Following the resolution of the longstanding formerly open question of whether the 1-strict order property, $\mathrm{SOP}_1$, is equal to $\mathrm{SOP}_2$, one of the main problems in model theory is to determine whether $\mathrm{SOP}_2$ is equal to $\mathrm{SOP}_3$. In this talk we discuss our very recent progress on this question, where we show, for $\mathcal{H}$ a hereditary class defined by finitely many forbidden weakly embedded substructures, that if the theory of every structure of which $\mathcal{H}$ is the age has $\mathrm{SOP}_2$, then the theory of every structure of which $\mathcal{H}$ is the age has $\mathrm{SOP}_3$. Crucially, this is a strict dichotomy: it is false if we replace $\mathrm{SOP}_2$ with the tree property (as demonstrated in examples of Conant and Kruckman, and Kruckman and Ramsey). We will start with some historical background on the problem of whether $\mathrm{NSOP}_2$ is equal to $\mathrm{NSOP}_3$, as well as the context for our theorem within the setting of $\mathrm{NSOP}_r$ theories for $r$ a real number. We will see how, by modifying recent work of Bodirsky, Bodor and Marimon, our general results reduce to what initially superficially appeared to be a mere verification for a concrete family of examples (the generic structures of Cherlin, Shelah and Shi). If time permits, we will give a very rough overview of the proof of this theorem, which involves harder versions of arguments on the real-valued $\mathrm{NSOP}_r$ hierarchy.

The twist flow on the character variety of a surface group arises in many different guises. For two, as the Fenchel—Nielsen twist flow on Teichmüller space and in constructing examples of quasi-Fuchsian groups by “bending” along a simple closed curve. We will discuss settings in which these flows “generically” connect arbitrary characters of a surface group and applications of this fact to the algebraic structure of character varieties, generalizing work of Rapinchuk, Benyash-Krivetz and Chernousov, and to positivity in higher-rank Teichmüller theory. This is joint work in progress with Jacques Audibert and Vasily Rogov.

In this 20-minute talk, I will present the historical context in which O-Minimality arose with a little timeline. After this, I will provide basic definitions and discuss many of the major theorems available to any who wish to use this tameness condition. Lastly, honing in on the tame topology and geometry available to those who employ O-Minimality, I will describe recent research of Pablo Guerrero in the direction of general O-Minimal topology.

Applying for postdoc positions can often be a daunting task. Matthew Harrison-Trainor will give a short presentation giving an overview of the basics of applying for postdocs: it will cover what materials and reference letters are needed; what the timeline is like; and some basic tips. The presentation will be followed by a question-and-answer panel with panelists Gabe Conant, Dhruv Mubayi, and Jagerynn Verano.

We study the problem of approximating the partition function of the transverse-field Ising model (TFIM), a widely studied quantum many-body model with important applications in quantum simulation and quantum annealing. Despite its fundamental importance, the algorithmic landscape for computing the TFIM partition function has remained poorly understood beyond restricted parameter regimes. We provide a precise characterization of the temperature regimes in which efficient approximation is possible, establishing a sharp computational phase transition. Let $J$ denote the symmetric interaction matrix and $\Delta(J) = \lambda_{\max}(J)-\lambda_{\min}(J)$ denote its spectral width. We show that, for all inverse temperatures $\beta \in [0,1/\Delta(J)]$, there exists an efficient classical randomized algorithm that approximates the partition function $\text{tr}(e^{-\beta H})$ to within an arbitrarily small multiplicative factor. To obtain this result, we apply the standard Trotter decomposition to map the quantum model to a classical spin system, and then leverage new techniques in Markov chain analysis to derive an efficient algorithm that samples from and computes the partition function of the resulting distribution.  This temperature threshold is tight: for $\beta > 1/\Delta(J)$, we show that approximating the partition function, even within an exponential factor, is NP-hard and thus is unlikely to admit an efficient classical or quantum algorithm. Joint work with Alistair Sinclair.

We study a continuum Lagrangian alignment system for interacting agents with weak initial data. We first prove global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We then pass from the Lagrangian description to an Eulerian one, and obtain an Euler-Reynolds-alignment system involving a nonnegative Reynolds stress and, in the nonlinear velocity-coupling case, an additional defect force caused by microscopic velocity fluctuations. Under a heavy-tailed interaction assumption, we show that these defect terms vanish asymptotically, leading to mono-kinetic closure at large times. In the linear velocity-coupling case, we further prove the global existence of weak solutions to the Euler-alignment system, including a sharp critical-threshold result in one dimension and a global existence result in higher dimensions under a large-coupling condition. We also establish mean-field convergence results for the underlying particle system, including uniform-in-time convergence in the linear case.