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**1 - 4**of**4**### Invariance principle on the slice

, 2015

"... The non-linear invariance principle of Mossel, O’Donnell and Oleszkiewicz establishes that if fpx1,..., xnq is a multilinear low-degree polynomial with low influences then the distribution of fpB1,...,Bnq is close (in various senses) to the distribution of fpG1,...,Gnq, where Bi PR t´1, 1u are indep ..."

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The non-linear invariance principle of Mossel, O’Donnell and Oleszkiewicz establishes that if fpx1,..., xnq is a multilinear low-degree polynomial with low influences then the distribution of fpB1,...,Bnq is close (in various senses) to the distribution of fpG1,...,Gnq, where Bi PR t´1, 1u are independent Bernoulli random variables and Gi „ Np0, 1q are independent standard Gaussians. The invariance principle has seen many application in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans–Williamson algorithm for MAX-CUT is optimal under the Unique Games

### Harmonicity and Invariance on Slices of the Boolean Cube

, 2015

"... In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree functions. Here we provide an alternative proof for general low-degree functions, with no constraints on the influences. We show that any real-valued function on the slice, whose de ..."

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In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree functions. Here we provide an alternative proof for general low-degree functions, with no constraints on the influences. We show that any real-valued function on the slice, whose degree when written as a harmonic multi-linear polynomial is o( n), has approximately the same distribution under the slice and cube measure. Our proof is based on a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales. While such decompositions have been used in the past for stationary reversible Markov chains, ours decomposition is applied in a non-reversible non-stationary setup. We also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof. Finally, we provide independent simple proofs for the facts that 1) one cannot distinguish between the slice and the cube based on functions of o(n) coordinates and 2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is o( n). ∗This material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the

### Research Statement

, 2012

"... torics. Of course, these two subjects are no strangers: combinatorics is often used as a tool in theoretical computer science. The questions in theoretical computer science that I find most attractive are those which have a strong combinatorial flavor. Moreover, for me, combinatorics has an innate a ..."

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torics. Of course, these two subjects are no strangers: combinatorics is often used as a tool in theoretical computer science. The questions in theoretical computer science that I find most attractive are those which have a strong combinatorial flavor. Moreover, for me, combinatorics has an innate appeal, and I pursue it in and of itself. Combinatorics is a vast subject. My research has concentrated on using spectral methods to answer two types of questions: those of extremal combinatorics, of the Erdős-Ko-Rado type, and those of the analysis of Boolean functions, following the seminal work of Friedgut-Kalai-Naor. Together with my coauthors, we have proved a decades-old conjecture in ex-tremal combinatorics concerning the maximal size of triangle-intersecting families of graphs. In more recent work, we have generalized Friedgut-Kalai-Naor to Boolean functions on Sn. My contributions in theoretical computer science (with various coauthors) span several areas. In the sequel, I will focus on three areas encompassing my main contributions. First, I have designed a combinatorial algorithm for monotone submodular maximization over a matroid. Second, I have generalized a simulation result in circuit complexity to a correspond-ing result in proof complexity. Finally, I have studied the complexity class of comparator circuits, constructing a universal comparator circuit, and proving oracle separation results. In the future, I hope to combine the two threads of my research. Analysis of Boolean functions has led in the past to deep results in hardness of approximation. I believe that my expertise puts me in an excellent position to pursue similar questions. 1

### THESIS FOR THE DEGREE OF LICENTIATE OF PHILOSOPHY

"... On properties of generalizations of noise sensitivity ..."

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