**Theorem** (Hopf and Pannwitz, 1934)**:** Let be a set of points in the plane in general position (no three points on a line) and consider line segments whose endpoints are in . Then there are two disjoint line segments.

**Micha Perles’s proof by Lice:**

Useful properties of lice: A louse lives on a head and wishes to lay an egg on a hair.

Think about the points in the plane as little heads, and think about each line segments between two points as a hair.

The proof goes as follows:

**Step one:** You take lice from your own head and put them on the points of $X$.

**Step two:** each louse examines the hairs coming from the head and lay eggs (on the hair near the head)

**Step three** (not strictly needed)**:** You take back the lice and put them back on your head.

To make it work we need a special type of lice: spoiled-left-wing-louse.

A spoiled-left-wing louse lays an egg on a hair if and only if the area near the head, 180 degrees to the right of this hair is free from other hairs.

**Lemma:** Every louse lays at most one egg.

**Proof of lemma: **As you see from the picture, if the louse lays an egg on one hair, this hair disturbs every other hair.

**Proof of theorem continued:** since there are line segments and only at most eggs there is a hair X between heads A and B with no eggs.

We look at this hair and ask:

Why don’t we have an egg near head A: because there is a hair Y in the angle 180° to the right.

Why don’t we have an egg near head B: because there is a hair Z in the angle 180° to the right.

Y and Z must be disjoint. Q. E. D.

**Remarks:** We actually get a ZIG formed by Y, X, and Z

If we use right-wing-spoiled lice we will get a ZAG.

We can allow the points not to be in general position as long as one hair from a head does not contain another hair from the same head.

The topological version of this problem is the infamous Conway’s thrackle conjecture. See also Stephan Wehner page about it.

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In the very short 2003 paper A simple algorithm for edge-coloring bipartite multigraphs, Information Processing Letters 85 (2003), 301-302, Noga Alon used a similar idea for algorithmic purposes. (He also observed the connection to the camel riddle). Here is how the extra camel idea is used for:

**Theorem:** A bipartite cubic graph has a perfect matching.

(A cubic graph is a 3-regular graph.)

**Proof:** Suppose that has vertices. Multiply each edge times ( large) so that the degree of each vertex is of the form . Now ask your neighbor to give you an additional perfect matching and add it to the graph which now is regular of degree . The new graph has an Eulerian cycle. (If not connected, every connected component has an Eulerian cycle.) When we walk on the Eulerian cycle and take either the even edges or the odd edges we get two subraphs of degree . At least one of them does not use all the edges of the neighbor. We move to this subgraph and give the unused edge back to the neighbor. We repeat, and in each step we move to a regular subgraph of degree a smaller power of two, and give back at least one edge to the kind neighbor. If is large enough to start with we will end with a perfect matching that uses only the original edges of our graph.

(Remark: We can take or . If we are a little more careful and in each step try to give many edges back to the kind neighbor we can use or so.)

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Being invited to give a plenary lecture at the 7ECM was a great honor and, as Keren Vogtmann said in her beautiful opening lecture on outer spaces, it was also a daunting task. I am thankful to Günter Ziegler for his introduction. When I ask myself in what way I am connected to the person I was thirty years ago, one answer is that it is my long-term friendship with Günter and other people that makes me the same person. My lecture deals with the analysis of Boolean functions in relation to expansion (isoperimetric) properties of subsets of the discrete n-dimensional cube. The lecture has made a subjective selection of some results, proofs, and problems from this area.

Yesterday, Leonid Polterovich and I were guests of the exhibition “Transcending Tradition: Jewish Mathematicians in German-Speaking Academic Culture.” I will start by briefly mentioning the great impact of German-Jewish mathematicians on the early history of the Einstein Institute of Mathematics and Physics at the Hebrew University of Jerusalem, my main academic home since the early seventies. In this picture you can see some early faces of our Institute.

Edmund Landau, the founder and first head of the Institute, moved to Jerusalem from Göttingen in 1927 and moved back to Göttingen a year and a half later. Abraham (Adolf) Halevi Fraenkel moved to Jerusalem from Kiel in 1928 and he can be seen as the father of logic, set theory, and computer science in Israel. My own academic great-grandfather is Michael Fekete, who immigrated to Jerusalem from Budapest in 1928.

