Seminars Sorted by Series

Getting the deterministic complexity closer to the best known randomized complexity is an important goal in algorithms and communication protocols. In this work, we investigate the case where instead of one input, the algorithm/protocol is given...

Recently, Borgs, Chayes, Lovasz, Sos, Szegedy and Vesztergombi introduced a natural metric on the space of dense graphs, and identified the completion of the metric space that arises. Independently and simultaneously, Janson, Riordan and I defined a...

The ``direct product code'' of a function $f$ gives its values on all $k$-tuples $(f(x_1),\dots,f(x_k))$. This basic construct underlies “hardness amplification” in cryptography, circuit complexity and PCPs. A recent breakthrough by Dinur and...

We study combinatorial group testing schemes for learning d-sparse boolean vectors using highly unreliable disjunctive measurements. We consider an adversarial noise model that only limits the number of false observations, and show that any noise...

Since the work of Karp and Lipton in the 1980s, complexity theorists know and love the /poly operator, which adds a polynomial-sized advice string to any complexity class. But it's interesting to consider much more general kinds of "advice objects"...

This series of three talks will give a nontechnical, high level overview of geometric complexity theory (GCT), which is an approach to the P vs. NP problem via algebraic geometry, representation theory, and the theory of a new class of quantum...

Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using...

We bound the time it takes for a group of birds to reach steady state in a standard flocking model. We prove that (i) within single exponential time fragmentation ceases and each bird settles on a fixed flying direction; (ii) the flocking network...

A local graph partitioning algorithm finds a set of vertices with small conductance (i.e. a sparse cut) by adaptively exploring part of a large graph $G$, starting from a specified vertex. For the algorithm to be local, its complexity must be...

We prove that NP is different from coNP and coNP is not a subset of MA in the number-on-forehead model of multiparty communication complexity for up to k=(1 - e)log(n) players, where e>0 is any constant. Prior to our work, the problem was open even...

The local ratio framework introduced by Bar Yehuda and Evan in 1981 provides a unified framework for many approximation algorithms. Recently, the local ratio technique has been used to provide efficient approximation algorithms for a number of...

A number of recent results on optimization problems involving submodular functions have made use of the "multilinear relaxation" of the problem. I will show a general approach to deriving inapproximability results in the value oracle model, based on...

This work attempts to broaden the foundations of public-key cryptography. We construct a new public key encryption based on two “hardness on average” assumptions: (1) it is hard to “learn parity with noise” for random sparse equations; and (2) it is...

A halfspace (a.k.a. threshold) is a boolean function h : {-1,1}^n --> {-1,1} of the form h(x) = sign(w_0 + w_1 x_1 + ... + w_n x_n), where the weights w_i are arbitrary real numbers. Halfspaces are studied extensively in the theories of complexity...

I will discuss a lower bound of $\omega(n^(k/4))$ on the size of constant-depth $(AC_0)$ circuits solving the k-clique problem on n-vertex graphs. This bound follows from a stronger result that $AC_0$ circuits of size $O(n^(k/4))$ almost surely fail...

Via four applications: distribution modeling, probability estimation, data compression, and classification, we argue that when learning from data, discrete values should be ignored except for just their appearance-order pattern. Along the way, we...

We prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae. A read-once boolean formula F is a propositional formula in which each variable appears exactly once. Such a formula...

I will present a construction of an n-point negative type metric such that every t-point sub-metric is isometrically L1-embeddable, but embedding the whole metric into L1 incurs distortion at least k, where both t and k are (\log\log\log n)^{\Omega...

On Feb. 1, in his blog, Tim Gowers proposed an open collaboration on a math problem. Specifically, he suggested working on a combinatorial proof of the Density Hales-Jewett Theorem. This theorem states that for every delta > 0, if n is sufficiently...

Every Boolean function on n variables can be expressed as a unique multivariate polynomial modulo p for every prime p. In this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We...

Ideally, one would want to base the security of standard cryptographic primitives (such as pseudorandom generators) on widely believed worst-case complexity assumptions like P does not equal NP. However, it is currently not known if this is possible...

I will talk about a recent work describing a deterministic polynomial time algorithm for blackbox identity testing for depth three circuits with bounded top fanin over the field of rational numbers. This resolves a question posed by Klivans and...

Constraint Satisfaction Problems appear everywhere. The study of their quantum analogues (in which the constraints no longer commute), has become a lively area of study, and various recent results provide interesting insights into quantum physics...

A PCP is a proof system in which the proofs that can be verified by a verifier that reads only a very small part of the proof. One line of research concerning PCPs is trying to reduce their soundness error (i.e., the probability of accepting false...

An XOR game is a very simple model of evaluating a distributed function f(x,y) . With probability p(x,y) a Verifier sends questions x, y to Alice and Bob, respectively. Without communicating, Alice and Bob then output a, b in {-1,+1} in the hope...

Sex has been called "the queen of problems in evolutionary biology" since it is pervasive in nature yet its functional significance has not been known. It has often been assumed that the shuffling of genes due to sex is an adaptation that...

Research in the area of privacy of data analysis has been flourishing recently, with a rigorous notion such as differential privacy regarding the desired level of privacy and sanitizing algorithms matching the definition for many problems. Most of...

We give new results on the number-on-the-forhead (NOF) communication complexity of the multiparty pointer jumping problem. The origional motivation for this problem comes from circuit complexity. Specifically, there is no explicit function known to...

The question whether or not parallel repetition reduces the soundness error is a fundamental question in the theory of protocols. While parallel repetition reduces (at an exponential rate) the error in interactive proofs and (at a weak exponential...

Algorithms spend time on performing arithmetic computations, but often more on moving data, between the levels of a memory hierarchy and between parallel computing entities. Judging by the hardware evolution of the last few decades, the fraction of...

I will be describing recent joint efforts with Tim Gowers to decompose a bounded function into a sum of polynomially structured phases and a uniform error, based on the recent inverse theorem for the U^k norms on F_p^n by Bergelson, Tao and Ziegler...

The Unique Games Conjecture (Khot, 2002) is a central open question about hardness of approximation. In recent years, a sequence of works showed that the truth of this conjecture would have profound implications: If the conjecture is true, then...

How many edges of the n-dimensional Boolean hypercube can be sliced by a degree-d polynomial surface? This question can be equivalently stated as "What is the maximum average sensitivity of any degree-d polynomial threshold function?" In 1994...

In this talk, I shall describe an ongoing project to develop a complexity theory for cryptographic (multi-party computations. Different kinds of cryptographic computations involve different constraints on how information is accessed. Our goal is to...

For four years now, we have been conducting "medium-scale" experiments in how human subjects behave in strategic and economic settings mediated by an underlying network structure. We have explored a wide range of networks inspired by generative...

In this survey I will present several extremal problems, and some solutions, concerning convex lattice polytopes. A typical example is to determine the smallest area that a convex lattice polygon can have if it has exactly n vertices.

Semidefinite programming bounds are widely used in combinatorial optimization, quantum computing and complexity theory. The first semidefinite programming bound to gain fame is the so-called theta number developed by Lov\'asz to compute the Shannon...