I'll discuss joint work with Alessio Caminata, Luca Schaffler, and Han-Bom Moon in which we study equations defining (the closure of) the locus of n points in projective space that lie on a rational normal curve and apply these equations to resolve a question of Lior Pachter and David Speyer from 2004 on the tropical geometry of the space of phylogenetic trees.

Consider all words over a finite alphabet that avoid a set of forbidden patterns, e.g. binary sequences with no two adjacent 1s. This can be viewed through the lens of ergodic theory, as a dynamical system (a 'shift space'); or combinatorial probability, as an iid sequence or a markov chain conditioned to avoid the forbidden set; or statistical physics, as the thermodynamic limit of a natural Gibbs measure. I will discuss connections between ideas from these worlds in the case where the forbidden set is of size one or two, including new results related to conjugacy of the underlying shifts, their entropies, and the asymptotic density of 1s.

A dilation surface is roughly a surface made up of polygons in the complex plane with parallel sides glued together by complex linear maps. The action of SL(2, R) on the plane induces an action of SL(2, R) on the collection of all dilation surfaces, aka the moduli space, which possesses a natural manifold structure. Curiously, an analogue of the Hodge theorem tells us that any vector field on a Riemann surface that vanishes at finitely many points P can be homotoped (through vector fields only vanishing at P) to the straight line foliation of a dilation surface. The first result that I will present, joint with Nick Salter, produces an SL(2, R)-invariant measure on the moduli space of dilation surfaces that is mutually absolutely continuous with respect to Lebesgue measure. The construction fundamentally uses the group cohomology of the mapping class group with coefficients in the homology of the surface. It also relies on joint work with Matt Bainbridge and Jane Wang, showing that the moduli space of dilation surfaces is a K(pi, 1) where pi is the framed mapping class group. The second result that I will present, joint with Bainbridge and Wang, is that an open and dense set of dilation surfaces have Morse-Smale dynamics, i.e. the horizontal straight lines spiral towards a finite set of simple closed curves in their forward and backward direction. A consequence is that any fully supported SL(2, R) invariant measure on the moduli space of dilation surfaces cannot be a finite measure.

We sketch the proof that if a Fuchsian group $\Gamma$ has finite index in its commensurator (i.e. is non-arithmetical), then the theory of its two sorted universal cover structure is axiomatized by a sentence in $L_{\omega_1,\omega}$ that is categorical in uncountable powers. This is a simpler version of the earlier result in the arithmetic case (Daw & Harris; Eterović). The argument is structured rather differently and clarifies the roles of model theory and geometry in the earlier case by replacing some `geometric/number theoretic' arguments by model theoretic ones. We point to the role of finite index in allowing these simplifications. Joint work with Joel Nagloo and incorporating some earlier work with Andres Villaveces in a survey of Zilber's `logically perfect structures are $L_{\omega_1,\omega}(Q)$-categorical in uncountable power.

In this talk, I will prove that the marked moduli space of any infinite type surface is contractible. The marked moduli space of an infinite type surface (equipped with an action of the big mapping class group) was introduced in my earlier work, as the generalisation of the usual Teichmüller space of a finite type surface. This result is analogous to that of Douady--Earle, who proved that the (quasiconformal) Teichmüller space of an arbitrary Riemann surface, whether of finite or infinite type, is contractible. Even though the marked moduli space reduces to the Teichmüller space in case the surface is of finite type, it is quite distinct from the Teichmüller space in case the surface is of infinite type. Nevertheless, we are able to adapt the Douady--Earle proof to the setting of the marked moduli space. A key difference is that in this setting, we use a Fréchet space topology on the vector space of (-1, 1)-forms (that is, Beltrami forms), rather than the usual Banach space topology.

Recent advances in dynamic treatment regimes (DTRs) provide powerful optimal treatment searching algorithms, which are tailored to individuals’ specific needs and able to maximize their expected clinical benefits. However, existing algorithms could suffer from insufficient sample size under optimal treatments, especially for chronic diseases involving long stages of decision-making. To address these challenges, we propose a novel individualized learning method which estimates the DTR with a focus on prioritizing alignment between the observed treatment trajectory and the one obtained by the optimal regime across decision stages. By relaxing the restriction that the observed trajectory must be fully aligned with the optimal treatments, our approach substantially improves the sample efficiency and stability of inverse probability weighted based methods. In particular, the proposed learning scheme builds a more general framework which includes the popular outcome weighted learning framework as a special case of ours. Moreover, we introduce the notion of stage importance scores along with an attention mechanism to explicitly account for heterogeneity among decision stages. We establish the theoretical properties of the proposed approach, including the Fisher consistency and finite-sample performance bound. Empirically, we evaluate the proposed method in extensive simulated environments and a real case study for COVID-19 pandemic.

In this talk I'll give a survey of some works studying the distribution of rational points on compactifications of semisimple groups. Time allowing, I'll state a conjecture about the distribution of Campana points on Fano varieties and discuss some evidence using homogeneous varieties. The newer results in this talk are joint work with Demirhan, Chow-Loughran-Tanimoto, and Tanimoto-Tschinkel.

In an effort to find all bound states supported by the nonlinear Schrödinger equation we discovered that as their frequency approaches infinity each bound state separates into peaks which behave like particles. More precisely each peak localizes at a point in space and the force exerted by the potential depends solely on its position while the forces exerted by the other peaks depend on their relative positions. For the bound state to exist all peaks must be stationary hence the forces acting on it must sum to zero. Thus we get a system of algbraic equations which has unique solutions near a local minima of the potential but has infinitely many solutions near a saddle or local maxima. For the latter cases we employ a computer assisted proof to determine and classify a large set of solutions. This is joint work with A. Zarnescu (BCAM) and D. Manea (U. Bucharest).

In 1954, Følner proved the following discrete analogue of Steinhaus's Theorem: If $A$ is a set of positive upper Banach density in an abelian group $G$, then $A-A$ almost (i.e., modulo zero Banach density) contains a neighborhood of the identity in the Bohr topology on $G$. An analogous result for countable discrete amenable groups was proved by Beiglbock, Bergelson, and Fish in 2010 using ergodic theory. In this talk, I will present a proof of this result which is valid for arbitrary discrete amenable groups and is directly inspired by fundamental ideas and facts from local stability theory in continuous logic. Familiarity with continuous logic (or any logic) will not be necessary.

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