I would like to say a few words about two remarkable documents written by Landau in 1925, both related to the inauguration ceremony of the Hebrew University of Jerusalem. You can read more about them in the paper *Zionist internationalism **through number theory: Edmund Landau at the Opening of the Hebrew University **in 1925* by Leo Corry and Norbert Schappacher . The first document is Landau’s toast for the opening ceremonies. Let me quote two sentences:

May great benefit emerge from this house dedicated to pure science, which does not know borders between people and people. And may this awareness emerge from Zion and penetrate the hearts of all those who are still far from this view.

The second document, also from 1925, is probably the first mathematical paper written in Hebrew in modern times. It is devoted to twenty-three problems in number theory and here are its concluding sentences.

At this number of twenty-three problems I want to stop, because

twenty-three is a prime number, i.e., a very handsome number for us. I am certain that I should not fear to be asked by you, for what purpose does one deal with the theory of numbers and what applications may it have. For we deal with science for the sake of it, and dealing with it was a solace in the days of internal and external war that as Jews and as Germans we fought and still fight today.

I wish to make two remarks: First, note that Landau moved from the very ambitious hopes and program of science as a bridge that eliminates borders between nations to a more modest and realistic hope that science and mathematics give comfort in difficult times. Juggling between very ambitious programs and sober reality is in the nature of our profession and we are getting paid both for the high hopes and aims, as well as for the modest results. Second, Landau is famous for his very rigorous and formal mathematical style but his 1925 lecture is entertaining and playful. I don’t know if his move to Jerusalem was the reason for this apparent change of style. Parts of Landau’s lecture almost read like stand-up comedy. Here is, word for word, what Landau wrote about the twin prime conjecture:

Satan knows [the answer]. What I mean is that besides God Almighty no one knows the answer, not even my friend Hardy in Oxford.

These days, ninety years after Landau’s lecture, we can say that besides God Almighty no one knows the answer and not even our friend James Maynard from Oxford. We can only hope that the situation will change before long.

Landau’s hopeful comments were made only nine years after the end of the terrible First World War. He himself died in 1938 in Berlin, after having been stripped of his teaching privileges a few years earlier. I don’t know to what extent the beauty of mathematics was a source of comfort in his last years, but we can assume that this was indeed the case. My life, like the lives of many others of my generation, was overshadowed by the Second World War and the Holocaust and influenced by the quest to come to terms with those horrible events.

Here is the videotaped lecture. (and the slides).

More sources: The home page of my Institute, and a page about its history; An article in the AMS Notices; A blog post in Hebrew about the 1925 Hebrew University events; Shaul Katz: Berlin roots – Zionist incarnation: The ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem. *Science in Context* 17(1/2), 199-234 (2004); A blog post about Yaakov Levitzki and the Amitzur-Levitzki theorem; Schappacher, Norbert: Edmund Landau’s Göttingen — From the life and death of a great mathematical center. *Mathematical Intelligencer* 13 (1991), 12-18. (Talk at the Dedication of the *Landau Center for Research in Mathematical Analysis*, Jerusalem, Feb. 28th, 1989). A recent post on Tao’s blog related to mathematics, science, scientific relations and recent events.

**(Below) First and last page of Hardy and Heilborn obituary on Landau.**

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Boolean functions, Influence, threshold, and Noise

Trying to follow an example of a 1925 lecture by Landau (mentioned in the lecture), the writing style is very much that of a lecture. It goes without saying that I will be very happy for corrections and suggestions of all kinds.

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I am sure that every one of the readers of this blog heard about Laci Babai’s quasi-polynomial algorithm for graph isomorphism and also the recent drama about it: A mistake pointed out by Harald Helfgott, a new sub-exponential but not quasi-polynomial version of the algorithm that Laci found in a couple of days, and then, a week later, a new variant of the algorithm again found by Laci which is quasi-polynomial. You can read the announcement on Babai’s homepage, three excellent Quanta magazine articles by Erica Klarreich** **(I,II,III), Blog posts over Harald’s blog (III,II,I) with links to the video and article (in French), and many blog posts all over the Internet (GLL4,GLL3,GLL2,GLL1,…).

