements of weak lensing of the CMB and how they can help us interpret intriguing discrepancies in cosmological parameters between the high and low redshift Universe.

Zoom link: https://pitp.zoom.us/j/94451033605?pwd=Tkx4dHZTblMxUFJlZENyblJQVFo2dz09