### <u>Charmed Roots and the Kroweras Complement</u> <iframe data-src="charmed_NN.html" data-preload width="950" height="950" style="border: 1px solid black;"></iframe> <!--<iframe data-src="title.html" data-preload></iframe> have the cube!--> <!--maybe make hover on bijections into animations?-->

<br><br><br><br><br> ### <u>Nonnesting vs. Noncrossing</u>

### <u>Roots</u> - Let $\Phi$ be an irreducible root system of rank $r$ with Weyl group $W$. - The <span style="color:#3498db">*positive root poset*</span> $\Phi^+$ is defined by $\alpha < \beta$ iff $\beta-\alpha \in \mathbb{Z}_{\geq 0} \Phi^+$. <iframe data-src="root_posets.html" data-preload width="750" height="750"></iframe>

### <u>Type $F_4$</u> <img data-src="Ringel1.png" width="500"><br> <img data-src="Ringel2.png" width="400">

### <u>Roots and Exponents</u> - The <span style="color:#3498db">*positive root poset*</span> $\Phi^+$ is defined by $\alpha < \beta$ iff $\beta-\alpha \in \mathbb{Z}_{\geq 0} \Phi^+$. - The minimal elements are the $r$ <span style="color:#3498db">*simple roots*</span>; - The maximal element is the <span style="color:#3498db">*highest root*</span>, with rank $h-1$; and - The <span style="color:#3498db">*exponents*</span> $e_1,e_2,\ldots,e_r$ are the dual partition of the rank sizes. <iframe data-src="root_posets.html" data-preload width="450" height="450"></iframe>

### <u>Roots and Exponents</u> > The exponents for type $A_n$ are $1,2,\ldots,n.$ > > The exponents for type $B_n$ are $1,3,\ldots,2n-1.$ > > The exponents for type $F_4$ are $1,5,7,11.$ <iframe data-src="root_posets.html" data-preload width="450" height="450"></iframe>

### <u>Roots and Exponents</u> <img data-src="kostant1.png" height="600">

### <u>Nonnesting Gauss</u> - Gauss's elementary school teacher's punishment: > Sum the exponents for $\Phi$ of type $A_{100}.$ <img data-src="10Deutschmarksfront.jpg" width="600">

### <u>Nonnesting Gauss</u> - Gauss's elementary school teacher's punishment: > Sum the exponents for $\Phi$ of type $A_{100}.$ - Gauss immediately noticed that $e_i + e_{r+1-i} = h$. <iframe data-src="root_posets_down.html" data-preload width="550" height="550"></iframe>

### <u>Nonnesting Gauss</u> - By definition of the exponents, $e_1+e_2+\cdots+e_r = |\Phi^+| = N.$ - Gauss's computation: $$\begin{array}{llllllllll} & e_1 &+&e_2 &+&\cdots &+&e_r &=& N \\\ +& e_r &+&e_{r-1} &+&\cdots &+&e_1 &=& N \\\ \hline & h &+&h &+&\cdots &+& h &=& 2N \\\ &&&&rh&&& &=& 2N \end{array}$$ - Therefore, $e_1+\cdots+e_r = N = \frac{rh}{2}.$ <iframe data-src="root_posets_down.html" data-preload width="350" height="350"></iframe>

### <u>Nonnesting Gauss</u> > $$\begin{aligned} \text{Type $A_n$}&: 1+2+\cdots+n &&= \frac{n(n+1)}{2} \\\ \\ \text{ Type $B_n$}&: 1+3+\cdots+(2n-1) &&= n^2.\end{aligned}$$ <iframe data-src="root_posets_down.html" data-preload width="550" height="550"></iframe>

### <u>Nonnesting Gauss</u> > $$\text{Type $F_4$}: 1+5+7+11 = 24.$$ <img data-src="Ringel2.png" height="400">

### <u>Noncrossing Gauss</u> <img data-src="Kostant3.png" width="500"> <br> <img data-src="kostant2.png" width="500">

### <u>Noncrossing Gauss</u> - A <span style="color:#3498db">*(standard) Coxeter element*</span> is a product of the $r$ simple reflections of $W$ (in any order). > In $S_n$, a Coxeter element is a long cycle with cycle notation $$(\mathbf{1} \textcolor{#3498db}{< a_2 < \cdots < a_{m-1} <} \mathbf{n} \textcolor{#3498db}{> a_{m+1} > \cdots> a_n}).$$ > In $S_9$, $(\mathbf{1} \textcolor{#3498db}{, 3, 4, 7,} \mathbf{9} \textcolor{#3498db}{, 8, 6, 5, 2})$ is a Coxeter element. - Because Dynkin diagrams are trees, all Coxeter elements are conjugate by a sequence of *initial* (or *final*) simple reflections $s$: $$c \to scs \to \cdots$$

