1. Open problems

Komlós conjecture
Problem 1.05.
There is a universal constant $c>0$ s.t. for every $v_1,v_2,...,v_n \in \mathbb{B_2}^d$, there exist signs $\epsilon_1,...,\epsilon_n\in \{1,1\}$ s.t. $ \sum_{i=1}^n \epsilon_i v_i_{\infty} \le c$ 
Constructive Komlós
Problem 1.1.
Find a polynomial time algorithm to find the signs above. 
BeckFiala conjecture and making it constructive
Problem 1.15.
Given a set system $(V,\mathcal{S})$ with $V=[n]$, $\mathcal{S}=S_1,...,S_m$. Each $i\in V$ occurs in at most $t$ sets from $\mathcal{S}$. Then there is a coloring to the elements of $V$ achieving a discrepancy of $O(\sqrt{t})$. Komlós conjecture implies this by treating the incidence vector of each $i\in V$ as a vector and scaling it down by $\sqrt{t}$. 
BeckFiala for random set system
Problem 1.2.
[Esther Ezra, Shachar Lovett] Each element $i\in [n]$ lies in $t$ randomly selected sets from $\mathcal{S}$. 
Hypercube set system
Problem 1.25.
[Shachar Lovett] Given a set system where the elements are the vertices of the hypercube $\{0,1\}^n$. Corresponding to each vertex $v$, there is a set $S_v$ which is generated by picking a random subset of the neighbours of $v$ in the hypercube. What is the discrepancy of this set system? 
Discrepancy of permutations
Problem 1.3.
a) discrepancy of kpermutations
For $k=3$, $[NNN'12]$ showed a lower bound of $\Omega(\log n)$ where $n$ is the number of elements in the ground set. For general $k$, we know that the answer lies in $[\sqrt{k}+\log n,\sqrt{k}\log n]$. An open question is to clsoe this gap.
b) discrepancy of krandom permutations
A lower bound of $\Omega(\sqrt{k})$ is known. 
Matrix Spencer
Problem 1.35.
Given matrices $A_1,...,A_n \in\mathbb{R}^{n\times n}$ satisfying $A_i_{op} \le 1$ where $._{op}$ is the operator norm of matrices. Do there exists signs $\epsilon_1,...,\epsilon_n\in\{1,1\}$ s.t. $\sum_{i=1}^n\epsilon_i A_i  =O(\sqrt{n})$. 
Steinitz conjecture
Problem 1.4.
Given vectors $v_1,...,v_n\in\mathbb{R}^d$ and a norm $X$ satisfying $\sum_{i=1}^n v_i=0$ and $v_i_X\le 1$ for all $i$. Define $S_n(X,d)=min_{\pi\in S^n}max_{k=1,...,n} \sum_{i=1}^k v_{\pi(i)}_X$ where $S^n$ is the set of all permutations of $[n]$. Let $S(X,d)=sup_n S_n(X,d)$. Then, $S(X,d) =O(\sqrt{d})$.
ii) If $X=l_p$ for $1< p < 2$, $S(X,d) =\Omega(d)$
iii) For $X=l_2$, Banaszczyk’13 gave a bound of $O(\sqrt{d}+\sqrt{\log n})$. A lower bound of $\sqrt{d}$ is known here.
iv) It is known that $S(X,d) \ge \alpha(X,X,d)$ [Chobanyan] (refer to vector balancing problem below for definition of $\alpha$) . Given in Barany’s epic document. Moreover, for a given set of vectors $v_1,..,v_n$ , given a signing of it achieving discrepancy $\alpha$, we can recover a permutation achieving a guarantee of $\alpha$ in the Steinitz setting [Harvey].
Remark. An algorithmic guarantee of $\sqrt{d}log\,n$ was worked upon in the workshop.


Vector Balancing
Let $X,Y$ be norms on $\mathbb{R}^d$. Given vectors $v_1,...,v_n\in \mathbb{R}^d$ satisfying $v_i_X\le 1$ for all $i$. We want to find signs $\epsilon_i \in \{1,1\}$ to minimize $ \sum_{i=1}^n\epsilon_iv_i _Y$. Call this the discrepancy of the vectors. Define $\alpha_n(X,Y,d)$ as the maximum of this discrepancy over all choice of $n$ vectors satisfying $v_i_X\le 1$ and $\alpha(X,Y,d) =sup_n \alpha_n(X,Y,d)$. Banaszczyk proved $\alpha(X,Y,d) \ge \sqrt{d}(\frac{vol(B_X)}{vol(B_Y)})^{1/d}$ where $B_X,B_Y$ are the unit balls of the corresponding norms.Problem 1.5.
Is this lower bound also an upper bound for $\alpha(X,Y,d)$ upto $\log d$ factors?
Remark. [Sasho Nikolov] In fact, this lower bound can be too small by a factor of $\sqrt{d}$, for example when $X = Y = \ell_1^d$. However, the following stronger lower bound may in fact also be an upper bound, up to logarithmic factors: $\max_{k = 1}^{d}\max_{W: \mathrm{dim}(W) = k} \min_{B_Y \cap W \subseteq E} \sqrt{d} \frac{vol(E)^{1/d}}{vol(B_X \cap W)^{1/d}}$, where the maximum is over subspaces $W$ of dimension $k$ and the minimum is over ellipsoids $E$ containing $B_Y \cap W$.

Remark. [Sasho Nikolov] The formula above should be $\max_{k = 1}^d \max_{W: \mathrm{dim}(W) = k} \min_{E: B_X \cap W \subseteq E} \sqrt{k} \frac{vol(E)^{1/k}}{vol(B_Y \cap W)^{1/k}}$.


Tusnady’s problem
Problem 1.65.
Let $d_n$ be the discrepancy of $n$ points in $\mathbb{R}^2$ wrt all alixaligned rectangles i.e. the set system comprises of all axisaligned rectangles in $\mathbb{R}^2$ as sets, maximised over all choices of $n$ points. Tusnady’s problem then asks for the exact value of $d_n$.
We can also ask for the same question in $\mathbb{R}^d$. 
Beating LLL
Problem 1.7.
Consider a matrix where both columns are rows are tsparse. LLL gives a discrepancy bound of $O(\sqrt{t\log t})$ here. If we assume the stronger property that every pair of rows intersect in at most one element, can we get a $O(\sqrt{t})$? 
Hardness of Komlos
Problem 1.75.
Can we prove some hardness statement about finding a coloring achieving Komlos guarantee or the Banaszczyk discrepancy guarantee? 
Reverse Banaszczyktype problems
Problem 1.8.
Say that a convex body $K$ has property B if for any set of vectors of length no more than $c$ (some small constant), there exists a signing of them s.t. this signed sum lies inside $K$.
What is the convex body with the smallest gaussian volume which has property $B$?
Remark. [Daniel Dadush] During the workshop it was shown that if $K$ has property $B$, then $\mathbb{E} \G\_K = (\log n)^O(1)$, where $G$ is a standard Gaussian random vector.


Polysize list of colorings
Problem 1.85.
Is there a list of colorings of polynomial size such that for any subset of the columns, one of these colorings gives it a discrepancy of O(h), where $h$ is the hereditary discrepancy?
Cite this as: AimPL: Hereditary discrepancy and factorization norms, available at http://aimpl.org/hereddiscrep.