## Sensitivity conjecture proved!

In a recent breakthrough, Hao Huang gave a 6 page paper proving the longstanding sensitivity conjecture. (Hat tip, Scott Aaronson and Gil Kalai. See this stackexchange post and this paper of Avishai for some links to the literature on this.)

The proof is beautiful and simple. I will write a few words here, but it is probably easier for you to just read the paper. The sensitivity conjecture was known to follow from the following statement: let be the *Boolean Cube* which is the degree graph on vertices identified with such that for every , if their Hamming distance is one. Then, the maximum degree of every subgraph of of size is at least .

Hao proves the above statement by showing that there is a *signing* of the adjacency matrix of that turns it into a matrix with eigenvalues equaling and eigenvalues equaling . That is, he shows (using a simple but clever inductive argument, see the 5 line proof of his Lemma 2.2) that there is an matrix with entries in whose nonzero entries correspond to the edges of the Boolean cube, and such that all the eigenvalues of are and they sum up to zero. (Note that this makes sense since should have the same *Frobenius norm* as the adjacency matrix of . The Frobenius norm squared is both the sum of squares of entries, which is for which is a degree graph, and also equal to the sum of squares of the eigenvalues, which is if all eigenvalues are .)

Once you have such a signing, the result follows from Cauchy’s Interlace Theorem that says that for every matrix and any matrix that is a principle sub-matrix of ,

where are the eigenvalues of and is the maximum eigenvalue of . A corollary of this (which is the only fact we need) is that if has its top eigenvalue with multiplicity (i.e., ), then every principle sub-matrix of order larger than will satisfy . (In fact, we only need .)

Indeed, suppose that is a subgraph of the Boolean cube of size . Then the principle submatrix of corresponding to the vertices of satisfies (since and the first eigenvalues of are ). But it’s easy to show that for every matrix the maximum eigenvalue of is upper bounded by the maximum norm of its rows, which in our case is the maximum degree of the graph .

## TCS Women at STOC (guest post by Virginia Williams)

*[Guest post by Virgi Vassilevska Williams on the TCS women program at STOC. In particular the TCS Women Spotlight workshop has a great program and is open to all. –Boaz]*

## Intro-TCS rebooted

This Spring and Summer I am doing some major editing to my text on introduction to theoretical computer science. I am adding figures (176 so far and counting..), examples, exercises, simplifying explanations, reducing footnotes, and mainly trying to make it more “user friendly” and less “idiosyncratic”. I am now adding in all chapters figures such as the following that outline the main results and how they are connected to one another.

Specifically, in the previous version of the book I used *programming languages *as the main computational models. While I still think this is the right way if we were to “start from scratch”, these idiosyncratic models made it harder for students to use other resources such as textbooks and lecture notes. They also make it more difficult for instructors to use individual chapters in their courses without committing to using the full book.

Hence in the new revision the standard models of **Turing Machines** and **Boolean Circuits** are front and center. We do talk about the programming-language equivalents as well, since I think they are important for the connection to practice and some concepts such as the duality of code and data are better explained in these terms. I also use the programming-language variants to demonstrate concepts to students in code including compilers from circuits to straightline programs, various “syntactic sugar” transformations, and the Cook-Levin Theorem and NP reductions.

One thing did not change – we still start with *Boolean Circuits *rather than automata as the initial computational model. Boolean circuits are closer to actual implementations of computing, are a finite (and hence simpler) model, but one that is non-trivial enough to allow showing some important theorems early in the course including existence of a circuit for computing every finite function, the existence of a circuit to evaluate other circuits, and the counting lower bound as well as the counting lower bound.

Circuits are also crucial for later material in the course since they make the proof of the Cook-Levin Theorem much simpler and cleaner, allow talking about results such as and Sipser-Gacs, and are crucial to be even able to state results in advanced topics such as derandomization, cryptography, and quantum computing.

We do cover automata as well, including the equivalence of regular expressions and deterministic finite automata. We also cover context-free grammars (though not pushdown automata) and the λ calculus, including its equivalence with Turing Machines and the Y combinator (see also this notebook)

I have also done some work on the technical side of producing the book. The book is written in markdown. Markdown has many advantages but it wasn’t designed for 600-page technical books full of equations and cross-references so I did need to use some extensions to it. I am using pandoc (and my own filter) to produce both the HTML and LaTeX/PDF versions of the book.

There is still more work to do. I plan to add a chapter on space complexity and on proofs and computation (including both interactive and zero knowledge proofs, as well as the “propositions as types” correspondence between proofs and programs). I need to add more examples and exercises. There are also still several chapters where the text is “rough around the edges”.

As usual, the latest version of the book is available on https://introtcs.org . If you see any typo, problem, etc.., please post an issue on the GitHub repository (you can also make a pull request for small typo fixes if you prefer)

We invite you to submit your papers to the 11th Innovations in

Theoretical Computer Science (ITCS). The conference will be held at

the University of Washington in Seattle, Washington from January 12-14,

2020.