### <u>Noncrossing Gauss</u> - Because Dynkin diagrams are trees, all Coxeter elements are conjugate by a sequence of <span style="color:purple">*initial*</span> (or <span style="color:purple">*final*</span>) simple reflections: <iframe data-src="quiver.html" data-preload width="800" height="800"></iframe>

### <u>Noncrossing Gauss</u> - Let $c \in W$ be a Coxeter element with reduced word $\mathbf{c}$ (of length $r$). - The <span style="color:#3498db">*$c$-sorting word*</span> $\mathbf{w}(\mathbf{c})$ is the leftmost reduced word for $w\in W$ in $\mathbf{c}^\infty$. - Let $w_\circ$ be the long element of $W$ (of length $N$). Up to commutations: $$\mathbf{w}\_\circ(\mathbf{c}) \mathbf{w}\_\circ(\widetilde{\mathbf{c}}) \simeq \mathbf{c}^h.$$ <!--, where $w_\circ$ is the long element of $W$.--> <iframe data-src="AR.html" data-preload width="250" height="250"></iframe> - Every root must appear twice in the inversion sequence (since $w_\circ^2=e$): $$2N = rh.$$ <!-- In type $A_n$ and $c=(1,2,\ldots,n)$, we recover Gauss's identity. > > But because we have choices for $c$, we get the identity in many ways.--> <!-- $$\mathbf{w}\_\circ(\mathbf{c})=(s_1s_2\cdots s_n) \cdot (s_1 s_2 \cdots s_{n-1}) \cdot \cdots \cdot (s_1 s_2) \cdot (s_1).$$--> <!--<iframe data-src="AR.html" data-preload></iframe>--> <!--animate picture showing that even in type A get new proofs-->

### <u>Noncrossing Gauss</u> > In type $A_n$ and $c=(1,2,\ldots,n)$, we recover Gauss's identity. > > But because we have choices for $c$, we get $2N=rh$ in many ways. <iframe data-src="AR.html" data-preload width="550" height="550"></iframe> <!-- $$\mathbf{w}\_\circ(\mathbf{c})=(s_1s_2\cdots s_n) \cdot (s_1 s_2 \cdots s_{n-1}) \cdot \cdots \cdot (s_1 s_2) \cdot (s_1).$$--> <!--<iframe data-src="AR.html" data-preload></iframe>--> <!--animate picture showing that even in type A get new proofs-->

<br><br><br><br><br> ### <u>Nonnesting Partitions</u>

### <u>Nonnesting Reiner + Postnikov</u> - The <span style="color:#3498db">*nonnesting partitions*</span> of type $A_n$ are the antichains <font size="5">(or order ideals/filters/etc.)</font> in the positive root poset $\Phi^+$. <iframe data-src="NN.html" data-preload width="800" height="800"></iframe> <!--<section data-background-iframe="NN.html" data-background-interactive></section>--> <!-- maybe throw in picture to draw arcs -->

### <u>Nonnesting Reiner + Postnikov</u> - The <span style="color:#3498db">*nonnesting partitions*</span> of type $W$ are the antichains <font size="5">(or order ideals/filters/etc.)</font> in the positive root poset $\Phi^+$. > Called <span style="color:#3498db">**nonnesting**</span> because in type $A_{n-1}$, antichains correspond to collections of <span style="color:#3498db">**nonnesting**</span> arcs on $n$ vertices. - $\mathrm{NN}(W)$ is counted by the Catalan number $$\mathrm{Cat}(W)=\prod_{i=1}^r \frac{h+1+e_i}{d_i}.$$

### <u>Nonnesting Reiner + Postnikov</u> <img data-src="Reiner2.png" width="500">

### <u>Nonnesting Reiner + Postnikov</u> > $$\begin{aligned}\mathrm{Cat}(A\_{n-1})&=\prod\limits\_{i=1}^{n-1} \frac{n+1+i}{i+1} = \frac{1}{n+1}\binom{2n}{n} = \mathrm{Cat}(n). \\\\ \mathrm{Cat}(B_n)&=\prod\limits_{i=1}^{n} \frac{2n+2i}{2i} = \binom{2n}{n}.\\\\ \mathrm{Cat}(F_4)&=105. \end{aligned}$$

