ABSTRACT: A space of countable extent, also called an omega_1-compact space, is one in which every closed discrete subspace is countable.  The axiom used in the following theorem is consistent if it is consistent that there is a supercompact cardinal.

Theorem 1  The LCT axiom implies that every hereditarily normal, omega_1-compact space
is sigma-countably compact,  i.e., the union of countably many countably compact subspaces.

Even for the specialized subclass of monotonically normal spaces, this is only a consistency result:

Theorem 2   If club, then there exists a locally compact, omega_1-compact monotonically
normal space that is not sigma-countably compact.

These two results together are unusual in that most independence results on
monotonically normal spaces depend on whether Souslin's Hypothesis (SH) is true,
and do not involve large cardinal axioms. Here, it is not known whether either
SH or its negation affect either direction in this independence result.

The following unsolved problem is also discussed:

Problem  Is there a ZFC example of a locally compact, omega_1-compact space
of cardinality aleph_1 that is not sigma-countably compact?

Due to Shor's algorithm, quantum computers are a severe threat for public key cryptography. This motivated the cryptographic community to search for quantum-safe solutions. On the other hand, the impact of quantum computing on secret key cryptography is much less understood. In this paper, we consider attacks where an adversary can query an oracle implementing a cryptographic primitive in a quantum superposition of different states. This model gives a lot of power to the adversary, but recent results show that it is nonetheless possible to build secure cryptosystems in it.
We study applications of a quantum procedure called Simon's algorithm (the simplest quantum period finding algorithm) in order to attack symmetric cryptosystems in this model. Following previous works in this direction, we show that several classical attacks based on finding collisions can be dramatically sped up using Simon's algorithm: finding a collision requires Ω(2n/2) queries in the classical setting, but when collisions happen with some hidden periodicity, they can be found with only O(n) queries in the quantum model.
We obtain attacks with very strong implications. First, we show that the most widely used modes of operation for authentication and authenticated encryption (e.g. CBC-MAC, PMAC, GMAC, GCM, and OCB) are completely broken in this security model. Our attacks are also applicable to many CAESAR candidates: CLOC, AEZ, COPA, OTR, POET, OMD, and Minalpher. This is quite surprising compared to the situation with encryption modes: Anand et al. show that standard modes are secure when using a quantum-secure PRF.
Second, we show that slide attacks can also be sped up using Simon's algorithm. This is the first exponential speed up of a classical symmetric cryptanalysis technique in the quantum model.

Quasimorphisms (QM) of groups to the reals are well studied and are linked to stable commutator length (scl) via Bavard Duality- Theorem. The notion of QM can be generalized to yield maps  between groups such that each QM from one group pulls back to a QM in the other.

We will give both a short overview of features of scl and investigate these generalized QMs with large scale properties of the commutator group. 

This is a survey (with some proofs) of chapter 2 of the notes http://renyi.mta.hu/~szamuely/beilintronew.pdf of T. Szamuely and G. Zabradi on Beilinson's approach to the p-adic Hodge decomposition theorem.

Given an integer $d$ such that $2 \leq d \leq 50$, we want to
answer the question: When is the sum of
$d$ consecutive cubes a perfect power? In other words, we want to find all
integer solutions to the equation
$(x+1)^3 + (x+2)^3 + \cdots + (x+d)^3 = y^p$. In this talk, we present some
of the techniques used to tackle such diophantine problems.

We consider families of fast-slow skew product maps of the form \begin{align*}x_{n+1}   = x_n+\eps^2 a_\eps(x_n,y_n)+\eps b_\eps(x_n)v_\eps(y_n), \quad

y_{n+1}   = T_\eps y_n, \end{align*} where $T_\eps$ is a family of nonuniformly expanding maps, $v_\eps$ is of mean zero and the slow variables $x_n$ lie in $\R^d$.  Under an exactness assumption on $b_\eps$ (automatically satisfied in the cases $d=1$ and $b_\eps\equiv I_d$), we prove convergence of the slow variables to a limiting stochastic differential equation (SDE) as $\eps\to0$.   Our results include cases where the family of fast dynamical systems

$T_\eps$ consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Similar results are obtained also for continuous time systems  \begin{align*} \dot x   =  \eps^2 a_\eps(x,y,\eps)+\eps b_\eps(x)v_\eps(y), \quad \dot y   =  g_\eps(y). \end{align*}

Here, as in classical Wong-Zakai approximation, the limiting SDE is of Stratonovich type $dX=\bar a(X)\,dt+b_0(X)\circ\,dW$ where $\bar a$ is the average of $a_0$

and $W$ is a $d$-dimensional Brownian motion.