ITCS seeks to promote research that carries a strong conceptual message

(e.g., introducing a new concept, model or understanding, opening a new

line of inquiry within traditional or interdisciplinary areas,

introducing new mathematical techniques and methodologies, or new

applications of known techniques). ITCS welcomes both conceptual and

technical contributions whose contents will advance and inspire the

greater theory community.

Submission deadline: September 9, 2019 (05:59pm PDT)

Notification to authors: October 31, 2019

Conference dates: January 12-14, 2020

See the website at http://itcs-conf.org/itcs20/itcs20-cfp.html for

detailed information regarding submissions.

Program committee

Nikhil Bansal, CWI + TU Eindhoven

Nir Bitansky, Tel-Aviv University

Clement Canonne, Stanford

Timothy Chan, University of Ilinois at Urbana-Champaign

Edith Cohen, Google and Tel-Aviv University

Shaddin Dughmi, University of Southern California

Sumegha Garg, Princeton

Ankit Garg, Microsoft research

Ran Gelles, Bar-Ilan University

Elena Grigorescu, Purdue

Tom Gur, University of Warwick

Sandy Irani, UC Irvine

Dakshita Khurana, University of Illinois at Urbana-Champaign

Antonina Kolokolova, Memorial University of Newfoundland.

Pravesh Kothari, Carnegie Mellon University

Rasmus Kyng, Harvard

Katrina Ligett, Hebrew University

Nutan Limaye, IIT Bombay

Pasin Manurangsi, UC Berkeley

Tamara Mchedlidze, Karlsruhe Institute of Technology

Dana Moshkovitz, UT Austin

Jelani Nelson, UC Berkeley

Merav Parter, Weizmann Institute

Krzysztof Pietrzak, IST Austria

Elaine Shi, Cornell

Piyush Srivastava, Tata Institute of Fundamental Research, Mumbai

Li-Yang Tan, Stanford

Madhur Tulsiani, TTIC

Gregory Valiant, Stanford

Thomas Vidick, California Institute of Technology (chair)

Virginia Vassilevska Williams, MIT

Ronald de Wolf, CWI and University of Amsterdam

David Woodruff, Carnegie Mellon University

## TCS Women

*[Guest post from Virginia Vassilevska Williams –Boaz]*

Barna Saha, Sofya Raskhodnikova and I are organizing the second annual TCS Women event at STOC’19. We had an event at STOC’18 and it went really well. We envision an exciting program for the TCS Women event at STOC 2019. The details about the program are forthcoming. The new TCS Women website is here: https://sigact.org/tcswomen/. There you can learn more about our initiative.

In addition to the event, we have secured generous funding (similar to last year) from the NSF and industrial partners such as Akamai, Amazon, Google and Microsoft for **travel grants** for women and underrepresented minority students and postdocs to attend STOC. These grants are separate from the STOC travel awards and you can apply to both.

The application deadline for the TCS Women travel scholarship is **April 25th, 2019**. The information on how to apply is available here https://sigact.org/tcswomen/tcs-women-travel-scholarship/. We hope to support as many women and underrepresented minority students and postdocs as possible all over the globe to come to STOC and FCRC. The participants will also have the opportunity to present their work at the STOC poster session.

If you are aware of eligible students (not only PhD) who are interested in attending STOC, please encourage them to apply.

Best wishes,

Virginia on behalf of the TCS Women organizers

## Donate to AddisCoder!

In 2011, as a graduate student Jelani Nelson founded the AddisCoder course on algorithms and coding for high schoolers in Addis Ababa Ethiopia. Since then the course has been offered twice more, and this summer it will occur again for the fourth time. Over 330 students have completed the course, and some of its alumni are now students in top institutions in the U.S., Ethiopia, and all over the world. I was a lecturer in this course in 2016 and will be again this summer (together with Jelani, Timnit Gebru, and Daniel Kang).

The students in AddisCoder come from all over Ethiopia, and many have never considered computer science as a potential field of study or even touched a computer. Once when meeting an alum, I asked whether he enjoyed the course. He told me that he didn’t just enjoy it, it was a lifechanging experience for him, and I believe the same holds for many other graduates.

The reason for this post is that AddisCoder is now a tax-exempt non profit organization, which you can donate to through our website. It is a fairly lean operation, but funds are needed for travel of teaching staff (with TA’s that come from all over the world). Please consider supporting AddisCoder yourself (any amount helps!), and forward this information to any friends or colleagues. If you work for a company that might be interested in supporting AddisCoder, you can contact Jelani at the email minilek+addiscoder@gmail.com .

## PCP Fest videos

In December I participated in the wonderful “PCP Fest” workshop in Tel Aviv. The videos from these workshops are now online on their youtube channel.

The channel contains not just videos of talks but also two wide ranging interviews of Alon Rosen with Avi Wigderson and Christos Papadimitriou, as well as a discussion between them. I think many people, especially students, might find these interviews inspiring. They cover Avi’s and Christos’ personal and intellectual journeys, that are similar in some ways and quite different in others.