**Paul Ellis, Rutgers University
2 invariants in knot theory **

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If I draw two knots on the chalk-board, can you tell me if one can be moved around to look like the other? If it cannot, could you prove it? I will show you two gadgets that will do this in certain cases, and also breifly discuss the idea of a complete invariant.

** DIMATIA REU students
What to expect in Prague **

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We will give a series of brief talks about the Czech Republic, the Czech language, Prague, the Charles University, and the DIMATIA center. We hope that the talks will interest all the REU students, even those who don't plan to go to Prague anytime soon.

**board game night! meet at Pad Thai (217 Raritan, Highland Park) at 6:30pm
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Meet at Pad Thai for dinner. Afterwards, we'll walk to Lara's house for a night of board games, cards, and other such fun.

** Eric Allender, Rutgers University
Some Fun Theorems in Computational Complexity Theory **

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Computational complexity theory tries to show that certain problems are hard to compute. However, does this even make sense? Can the notion of being "hard to compute" be made mathematically precise? There are some surprising difficulties that need to be overcome, in order to obtain a meaningful concept of computational complexity; many intuitive notions (such as the idea of an "optimal algorithm" for a given problem) have to be abandoned in the general case. This talk will describe some of the main triumphs of the field of computational complexity theory, as well as discussing some of the limitations that are inherent in any theory of computational complexity.

**LABYRINTH meet outside the DIMACS 4th floor lounge at 7pm
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Labyrinth is a night game usually for teams with two members. The labyrinth consists of several important positions chosen by orginazers in advance. Each team begins at a starting position and its task is to leave the labyrinth (to find the final position).

Positions are marked with letters. At every position you will find instructions that tell you where you can go next. Typically, there is also a question that should help you decide where to go. If you give the correct answer it leads you to the finish in the shortest possible way. Otherwise, the way will be probably longer.

Positions are located in an area of few square kilometers and the game usually takes 2-8 hours depenidng on teams abilities to give correct answers.

** Nina Fefferman, DIMACS/Tufts University
When females should stop supporting lazy males: mathematics and honey bees **

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Honey bees live in complicated and highly organized societies. Female workers provide in every way for the needs of the colony while males lounge around being fed, groomed and generally taking up resources and getting in the way. They present a huge cost to the hive, all in the hope that a few of them will get the chance to mate. However, at some point, the males overstay their welcome and the workers will forcibly eject them from their hives, leaving them to die alone in the cold cruel world. So a few natural questions arise: are worker bees efficiently making the decision to eject these drones? How do we define 'efficiency' in this case: locally or globally? How can we determine whether or not the observed behaviors match up? To answer these questions, we use actual experimental hives, simulation experiments and linear programming (a method of optimization where the objective function and all related constraints are linear). This work is ongoing, so don't expect too many final answers.

** Robert Wilson, Rutgers Department of Mathematics
Eric Allender, Rutgers Department of Computer Science
Panel on Graduate School in Mathematics and Computer Science **

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This will be a panel to help answer questions about applying to graduate school and what graduate school is like. Come with lots of questions!

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Capture the Flag **

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Capture the Flag is a game that goes something like this: http://www.usscouts.org/games/game_cf.html. Meet outside CoRE, and we'll divide up territory, choose teams, and have some fun!

** József Beck, Rutgers University
Quadratic Forms -- Old and New Results **

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Modern number theory started with the conjecture of Fermat (and Girard) that the Pythagorean equation p=x^2+y^2 is solvable to an odd prime p if and only if p is of the form 4k+1. The first published proof was given by Euler about 100 years later. Since then there is a huge interest in understanding similar questions. Shockingly, most of them are still open. In particular, I will explain what we know about a particular question: Euler's Numerus Idoneus Problem.

** Chung-chieh Shan, Rutgers University
Language machines **

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If the mind is like a computer, how does it listen and speak? For decades, the studies of human and computer languages have informed each other. For example, formal-language theory in computer science originated from models of natural-language syntax. More recently, the same tools are useful for explaining apparent noncompositionality in both kinds of languages.

** Scott Schneider, Rutgers University
How to Give a Talk **

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We will discuss useful tips for giving research talks. Important information about your first REU presentations (June 20-21) will be provided.

** Andrew Sills, Rutgers University
An Introduction to Integer Partitions **

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A partition of an integer n is a representation of n as a sum of positive integers where the order of the summands is considered irrelevant. Thus there are five partitions of the integer 4, namely 1+1+1+1, 2+1+1, 2+2, 3+1, and 4 itself. Partitions were first studied systematically by Euler in the eighteenth century; nonetheless partitions continue to be an active field of research today, employing techniques from combinatorics, classical analysis, the theory of modular forms, and other areas of mathematics. I plan to talk about some of the classical results which can be obtained by elementary methods.