Open problems in Mathematics stemming from research papers

Make 1000 USD from simple combinatorics (M. Talagrand)

Make 1000 USD from simple combinatorics. This is Research Problem 12.5.6 of the book Upper and Lower Bounds. You have to have a look at the book to get the connection between this problem and stochastic processes, but the problem can be understood without knowing anything about this book. ...
- Michel Talagrand
Open
Posted online: 2024-03-21 13:07:56Z by SciLag Admin95
- Combinatorics
Cube always hitting a lattice point

Is there a universal constant $c$ such that all translation and rotations of a cube $Q \subset \mathbb{R}^n$ with side-length $c$ always contains a point from the $\mathbb{Z}^n$? ...
- Martin Henk,
- George Tsintsifas
Open
Posted online: 2024-03-20 15:50:20Z by Stefan Steinerberger36
- Combinatorics
- Classical Analysis and ODEs
Cosine Sign Correlation

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of $n$ distinct positive integers. Let $X$ be a uniformly distributed random variable on $[0,2\pi]$. How small can the probability ...
Open
Posted online: 2024-03-20 15:31:29Z by Stefan Steinerberger39
- Classical Analysis and ODEs
- Probability
The Size of Intervals in the Hardy-Littlewood Maximal Function

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function (more smoothness can be assumed if deemed necessary). A classical object is the Hardy-Littlewood maximal function $\mathcal{M}f: \mathbb{R} \rightarrow \mathbb{R}$ defined via $$ (\mathcal{M} f)(x) = \sup_{r >0} \frac{1}{2r} \int_{x-r}^{x+r} |f(z)| dz.$$ ...
- Stefan Steinerberger
Open
Posted online: 2024-03-20 15:17:20Z by Stefan Steinerberger33
- Classical Analysis and ODEs
Littlewood's Cosine Root Problem

Let $A \subset \mathbb{N}$ be a set of positive integers. One basic intuition is that the function

$$ f(x) = \sum_{k \in A} \cos(kx)$$

should have many roots on $[0, 2\pi]$ when the set $A$ is large. Littlewood conjectured that the number of roots is probably something like $\#A - 1$. This was disproven in 2008 by Borwein, Erdelyi, Ferguson, Lockhart (Annals). The best bounds are ...
- J. E. Littlewood
Open
Posted online: 2024-03-20 15:04:36Z by Stefan Steinerberger21
- Classical Analysis and ODEs
Symbolic dynamics for real Hénon and Lozi maps (By S. \v Stimac)

Kneading theory is a combinatorial tool to understand the dynamics of a piecewise monotone map from the interval to itself and was developed by Milnor and Thurston (Lecture Notes 1988).

The set of kneading sequences, denoted as $\mathfrak{K}$, consists of all sequences $\bar{k}^n$ for $n \in \mathbb{Z}$. Each kneading sequence $\bar{k} = \bar{k}^n$, where $n \in \mathbb{Z}$, follows the structure: ...
- Eric Bedford,
- Pierre Berger,
- Fabrizio Bianchi,
- Xavier Buff,
- Sylvain Crovisier,
- Tien-Cuong Dinh,
- Romain Dujardin,
- Charles Favre,
- Tanya Firsova,
- Patrick Ingram,
- Yutaka Ishii,
- Liviana Palmisano,
- Enrique Pujals,
- Jasmin Raissy,
- Sonja Štimac,
- Gabriel Vigny
Open
Posted online: 2024-03-18 19:41:03Z by SciLag Admin28

View the group
- Dynamical Systems
Dissipative real Hénon maps (by S. Crovisier and E. Pujals)

Consider the quadratic real Hénon maps $$f_{c, b} (x, y):= (x^2+c-by, x).$$

The map $f_{c,b}$ is called mildly dissipative if $|b|< 1$ and if for any ergodic invariant measure $\mu$, not supported on a sink, and for $\mu$-almost every point $x$, both components of the stable curve $W^s(x)\setminus \{x\}$ are unbounded. ...
- Eric Bedford,
- Pierre Berger,
- Fabrizio Bianchi,
- Xavier Buff,
- Sylvain Crovisier,
- Tien-Cuong Dinh,
- Romain Dujardin,
- Charles Favre,
- Tanya Firsova,
- Patrick Ingram,
- Yutaka Ishii,
- Liviana Palmisano,
- Enrique Pujals,
- Jasmin Raissy,
- Sonja Štimac,
- Gabriel Vigny
Open
Posted online: 2024-03-18 19:41:03Z by SciLag Admin17

View the group
- Dynamical Systems

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