Babai’s result is an off-scale scientific achievement, it is wonderful in many respects, and I truly admire and envy Laci for this amazing breakthrough. I also truly admire Harald for his superb job as a Bourbaki expositor.

Tel Aviv University: Sackler distinguished lectures in Pure Mathematics Wednesday, January 18 (Poster. Sorry, too late, I heard it was very inspiring, don’t miss the other talks!)

Tel Aviv University Combinatorics seminar: Sunday, Jan. 22, 10:00-11:00, Location: Melamed (Shenkar building, ground floor, room 6)

Title: **Canonical partitioning and the emergence of the Johnson graphs:** **Combina****torial aspects of the Graph Isomorphism problem **

(The talk does not depend on Wednesday’s talk)

Hebrew University Colloquium San. Jan 22, 16:00-17:00 Title: **Graph isomorphism and coherent configurations: The Split-or-Johnson routine**

Lecture room 2, Manchester building (Mathematics)

*Local versus global symmetry and the Graph Isomorphism problem I–III*

Lecture I: Monday, January 23, 2017 at 15:30

Lecture II: Tuesday, January 24, 2017 at 15:30

Lecture III: Thursday, January 26, 2017 at 15:30

All lectures will take place at Auditorium 232, Amado Mathematics Building, Technion (Website)

Pekeris lecture, Jan 29, 11:00-12:00 **Hidden irregularity versus hidden symmetry **

EBNER AUDITORIUM (webpage)

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The purpose of this post is to (belatedly) formally announce that the project has ended, to give links to the individual posts and to briefly mention some advances and some thoughts about it.

The posts were

- Polymath10: The Erdos Rado Delta System Conjecture, Posted Nov 2, 2015. (138 comments)
- Polymath10, Post 2: Homological Approach, Posted Nov 10, 2015. (125 comments.)
- Polymath 10 Post 3: How are we doing?, Posted Dec 8, 2015. (103 comments.)
- Polymath10-post 4: Back to the drawing board?, Posted Jan 31, 2016. (11 comments.)
- Polymath 10 Emergency Post 5: The Erdos-Szemeredi Sunflower Conjecture is Now Proven. Posted May 17, 2016. (35 comments.)
- Polymath 10 post 6: The Erdos-Rado sunflower conjecture, and the Turan (4,3) problem: homological approaches, Posted on May 27, 2016. (5 comments.)

The problem was not solved and we did not come near a solution. The posts contain some summary of the discussions, a few results, and some proposals by the participants. Phillip Gibbs found a remarkable relation between the general case and the balanced case. Dömötör Palvolgyi shot down quite a few conjectures I made, and Ferdinand Ihringer presented results about some Erdos-Ko-Rado extensions we considered (In term of upper bounds for sunflower-free families). Several participants have made interesting proposals for attacking the problem.

I presented in the second post a detailed homological approach, and developed it further in the later threads with the help of Eran Nevo and a few others. Then, after a major ingredient was shot down, I revised it drastically in the last post.

Participants made several computer experiments, for sunflower-free sets, for random sunflower-free sets, and also regarding the homologica/algebraic ideas.

The posts (and some comments) give some useful links to literature regarding the problem, and post 5 was devoted to a startling development which occurred separately – the solution of the Erdos-Szemeredi sunflower conjecture for sunflowers with three petals following the cup set developments. (The Erdos-Szemeredi sunflower conjecture is weaker than the Erdos-Rado conjecture.)

A (too) strong version of the homological conjecture appeared in my 1983 Ph. D. thesis written in Hebrew. The typesetting used the Hebrew version of Troff.

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Five years ago I wrote a post entitled Is Backgammon in P? It was based on conversations with Peter Bro Miltersen and Uri Zwick (shown together in the above picture) about the computational complexity of computing the values (and equilibrium points) of various stochastic games, and also on some things I learned from my game theory friends over the years about proving that values exist for some related games. A few weeks ago two former students of Peter, Rasmus Ibsen-Jensen and Kristoffer Arnsfelt Hansen visited Israel and I had a chance to chat with them and learn about some recent exciting advances.

Is there a polynomial time algorithm for chess? Well, if we consider the complexity of chess in terms of the board size then it is fair to think that the answer is “no”. But if we wish to consider the complexity in terms of the number of all possible positions then it is easy to go backward over all positions and determine the outcome of the game when we start with each given position.

