Related papers: On two problems of Erdos and Hechler: New methods …
We investigate the interior of the moduli space of four-dimensional $\mathcal{N}=1$ theories of gravity arising from compactifications of the $E_8\times E_8$ heterotic string on Calabi-Yau threefolds. By studying the threshold corrections…
This thesis analyses extenders in fine structural mice. Kunen showed that in the inner model for one measurable cardinal, there is a unique measure. This result is generalized, in various ways, to mice below a superstrong cardinal. The…
A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase $(\gamma < \sqrt{2d})$ and…
Let $x$ denote a Laver real over $L$. We prove that in $L[x]$ there is a $\Pi^1_1$ infinite mad family. Since $\Pi^1_1$ and $\Sigma^1_2$ sets are Laver measurable in $L[x]$, this shows that there are examples of well-behaved classical…
We prove that every MAD family can be destroyed by a proper forcing that preserves $P$-points. With this result, we prove that it is consistent that $\omega_{1}=\mathfrak{u}<\mathfrak{a,}$ solving a nearly 20 year old problem of Shelah and…
In this paper we provide a formalism, Sudoku logic, in which a solution is logically deducible if for every cell of the grid we can provably exclude all but a single option. We prove that the deductive system of Sudoku logic is sound and…
In this paper we address two problems concerning a family of domains $M_{\Omega}(\mu) \subset \C^n$, called Cartan-Hartogs domains, endowed with a natural Kaehler metric $g(\mu)$. The first one is determining when the metric $g(\mu)$ is…
We prove the holomorphic rigidity conjecture of Teichm\"{u}ller space which loosely speaking states that the action of the mapping class group uniquely determines the Teichm\"{u}ller space as a complex manifold. The method of proof is…
We characterize the situation of having many normal measures on a measurable cardinal. We show the plausibility of having many normal measures on each compact cardinal.
New sets (typically found by computer search) with Sidon constant equal to the square root of their cardinalities are given. For each integer $N$ there are only a finite number of groups of prime order containing $N$-element extreme sets.…
Much of the theory of large cardinals beyond a measurable cardinal concerns the structure of elementary embeddings of the universe of sets into inner models. This paper seeks to answer the question of whether the inner model uniquely…
We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models or existence of large cardinals). We prove (assuming a weak version of GCH…
Using a geometric realization of the $SU(2)_R$ symmetry and a procedure of factorisation of the gauge and $SU(2)_R$ charges, we study the small instanton singularities of the Higgs branch of supersymmetric $U(1)^r$ gauge theories with eight…
We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $\kappa$ is indestructible by the higher random forcing $\mathbb Q_\kappa$. We then use this characterisation to show that…
In this paper, we construct a class of random measures $\mu^{\mathbf{n}}$ by infinite convolutions. Given infinitely many admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{\infty}$ and a positive integral sequence…
All spaces below are $T_0$ and crowded (i.e. have no isolated points). For $n \le \omega$ let $M(n)$ be the statement that there are $n$ measurable cardinals and $\Pi(n)$ ($\Pi^+(n)$) that there are $n+1$ (0-dimensional $T_2$) spaces whose…
An old idea for explaining the hierarchy is strong gauge dynamics. We show that such dynamics {\it also} stabilises the moduli in $M$ theory compactifications on manifolds of $G_2$-holonomy {\it without} fluxes. This gives stable vacua with…
Working in the context of $\mu$-abstract elementary classes ($\mu$-AECs) - or, equivalently, accessible categories with all morphisms monomorphisms - we examine the two natural notions of size that occur, namely cardinality of underlying…
A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on…
We observe that the nonstandard finite cardinality of a definable set in a strongly minimal pseudofinite structure D is a polynomial over the integers in the nonstandard finite cardinality of D. We conclude that D is unimodular, hence also…