### Potential graduate students or postdocs

I am currently NOT looking for any graduate students or postdocs with anticipated start date of January 2019 or earlier. If you are planning to begin studies in September 2019 or later, read on.

If you are interested in pursuing a M.Sc. or Ph.D. degree under my supervision on a thesis topic related to my research interests, contact me by email and include the following: a brief descriptions of topics or specific research problems that interest you, an overview of any research experience you have, your curriculum vitae, your grade point average and a copy of your transcripts (in English). You might not get a reply if you do not follow the above guidelines.

You are encouraged to contact me as early as 18-24 months before your intended arrival to the UofM. Then, in case of mutual interest and agreement, we can work on a small research project which would be supervised remotely (for 6-12 months). If we are both satisfied with the outcome/workflow of that project, you would be able to apply formally in timely manner for graduate studies at the UofM.

Please visit the Graduate Studies section of the Department of Mathematics website for more information on application procedures and funding opportunities.

PIMS postdoctorall fellowship is described here.

### Sample problems

Let $K\subset\mathbb{R}^d$ be a convex compact set with non-empty interior. The problem is to find finite subsets $A_n\subset K$ such that $A_n$ has at most $cn^d$ elements while for any polynomial $P$ of total degree $\le n$ we have $\max_{x\in K}|P(x)|\le c \max_{x\in A_n}|P(x)|$. This was recently solved for $d=2$ in [A. Kroo, *On the existence of optimal meshes in every convex domain on the plane*, J. Approx. Th., (2017) http://dx.doi.org/10.1016/j.jat.2017.02.004] and is known to be true for higher $d$ under additional smoothness requirements on the boundary of $K$. A solution will probably require some delicate multivariate Bernstein-type inequalities which may be of independent interest.

Hyperbolic cross in $\mathbb{R}^d$ is $\Gamma(N):=\{\mathbf{k}\in\mathbb{Z}^d:\prod_{j=1}^d\max\{|k_j|,1\}\le N\}$. We consider periodic polynomials with frequences from hyperbolic cross: $\mathcal{T}(N):=\{t:t(\mathbb{x})=\sum_{\mathbf{k}\in\Gamma(N)}c_{\mathbf{k}}e^{i(\mathbf{k},\mathbf{x})}\}$. The problem is to determine the order of the quantity $b(N,r):=\sup_{t\in\mathcal{T}(N)}\frac{\|t^{(r,\dots,r)}\|_{L_1}}{\|t\|_{L_1}}$. For any $d\ge2$ it is known that $c N^r\le b(N,r)\le c N^r (\log N)^{d-1}$. For $d=2$, the upper bound was improved to $b(N,r)\le c N^r (\log N)^{1/2}$ in [V.N. Temlyakov, *On two problems in the multivariate approximation*, East J. Approx.,
**4** (1998), 505-514]. That's everything what's known for $L_1$ case, while the problem is completely solved in $L_p$, $1\lt p\le\infty$. See [D. Dung, V.N. Temlyakov, T. Ullrich, *Hyperbolic cross approximation*, download (arXiv)] for a survey and other problems related to hyperbolic cross approximation.

If $\mathbb{C}^d$ is equipped with the usual Hermitian inner product $\langle x,y\rangle$, two orthonormal bases $\{x_1,\dots,x_d\}$ and $\{y_1,\dots,y_d\}$ of $\mathbb{C}^d$ are called unbiased if $|\langle x_j, y_k \rangle|=1/\sqrt{d}$ for all $j$ and $k$. A collection of orthonormal bases is a set of mutually unbiased bases (MUBs) if any pair of the bases is unbiased. There are at most $d+1$ elements in any collection of MUBs in $\mathbb{C}^d$. Understanding what is the actual largest size $M(d)$ of a set of MUBs in $\mathbb{C}^d$ is an important open question in quantum information theory. The smallest unresolved $d$ is $d=6$. When $d$ is prime or a power of a prime, there are largest (complete) MUBs, i.e., $M(d)=d+1$, see, e.g. [A. Roy, A. J. Scott, *Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements*, J. Math. Phys., **48** (2007) download (arXiv)] and [C. Godsil, A. Roy, *Equiangular lines, mutually unbiased bases, and spin models*, European Journal of Combinatorics, **50** (2009) 246-262 download (arXiv)]. Taking tensor products, one observes $M(d)\ge\min\{M(p_1^{\alpha_1})\dots M(p_k^{\alpha_k})\}$ for prime factorization $d=p_1^{\alpha_1}\dots p_k^{\alpha_k}$. A totally different approach by [P. Wocjan, T. Beth, *New Construction of Mutually Unbiased Bases in Square Dimensions*, Quantum Inf. Comput., **5** (2005) 93-101 download (arXiv)] establishes $M(s^2)\ge k+2$ if there exist $k$ mutually orthogonal latin squares of size $s$. It is interesting that $M(d)\ne d$, see [M. Weiner, *A gap for the maximum number of mutually unbiased bases*, Proc. Amer. Math. Soc., **141** (2013) 1963-1969 download (arXiv)]. For more information, see also MUB problem page at quantum information open problems wiki.