Now, **what about backgammon? ** Like chess, backgammon is a game of complete information. The difference between backgammon and chess is the element of luck; at each position your possible moves are determined by a roll of two dice. This element of luck increases the computational skill needed for playing backgammon compared to chess. It can easily be seen that optimal strategy for players in backgammon need not involve any randomness.

**Problem 1: **Is there a polynomial time algorithm to find the optimal strategy (and thus the value) of a stochastic zero sum game with perfect information? (Like backgammon)

This question (raised by Ann Condon in 1998) represents one of the most fundamental open problem in algorithmic game theory.

Heads-up poker is just a poker game with two players. To make it concrete you may think about heads-up Texas hold’em poker. This is not a game with complete information, but by according to the minmax theorem it still has a value. The optimal strategies are mixed and involve randomness.

**Problem 2: **Is there a polynomial time algorithm to find the optimal strategy (and thus the value) of a stochastic zero-sum game with incomplete information? (like heads-up Texas hold’em poker).

It will be very nice to find even a sub-exponential algorithm for a stochastic zero-sum game with incomplete information like poker.

**Problem 2′: **Is there a subexponential-time algorithm to find the optimal strategy (and thus the value) of a stochastic zero-sum game with incomplete information?

For games with complete information like backgammon, a subexponential algorithm was found by Walter Ludwig and in greater generality by Sergei Vorobyov, Henrik Björklund, and Sven Sandberg. It is related to subexponential simplex-type algorithms for linear programming called RANDOM-FACET, found in the early 90s by Matousek, Sharir and Welzl and myself.

Kristoffer Arnsfelt Hansen (see abstract below) presented a polynomial-time algorithm for 2-persons zero sum stochastic games, when the games have a bounded number of states. (Earlier algorithms were exponential.) The paper is: Exact Algorithms for Solving

Stochastic Games by Kristoffer Arnsfelt Hansen, Michal Koucky, Niels Lauritsen,

Peter Bro Miltersen, and Elias P. Tsigaridas. Slides of the talk are linked here.

As for backgammon there are very good computer programs. (We talked about chess-playing computers in this guest post by Amir Ban and since that time Go-playing computers are also available.) The site Cepheus Poker Project and this science paper Heads-up limit hold’em poker is solved are good sources on major achievements by a group of researchers from Alberta regarding two players poker.

**Problem 3: **Is there a polynomial time algorithm to find Nash equilibrium point (or another form of optimal strategy) of a stochastic n-player game with incomplete information? (like Texas holdem poker.) Here *n* is fixed and small.

I think that people are optimistic that even the answer to problem 3 is yes. (There are hardness results for finding equilibrium points in matrix games but the relevance to our case is not clear.) If we want an algorithm which optimally plays poker, it is not clear that finding a Nash equilibrium is the way to go.

**Problem 4:** Find an algorithm for playing Texas hold’em poker when there are more than two players.

When the objective is to maximize revenues against human players I expect that it will be possible to develop computer programs for playing poker better than humans.

**Problem 5:** How to play the game MEDIAN of the previous post?

**Matching pennies** is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match (both heads or both tails), then Even keeps both pennies, so wins one from Odd (+1 for Even, −1 for Odd). If the pennies do not match (one heads and one tails) Odd keeps both pennies, so receives one from Even (−1 for Even, +1 for Odd) (source wikiPedia)

Variants of this game have been played since ancient times. In Hebrew matching pennies is called ZUG O PERET (even or odd; זוג או פרט). It is played like this: There are two players. Each player in his turn makes an announcement “even” or “odd”. Then each of the two players shows (simultaneously) some number of fingers and the announcing player wins if the sum of fingers has the announced parity.

The big match is a drastic repeated version of matching pennies. The game is played between players Even and Odd. Each player has a penny and in each stage must secretly turn the penny to heads or tails and the payoffs are the same as in matching pennies. If Even plays “head” the game continues to the next stage. However if Even plays “tails” (or tries for the “big match” as it is called) then the payoff in that round is repeated for all future rounds: Namely, if the pennies match Even will get 1 for all future rounds, and if the pennies do not match Even will pay one for all future rounds.

