It was implicitly conjectured by Hambly-Lyons in 2010, which was made explicit by Chang-Lyons-Ni in 2018, that the length of a tree-reduced path with bounded variation can be recovered from its signature asymptotics. Apart from its intrinsic elegance, understanding such a phenomenon is also important for the study of signature lower bounds and may shed light on more general signature inversion properties. In this talk, we discuss how the idea of path development onto suitably chosen Lie groups can be used to study this problem as well as its rough path analogue.

Let us say that a theory $T$ in the language of set theory is $\beta$-consistent at $\alpha$ if there is a transitive model of $T$ of height $\alpha$, and let us say that it is $\beta$-categorical at $\alpha$ iff there is at most one transitive model of $T$ of height $\alpha$. Let us also assume, for ease of formulation, that there are arbitrarily large $\alpha$ such that $\mathrm{ZFC}$ is $\beta$-consistent at $\alpha$.

The sentence $\mathrm{VEL}$ has the feature that $\mathrm{ZFC}+\mathrm{VEL}$ is $\beta$-categorical at $\alpha$, for every $\alpha$. If we assume in addition that $\mathrm{ZFC}+\mathrm{VEL}$ is $\beta$-consistent at $\alpha$, then the uniquely determined model is $L_\alpha$, and the minimal such model, $L_{\alpha_0}$, is model of determined by the $\beta$-categorical theory $\mathrm{ZFC}+\mathrm{VEL}+M$, where $M$ is the statement "There does not exist a transitive model of $\mathrm{ZFC}$."

It is natural to ask whether $\mathrm{VEL}$ is the only sentence that can be $\beta$-categorical at $\alpha$; that is, whether, there can be a sentence $\phi$ such that $\mathrm{ZFC}+\phi$ is $\beta$-categorical at $\alpha$, $\beta$-consistent at $\alpha$, and where the unique model is not $L_\alpha$.  In the early 1970s Harvey Friedman proved a partial result in this direction. For a given ordinal $\alpha$, let $n(\alpha)$ be the next admissible ordinal above $\alpha$, and, for the purposes of this discussion, let us say that an ordinal $\alpha$ is minimal iff a bounded subset of $\alpha$ appears in $L_{n(\alpha)}\setminus L_\alpha$. [Note that $\alpha_0$ is minimal (indeed a new subset of $\omega$ appears as soon as possible, namely, in a $\Sigma_1$-definable manner over $L_{\alpha_0+1}$) and an ordinal $\alpha$ is non-minimal iff $L_{n(\alpha)}$ satisfies that $\alpha$ is a cardinal.] Friedman showed that for all $\alpha$ which are non-minimal, $\mathrm{VEL}$ is the only sentence that is $\beta$-categorical at $\alpha$. The question of whether this is also true for $\alpha$ which are minimal has remained open.

In this talk I will describe some joint work with Hugh Woodin that bears on this question. In general, when approaching a "lightface" question (such as the one under consideration) it is easier to first address the "boldface" analogue of the question by shifting from the context of $L$ to the context of $L[x]$, where $x$ is a real. In this new setting everything is relativized to the real $x$: For an ordinal $\alpha$, we let $n_x(\alpha)$ be the first $x$-admissible ordinal above $\alpha$, and we say that $\alpha$ is $x$-minimal iff a bounded subset of $\alpha$ appears in $L_{n_x(\alpha)}[x]\setminus L_{\alpha}[x]$.

Theorem. Assume $M_1^\#$ exists and is fully iterable. There is a sentence $\phi$ in the language of set theory with two additional constants, \r{c} and \r{d}, such that for a Turing cone of $x$, interpreting \r{c} by $x$, for all $a$

1. if $L_\alpha[x]\vDash\mathrm{ZFC}$ then there is an interpretation of \r{d}  by something in $L_\alpha[x]$ such that there is a $\beta$-model of $\mathrm{ZFC}+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$, and
2. if, in addition, $\alpha$ is $x$-minimal, then there is a unique $\beta$-model of $\mathrm{ZFC}+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$.