This is a very hard question in geometry where even partial results could be of interest. By $|K|$ we denote the $d$-dimensional volume of a convex body $K$, and by $conv\{x_1,\dots,x_n\}$ the convex hull (all linear combinations with non-negative coefficients and sum $1$) of the points $x_1,\dots,x_n$. For a convex body $K\subset\mathbb{R}^d$, and fixed $p\ge1$, let

$\qquad\displaystyle\mathbb{E}_n^p(K)=|K|^{-n-p}\int_K\dots\int_K|conv\{x_1,\dots,x_n\}|^pdx_1\dots dx_n$

be the expected value of the $p$-th power of the volume of the convex hull of $n$ random points from $K$.

Simplex conjecture is, in particular, the following: for any convex body $K\subset\mathbb{R}^d$, the inequality

$\qquad\displaystyle\mathbb{E}_{d+1}^p(K)\le\mathbb{E}_{d+1}^p(T^d)$

holds, where $T^d$ is a simplex in $\mathbb{R}^d$. In other words, $\mathbb{E}_{d+1}^p(K)$ attains its largest value when $K$ is a simplex. This has been verified for $d=2$ only. If proved (even for a particular value of $p$), simplex conjecture would imply the well-known slicing conjecture (largest $d-1$-volume of a $d-1$-section of any convex body in $\mathbb{R}^d$ of volume $1$ is bounded from below by a constant independent of $d$).

An asymptotic version of simplex conjecture has been proved in [I. Barany and C. Buchta, *Random polytopes in a convex polytope, independence of shape, and concentration of vertices*, Math. Ann. **297** (1993), no. 3, 467-497], namely, if $K$ is not a simplex, then for some $n_K$ one has $\mathbb{E}_{n}^1(K)< \mathbb{E}_{n}^1(T^d)$ for all $n>n_K$. However, it is quite likely that the simplex conjecture is wrong for large $d$. It can possibly be true for $d=3$, which seems to be an open question. See [G. Ambrus, K.J. Boroczky, *Stability results for the volume of random simplices*, Amer. J. of Math., **136** (2014), no. 4, 833-857] for a discussion of similar questions and other known related results. When $p=2$, the quantity $\mathbb{E}_{d+1}^2(K)$ can be immediately related to the isotropic constant of $K$. Also, if $K$ is in isotropic position, then the square of the isotropic constant equals to the dimensionless second moment of $K$ which is computed for many convex polytopes in Section 21.2 of [J.H. Conway and N.J.A. Sloane, *Sphere packings, lattices and groups*, A series of comprehensive studies in mathematics, vol. 290, Springer, 1999].

Here is an open question which does not require much background to be stated. It is a variant of a Whitney-type inequality, which is used in estimates on approximation by splines. Namely, for each positive integer $d$, find the smallest possible positive constant $c(d)$ such that for any continuous function $f:[0,1]^d\to\mathbb{R}$ there exists a linear function $l(x_1,\dots,x_d)=\alpha_0+\alpha_1x_1+\dots+\alpha_dx_d$ satisfying

$
\qquad\max_{x\in[0,1]^d}|f(x)-l(x)|\le w(d) \max_{x,y\in[0,1]^d}|f(x)+f(y)-2f((x+y)/2)|.
$

It was conjectured by Brudnyi and Kalton that $w(d)\le2$, and they were able to prove that $w(d)\le802$. It has recently been announced that $w(d)\le 26.3=2\cdot 12.65+1$, see [U. Feige, M. Feldman, I. Talgam-Cohen, *Approximate modularity revisited* download (arXiv)]. The preivously known best bound was $w(d)\le73$, established in [A. Bondarenko, A. Prymak, D. Radchenko, *On concentrators and related approximation constants*, J. Math. Anal. Appl., **402** (2013) 234-241 download (arXiv)]. The known approach shows an interesting connection between analysis and graph theory, and also uses some probabilistic arguments. More precise bounds for small $d$ are of interest as well, here is what's known for $d\le10$:

$w(1)=\frac12$, $w(2)=1$, $w(3)\le\frac54$, $w(3)\le\frac32$, $w(3)\le\frac74$, $w(6)\le2$, $w(7)\le2$, $w(8)\le2$, $w(9)\le\frac94$, $w(10)\le\frac52$.