By playing heads with probability 1/2 and tails with probability 1/2, Odd can guarantee an expected payoff of 0. But what about Even? Can he also guarantee an expected payoff of 0? This was an open question for quite some time. The big match was introduced in 1957 by Dean Gillette who asked if the game has a value, namely if Even has a strategy to guarantee a payoff of 0.

**Problem 7:** Does big match has a value?

Here is a blog post on the big match by Presh Talwalkar on his blog “mind your decisions.”* *You also can read about the big match in this post of Kevin Bryan’s economics blog “a fine theorem.”

In 1968, David Blackwell and Thomas S. Ferguson settled Gillete’s question and proved that even can guarantee a zero payoff and thus big match did in fact have a value. This was the first step to showing all zero-sum stochastic games have value under limiting average payoff, which was proven in 1982 by Mertens and Neyman.

Rasmus Ibsen-Jensen presented both positive and negative results on attaining the value for the big match with limited types of strategies and also on complexity issues regarding other stochastic games. Here are the slides for Rasmus’ talk (see full abstract below). Part of the talk is based on the paper The Big Match in Small Space by Kristoffer Arnsfelt Hansen, Rasmus Ibsen-Jensen, and Michal Koucky.

This is a remarkable story with very important results and open questions. Here is the Wikipedia article on stochastic games and this short paper by Eilon Solan. I see now that the post is becoming too long and I will have to talk about it in a different post.

**Problem 8** (informal): Does every stochastic game have ~~a value~~ an equilibrium?

Following a major step by Truman Bewley and Elon Kohlberg (1976), Jean-François Mertens and Abraham Neyman (1981) proved that every two-person zero-sum stochastic game with finitely many states and actions has a uniform value. Nicolas Vieille (2000) has shown that all two-person stochastic games with finite state and action spaces have a limiting-average equilibrium payoff. The big question is to extend Vieille’s result to games with many players.

Kristoffer, Rasmus and Abraham (Merale) Neyman.

Exact algorithms for solving stochastic games

Speaker: Kristoffer Arnsfelt Hansen, Aarhus University

==================================================

In this talk we consider two-player zero sum stochastic games

with finite state and action space from an algorithmic

perspective. Prior to our work, algorithms for solving

stochastic games relied either on generic reductions to decision

procedures for the first order theory of the reals or on value or

strategy iteration. For all these algorithms, the complexity is

at least exponential even when the number of positions is a

constant and even when only a crude approximation is required

We will present an exact algorithm for solving these games based

on a simple recursive bisection pattern. The algorithm runs in

polynomial time when the number of positions is constant and our

algorithms are the first algorithms with this property. While the

algorithm is not based directly on real algebraic geometry, our

algorithm depends heavily on results from the field.

Based on joint work with Michal Koucký, Niels Lauritzen,

Peter Bro Miltersen, and Elias P. Tsigaridas published at STOC’11.

Abstract: The talk will attempt to characterize good strategies for some special cases of stochastic games. For instance, the talk will argue that there might always be a good strategy with a certain property for all games in a special case of stochastic games or that no good strategy exists that has some property for some game. Concretely,

1) for the stochastic game the Big Match, no good strategy (for lim inf) exists that only depends on how long the game has been playing and a finite amount of extra memory (when the extra memory is updated deterministically).

2) for the Big Match there is a good strategy that uses only a single coin flip per round and exponentially less space then previous known good strategies.

3) let x be the greatest reward in a stochastic game. The talk will next give a simple characterization of the states of value equal to x for which there exists either (a) an optimal strategy; (b) for each epsilon>0, a stationary epsilon-optimal strategy; or (c) for each epsilon>0, a finite-memory epsilon-optimal strategy (when the memory is updated deterministically) . The characterization also gives the corresponding strategy.

4) the talk will then consider stochastic games where there exists epsilon-optimal stationary strategies for all epsilon>0. It will argue that the smallest positive probability in a stationary epsilon-optimal strategy must be at least double exponential small for some sub-classes of stochastic games, while for other classes exponential small probabilities suffices.