The sentence $\phi$ asserts the existence of an object which is external to $L_\alpha[x]$ and which, in the case where $\alpha$ is minimal, is canonical. The object is a branch $b$ through a certain tree in $L_\alpha[x]$, and the construction uses techniques from the HOD analysis of models of determinacy.

In this talk I will sketch the proof, describe some additional features of the singleton, and say a few words about why the lightface version looks difficult.

We survey some recent progress in understanding stationary reflection at successors of singular cardinals and its influence on cardinal arithmetic:

1) In joint work with Yair Hayut, we reduced the consistency strength of stationary reflection at $\aleph_{\omega+1}$ to an assumption weaker than $\kappa$ is $\kappa^+$ supercompact.

2) In joint work with Yair Hayut and Omer Ben-Neria, we prove that from large cardinals it is consistent that there is a singular cardinal $\nu$ of uncountable cofinality where the singular cardinal hypothesis fails at nu and every collection of fewer than $\mathrm{cf}(\nu)$ stationary subsets of $\nu^+$ reflects at a common point.

The statement in the second theorem was not previously known to be consistent. These results make use of analysis of Prikry generic objects over iterated ultrapowers.

Two classical results of Magidor are: 

(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and 

(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.

These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes.

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH.  Then we obtain this situation at $\aleph_\omega$. This is joint work with Alejandro Poveda and Assaf Rinot.

Self-similar fragmentation processes are random models for particles that are subject to successive fragmentations. When the index of self-similarity is negative the fragmentations intensify as the masses of particles decrease. This leads to a shattering phenomenon, where the initial particle is entirely reduced to dust - a set of zero-mass particles - in finite time which is what we call the extinction time. Equivalently, these extinction times may be seen as heights of continuous compact rooted trees or scaling limits of heights of sequences of discrete trees. Our objective is to set up precise bounds for the large time asymptotics of the tail distributions of these extinction times.

A uniform spanning tree of $\mathbb{Z}^4$ can be thought of as the "uniform measure" on trees of $\mathbb{Z}^4$. The past of 0 in the uniform spanning tree is the finite component that is disconnected from infinity when 0 is deleted from the tree. We establish the logarithmic corrections to the probabilities that the past contains a path of length $n$, that it has volume at least $n$ and that it reaches the boundary of the box of side length $n$ around 0. Dimension 4 is the upper critical dimension for this model in the sense that in higher dimensions it exhibits "mean-field" critical behaviour. An important part of our proof is the study of the Newtonian capacity of a loop erased random walk in 4 dimensions. This is joint work with Tom Hutchcroft.

We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits. More precisely, we exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible 'at infinity'. We then transfer this result to finite graphs via local weak convergence and a relative compactness argument. We believe that this 'local weak limit' approach to mixing properties of Markov chains will have many other applications.

We investigate a graph theoretic analog of geodesic geometry. In a graph $G=(V,E)$ we consider a system of paths $P=\{P_{u,v}| u,v\in V\}$ where $P_{u,v}$ connects vertices $u$ and $v$. This system is consistent in that if vertices $y,z$ are in $P_{u,v}$, then the sub-path of $P_{u,v}$ between them coincides with $P_{y,z}$. A map $w:E\to(0,\infty)$ is said to induce $P$ if for every $u,v\in V$ the path $P_{u,v}$ is $w$-geodesic. We say that $G$ is metrizable if every consistent path system is induced by some such $w$. As we show, metrizable graphs are very rare, whereas there exist infinitely many 2-connected metrizable graphs.
 

I will introduce product structure theory of graphs and show how families of graphs that have such a structure admit short adjacency labeling scheme and small induced universal graphs. Time permitting, I will talk about another recent application of product structure theory, namely vertex ranking (coloring).

It is sometimes possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which $P$ can be obtained as a (linear) projection. In this talk, we discuss some results on the extension complexity of random $d$-dimensional polytopes (obtained as convex hulls of random points on either on the unit sphere or in the unit ball), and on the extension complexity of polygons with all vertices on a common circle. Joint work with Matthew Kwan and Yufei Zhao