<br><br><br><br> ### <u>Noncrossing Partitions</u>

### <u>Noncrossing Reiner + Bessis</u> <img data-src="Reiner1.png" width="400"> <br> <img data-src="Bessis1.png" width="400">

### <u>Noncrossing Reiner + Bessis</u> - Fix $c$ a Coxeter element of $W$. - The <span style="color:#3498db">*$c$-noncrossing*</span> partitions of type $W$ are collections of $r$ reflections in $\mathrm{inv}(\mathbf{c}^h)$ that product to $c$. <font size="5">(Restricting to the first $\mathbf{w}_\circ(\mathbf{c})$ gives an element $\leq_T c$.)</font> > Called <span style="color:#3498db">**noncrossing**</span> because in type $A_n$, these can be drawn as collections of <span style="color:#3498db">**noncrossing**</span> blocks. - $\mathrm{NC}\_{c}(W)$ is counted by $\mathrm{Cat}(W)$. <iframe data-src="NCnoKrew.html" data-preload width="300" height="300"></iframe>

### <u>Noncrossing Reiner + Bessis</u> - The <span style="color:#3498db">*$c$-noncrossing partitions*</span> of type $A_n$ are set partitions whose blocks are noncrossing when drawn on the cycle notation of $c$. <iframe data-src="NCnoKrew.html" data-preload width="800" height="800"></iframe> <!--<section data-background-iframe="NN.html" data-background-interactive></section>--> <!-- maybe throw in picture to draw arcs -->

<br><br><br><br> ### <u>Nonnesting and Noncrossing Partitions</u>

### <u>NN = NC ?</u> <img data-src="Drew1.png" width="500"> <br> <img data-src="Drew2.png" width="500">

### <u>NN = NC ?</u> Last(?) open(?) problem in Coxeter-Catalan combinatorics (since ~1995): > Find a <span style="color:#3498db">**uniform**</span> bijection between $\mathrm{NN}(W)$ and $\mathrm{NC}\_c(W).$ <iframe data-src="charmed_bij1.html" data-preload width="600" height="600"></iframe> <!--<section data-background-iframe="bijection1.html" data-background-interactive></section>-->

### <u>Open?</u> <img data-src="AST.png" width="600">

### <u>NN = NC ?</u> |year|authors|type|result| |---|---|---|---| |???|everybody|type $A$| resolve crossings into nestings| |~1998|Athanasiadis|type $A$| bijections for specific $c$ (type)| |~2008|Fink, Giraldo|classical| bijections for specific $c$ (type)| |~2009|Stump, Ruby|classical| specific $c$ (openers-closers)| |~2011| Armstrong, Stump, Thomas| all types| inductive characterization for $c$ bipartite via support and rowmotion/the Panyushev map| |~2013| W| all types| (no proofs) conjectures for all $c$, checked by computer in low rank| |2023| DFISATW| type $A$| all $c$| <!-- BENJAMIN DEQUENE, GABRIEL FRIEDEN, ALESSANDRO IRACI, FLORIAN SCHREIER-AIGNER, HUGH THOMAS, AND NATHAN WILLIAMS --> <!-- - (~2008) Fink, Giraldo: bijections in the classical types $A,B/C,D$ using combinatorial models for noncrossing partitions. - (~2011) Armstrong, Stump, Thomas (AST): inductive characterization for $c$ bipartite via support and the Panyushev map. - (~2013) W: some conjectures, no proofs.--> <!--illustrate the usual bijection in linear type A-->

### <u>NN $\neq$ NC: Discrepencies</u> <!--conjugation by initial simplesTwo discrepencies between noncrossing and nonnesting objects:--> 1. $\mathrm{NC}\_c(W)$ requires the choice of a Coxeter element; $\mathrm{NN}(W)$ does not. - This choice doesn't change the combinatorics of noncrossing partitions, since there is an easy bijection between $\mathrm{NC}\_c(W)$ and $\mathrm{NC}\_{scs}(W)$: $$\alpha\_{c,s} : \textcolor{purple}{\pi} \mapsto \textcolor{purple}{s\pi s}.$$ <iframe data-src="NCnoKrew.html" data-preload width="500" height="500"></iframe> <!-- **Fix:** rename to $\mathrm{NN}\_c(W)$. But...why?-->