1) and 2) is based on “The Big Match in Small Space”, 3) is based on “The Value 1 Problem Under Finite-memory Strategies for Concurrent Mean-payoff Games” 4) is based on “Strategy Complexity of Concurrent Stochastic Games with Safety and Reachability Objectives” and “The complexity of ergodic mean-payoff games“. All papers can be found in http://Rasmus.Ibsen-Jensen.com

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Ehud Friedgut reminded me of the game MEDIAN which I proposed many years ago.

There are three players and they play the game for eight rounds. In every round all players simultaneously say a number between 1 and 8. A player whose number is (strictly) between the other two get a point. At the end of the game the winner is the player whose number of points is strictly between those of the others.

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The institute was inaugurated in 1925 by a lecture of Edmund Landau, who later served as one of the first heads of the department. It has since developed into a defining and leading place in mathematics research, with world renowned research faculty working in diverse areas of up-to-date research.

Our graduate program gives students the chance to develop into researchers that shape mathematics of the future. The department offers a uniquely attractive environment to learn and work, with weekly seminars, frequent special lecture series on current topics in mathematics and scientific exchange with visiting researchers from around the world.

This is enriched further by the Israel Institute of Advanced Studies situated at Hebrew University that organizes thematic years on state-of-the art advances in science, and the close collaboration with the renowned departments of physics and computer science and engineering. You can venture even further and visit the nearby University of Tel Aviv, the Technion, the Weizmann Institute, Bar Ilan University, Ben Gurion University or the University of Haifa, that contribute to the active research environment and that we here enjoy a frequent and close scientific exchange with.

Accepted graduate students are expected to take up the standard course load in the department (12 credit points — each credit point is roughly 1 hour per week for a semester long course) — but are otherwise free to pursue their research.

The Hebrew University is a unique place that unites students and researchers of all faiths and origins to work together and advance science in a secular and inclusive atmosphere for the betterment of our world. It is consistently ranked among the top universities worldwide. Advanced courses for PhD students as well as all research seminars and the Colloquium lectures are typically given in English.

You can explore the unique cultural environment the city has to offer, uniting a rich past with a vibrant youth culture, allowing you to witness history as well as one of the many street concerts.

Our admissions procedure looks for students with a record of excellent academic achievements. We ask you to submit a CV, a brief outline of your research interests, and scans of official university transcripts (as PDF files), as well as names of three possible advisors at our department. These names are not binding, but help us get a feeling what your goals are. You also need to arrange for two recommendation letters to be sent to us directly by the letter writers to math.gradschool@mail.huji.ac.il . To ensure we can properly access the recommendations, please ensure the subject of the email will be “Recommendation letter for Last Name, First Name”. For the application material, please ensure the subject of the email will be “Application Last Name, First Name”. As part of our admissions process, applicants who pass our initial screening would typically be either invited for an interview or be interviewed over skype.

The deadline for receiving all application materials, including recommendation letters, is January 31. Applications submitted after this date will be considered on a case by case basis.

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Monday 10-11:45 (Combinatorics seminar) **Adam Shefer – Geometric Incidences and the Polynomial Method**

Location: Rothberg (CS) B220

On Monday afternoon we will have four talks at the library of Belgium house by

**13:15-14:00 Peter Pach, Progression-free sets. New: SLIDES**

**14:10-14:55 Shoham Letzter,**

**15:15-16:00 Jordan Ellenberg, **

**16:10- 16:55 Fedya Petrov, Group rings vs. polynomials. New: SLIDES**

and a** problem session** moderated by Jordan starting at 16:55. New: PROBLEMS.

On Tuesday we start at 9:30 and will have four talks at the library of Belgium house:

**9:30-10:15 Noga Alon, Combinatorial Nullstellensatz and its algorithmic aspects. New: SLIDES**

** 10:35-11:20 Olga Holtz, A potpourri on power ideals, hyperplane
arrangements, graphs, and zonotopes (NEW: SLIDES)**

( lunch)

**UPDATES—Changes**

**Wednesday 9:30-10:15, ** **Anurag Bishnoi, zeros of polynomials over a finite grid. NEW:SLIDE.**

**Thursday 11:00-12:00 Seva Lev, Avoiding 3AP with differences in Room 209 Mathematics.**

Further informal discussions and talks may continue on Wednesday/Thursday.

The Thursday 14:30 Colloquium by **Jordan Ellenberg **will be on **The cap set problem**.

I will update titles as they come along.

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