### <u>NN $\neq$ NC: Discrepencies</u> <!--Two discrepencies between noncrossing and nonnesting objects:--> <!--1. $\mathrm{NC}\_c(W)$ requires the choice of a Coxeter element $c$; $\mathrm{NN}(W)$ does not. $$\alpha\_{c,s} : \pi \mapsto s\pi s.$$--> 2. $\mathrm{NC}\_c(W)$ has the <span style="color:#3498db">*Kreweras complement*</span> (of order $2h$); $\mathrm{NN}(W)$... $$\mathrm{Krew}\_c : \textcolor{purple}{\pi} \mapsto \textcolor{green}{\pi^{-1}c}.$$ <iframe data-src="NC.html" data-preload width="650" height="650"></iframe> <!-- These two maps on $\mathrm{NC}\_c(W)$ are compatible, in the sense that $$\alpha\_{c,s} \circ \mathrm{Krew}\_c = \mathrm{Krew}\_{scs} \circ \alpha\_{c,s}.$$-->

### <u>NN $\neq$ NC: Discrepencies</u> <!--Two discrepencies between noncrossing and nonnesting objects:--> > These two maps on $\mathrm{NC}\_c(W)$ are compatible, in the sense that $$\alpha\_{c,s} \circ \mathrm{Krew}\_c = \mathrm{Krew}\_{scs} \circ \alpha\_{c,s}.$$ <img data-src="NC.png" height="400">

### <u>NN $\neq$ NC: Discrepencies</u> <!--Two discrepencies between noncrossing and nonnesting objects:--> <!--These two maps on $\mathrm{NC}\_c(W)$ are compatible, in the sense that $$\alpha\_{c,s} \circ \mathrm{Krew}\_c = \mathrm{Krew}\_{scs} \circ \alpha\_{c,s}.$$--> > So we want two maps on $\mathrm{NN}\_c(W)$ that are similarly compatible... $$\beta\_{c,s} \circ \mathrm{Krow}\_c = \mathrm{Krow}\_{scs} \circ \beta\_{c,s},$$ <img data-src="NN.png" height="400">

### <u>NN $\neq$ NC: Discrepencies</u> <!-- So we want two maps on $\mathrm{NN}\_c(W)$ that are similarly compatible: $$\beta\_{c,s} \circ \mathrm{Krow}\_c = \mathrm{Krow}\_{scs} \circ \beta\_{c,s},$$--> > ...and a bijection $\mathrm{Charm}_c$ between noncrossing and nonnesting partitions such that $$ \mathrm{Charm}\_c \circ \mathrm{Krew}\_c = \mathrm{Krow}\_c \circ \mathrm{Charm}\_c.$$ <img data-src="NNNC.png" height="300">

<br><br><br><br> ### <u>The K**row**eras complement</u>

### <u>The K**row**eras complement</u> | NN/NC | discoverer | statement | | -------- | ------- | ------- | |$\mathrm{NC}_c(W)$ | ??? | The Kreweras complement has order $2h$| |$\mathrm{NN}(W)$ | Panyushev + AST | *Rowmotion* has order $2h$| |$\mathrm{NN}(S_n)$ | Striker and W + White | SW *promotion* has order $2n$| |$\mathrm{NC}_c(W)$ | Thomas and W | $\mathrm{Krew}_c$ can be computed by associahedron flips (*slow motion*) in $\mathrm{inv}(\mathbf{w}\_\circ(\mathbf{c}))$ order | | $\mathrm{NN}(n)$ | Thomas | SW *promotion* toggles in $\mathrm{inv}(\mathbf{w}_\circ(\mathbf{c}))$ order for $c=(1,2,3,\ldots,n)$| | $\mathrm{NN}(W)$ | Thomas and W | Toggling in $\mathrm{inv}(\mathbf{w}_\circ(\mathbf{c}))$ order *appears* to always have order $2h$ | <!--NN has promotion in linear type A that matches. More general? Remark 4.5. For the linear Coxeter element c1 = (1, 2, . . . , n), Krowc1 toggles roots in lexicographic order [(1, 2),(1, 3), . . . ,(n − 1, n)], which is equivalent under commuting toggles to the sequence that toggles the elements of the root poset from left to right. In [SW12], this map was termed promotion because—viewing a nonnesting partition as a standard Young tableau of shape 2 × n— Krowc1 recovers Sch¨utzenberger promotion on standard Young tableaux. The main result of [SW12] showed that promotion on nonnesting partitions was conjugate by toggles to rowmotion. Our Krow-eras complement is a reference to this second map.--> <!--pic here?--> <!--NN has Panyushev in bipartite type that matches. (Bipartite c comes from Panyushev and rowmotion! AST)-->

### <u>The K**row**eras complement</u> - The <span style="color:#3498db">*Kroweras complement*</span> toggles in $\mathrm{inv}(\mathbf{w}_\circ(\mathbf{c}))$ order. <!--- To have a commuting picture, we need: - a bijection $\beta\_{c,s}: \mathrm{NN}\_c(W) \to \mathrm{NN}\_{scs}(W)$ such that $$\beta\_{c,s} \circ \mathrm{Krow}\_c = \mathrm{Krow}\_{scs} \circ \beta\_{c,s}, \text{ and }$$ - a bijection $\mathrm{Charm}_c: \mathrm{NN}_c(W) \to \mathrm{NC}_c(W)$ such that $$\mathrm{Krew}_c \circ \mathrm{Charm}_c = \mathrm{Charm}_c \circ \mathrm{Krow}_c.$$--> <iframe data-src="AR.html" data-preload width="600" height="600"></iframe> <!--Remark 4.4. The map Krowc does not depend on the choice of reduced word c for c: all reduced words for c are related by a sequence of commutations of non-adjacent transpositions, and each commutation only affects the sequence α (k) by interchanging pairs of roots that are not adjacent in the root poset. By the remark after Equation (8), this, therefore, does not change the composition of toggles. We may therefore view Krowc as the composition of toggles according to any linear extension of the poset obtained from the transitive closure of the Auslander–Reiten quiver AR(c).--> <!--- Still need $\beta_{c,s}$ and $\mathrm{Charm}_c$. - The bijection $\beta_{c,s}$ can be explicitly constructed in the *toggle group*. - So we'll only look at $\mathrm{Charm}_c$ today. -->

### <u>The K**row**eras complement</u> <!--- To have a commuting picture, we need: - a bijection $\beta\_{c,s}: \mathrm{NN}\_c(W) \to \mathrm{NN}\_{scs}(W)$ such that $$\beta\_{c,s} \circ \mathrm{Krow}\_c = \mathrm{Krow}\_{scs} \circ \beta\_{c,s}, \text{ and }$$ - a bijection $\mathrm{Charm}_c: \mathrm{NN}_c(W) \to \mathrm{NC}_c(W)$ such that $$\mathrm{Krew}_c \circ \mathrm{Charm}_c = \mathrm{Charm}_c \circ \mathrm{Krow}_c.$$--> <img data-src="NNNC.png" height="300"> <!--Remark 4.4. The map Krowc does not depend on the choice of reduced word c for c: all reduced words for c are related by a sequence of commutations of non-adjacent transpositions, and each commutation only affects the sequence α (k) by interchanging pairs of roots that are not adjacent in the root poset. By the remark after Equation (8), this, therefore, does not change the composition of toggles. We may therefore view Krowc as the composition of toggles according to any linear extension of the poset obtained from the transitive closure of the Auslander–Reiten quiver AR(c).--> - The <span style="color:#3498db">*Kroweras complement*</span> toggles in $\mathrm{inv}(\mathbf{w}_\circ(\mathbf{c}))$ order. - The bijection $\beta_{c,s}$ can be explicitly constructed in the <span style="color:#3498db">*toggle group*</span>. - So we'll concentrate on $\mathrm{Charm}_c$ today.

<br><br><br><br> ### <u>Charmed Bijections</u>

### <u>Charmed bijections</u> <iframe data-src="charmed_bij1.html" data-preload width="800" height="800"></iframe>

### <u>$c$-Charmed roots</u> For $c = (\mathbf{1} \textcolor{#3498db}{< a_2 < \cdots < a_{m-1} <} \mathbf{n} \textcolor{#3498db}{> a_{m+1} > \cdots> a_n}),$ define $L_c = \\{\textcolor{#3498db}{a_2,\ldots,a_{m-1}}\\}$ and $R_c = \\{\textcolor{#3498db}{a_{m+1}, \ldots, a_n}\\}.$ > **Def.** A root $(i, j)$ is **$c$-charmed** if $\begin{cases} i \in L_c \\\ j \in R_c \end{cases}$ or $\begin{cases} i \in R_c \\\ j \in L_c\end{cases}.$ <iframe data-src="ARcharmed.html" data-preload width="400" height="400"></iframe> <!-- In $S_9$ with $c=(\mathbf{1}\textcolor{#3498db}{,3,4,7,}\mathbf{9}\textcolor{#3498db}{,8,6,5,2})$, > > $L\_c = \\{\textcolor{#3498db}{3,4,7}\\}$ and $R\_c = \\{\textcolor{#3498db}{8,6,5,2}\\}.$ <iframe data-src="ARcharmed.html" data-preload width="600" height="600"></iframe>-->

### <u>$c$-Charmed roots</u> <iframe data-src="ARcharmed.html" data-preload width="800" height="800"></iframe>

### <u>$c$-Charmed bijections</u> > **Thm (DFISATW).** For any standard Coxeter element $c \in S_n$, there is a unique (support-preserving) bijection $$\mathrm{Charm}\_c: \mathrm{NC}_c(n) \to \mathrm{NN}(S_n)$$ satisfying $$ \mathrm{Charm}\_c \circ \mathrm{Krew}\_c = \mathrm{Krow}\_c \circ \mathrm{Charm}\_c.$$ <img data-src="NNNC.png" height="180">

### <u>$c$-Charmed bijections</u> <iframe data-src="c_charmed_bij.html" data-preload width="800" height="800"></iframe>

### <u>$c$-Charmed bijections: Proof</u> <img data-src="NNNC.png" height="400"> <img data-src="proof.png" width="600">

### <u>THANK YOU</u> <iframe data-src="charmed_NN.html" data-preload width="950" height="950" style="border: 1px solid black;"></iframe>

### <u>Questions</u> > - <u>Question</u>: What about other types? > > - <u>Answer</u>: Yes, this would be awesome! We do type $B$ by folding. > - <u>Question</u>: Explain again about the question mark on NN=NC? > > - <u>Answer</u>: No. (Come see me after.) > - <u>Question</u>: Are there applications of general charmed bijections? > > - <u>Answer</u>: I don't know.

<!--<img data-src="fpsac.png" width="600">--> ### Thanks to <u>**all of you**</u> for making this another wonderful FPSAC!<br><br><br><br><br><br><br><br> <!--<iframe data-src="charmed_NN.html" data-preload width="950" height="950" style="border: 1px solid black;"></iframe>-->

<!--<img data-src="fpsac.png" width="600">--> ### And thank you to the <u>**program and organizing committees**</u> for all their work behind the scenes! <!--<iframe data-src="charmed_NN.html" data-preload width="950" height="950" style="border: 1px solid black;"></iframe>-->

<!--<img data-src="fpsac.png" width="600">--> ### And, of course, to <u>**all**</u> the committees. <!--<iframe data-src="charmed_NN.html" data-preload width="950" height="950" style="border: 1px solid black;"></iframe>-->

<!--<img data-src="fpsac.png" width="600">--> ### Especially to <u>**Christian G., K., and S.**</u> <!--<iframe data-src="charmed_NN.html" data-preload width="950" height="950" style="border: 1px solid black;"></iframe>-->

### And let's please give a *very warm and extended* thank you to all the <u>**local organizers**</u> for making sure everything ran so smoothly this week! | | | | -------- | ------- | Ashleigh Adams | Kudret Bostanci Florian Buck | Nathan Chapelier Galen Dorpalen-Barry | Thomas Gerber Lorenzo Giordani | Lydia Gösmann Hy Khang Hoang | Elena Hoster Max Osterlitz | Timm Peerenboom Lars Rupieper | Annika Schulte Christian Stump | Laura Voggesberger Sven Wiesner | Mike Zabrocki <!--Help desk: Khang, Florian, Lars and Max--> <!--- Kudret Bostanci - Florian Buck - Nathan Chapelier - Galen Dorpalen-Barry - Lorenzo Giordani - Lydia Gösmann - Hy Khang Hoang - Elena Hoster - Max Osterlitz - Timm Peerenboom - Lars Rupieper - Annika Schulte - Christian Stump - Laura Voggesberger - Sven Wiesner Help desk students: Khang, Florian, Lars and Max-->