This talk is based on a joint work with Yang Li. Motivated by collapsing Calabi-Yau 3-folds and G2-manifolds with Lefschetz K3 fibrations in the adiabatic setting, Donaldson and Scaduto conjectured the existence of a special Lagrangian pair-of-pants in the Calabi-Yau 3-fold $X \times \mathbb R^2$, where $X$ is either a hyperkähler K3 surface (global version) or an A2-type ALE hyperkähler 4-manifold (local version). After a brief introduction to the subject, we discuss the significance of this conjecture in the study of Calabi-Yau 3-folds and G2-manifolds, and then prove the local version of the conjecture, which in turn implies the global version for an open subset of the moduli of K3 surfaces.
We consider the classical 2-opinion dynamics known as the voter model on finite graphs. It is well known that this interacting particle system is dual to a system of coalescing random walkers and that under so-called mean-field geometrical assumptions, as the graph size increases, the characterization of the time to reach consensus can be reduced to the study of the first meeting time of two independent random walks starting from equilibrium.
As a consequence, several recent contributions in the literature have been devoted to making this picture precise in certain graph ensembles for which the above mentioned meeting time can be explicitly studied. I will first review this type of results and then focus on the specific geometrical setting of random regular graphs, both static and dynamic (i.e. edges of the graphs are rewired at random over time), where in recent works we study precise first order behaviour of the involved observables. We will in particular show a quasi-stationary-like evolution for the discordant edges (i.e. with different opinions at their end vertices) which clarify what happens before the consensus time scale both in the static and in the dynamic graph setting. Further, in the dynamic geometrical setting we can see how consensus is affected as a function of the graph dynamics.
Based on recent and ongoing joint works with Rangel Baldasso, Rajat Hazra, Frank den Hollander and Matteo Quattropani.
We derive the general structure for returning to the steady macroscopic nonequilibrium condition, assuming a first-order relaxation equation obtained as zero-cost flow for the Lagrangian governing the dynamical fluctuations. The main ingredient is local detailed balance from which a canonical form of the time-symmetric fluctuation contribution (aka frenesy) can be obtained. That determines the macroscopic evolution. As a consequence, the linear response around stationary nonequilibrium gets connected with the small dynamical fluctuations, leading to fluctuation-response relations. Those results may be viewed as nonequilibrium extension of the well-known structure where the relaxation to equilibrium is characterized by a (dissipative) gradient flow on top of a Hamiltonian motion.
No-go theorems (Bell, Kochen–Specker, …) formally show the departure of quantum theory from the classical worldview. These are formulated in the framework of ontological models and, if one accepts such framework, entail that quantum theory involves problematic (“fine-tuned”) properties. I will argue that the lesson to take from the no-go theorems is to abandon the framework of ontological models as the way to model reality. I will analyze what I believe to be the unnatural assumptions of such framework and I will propose a way to change it. The basic principle of the new notion of reality I propose is that for something to exist is for something to be recorded. I will motivate the principle and explore its consequences. In order to implement such proposal into a precise theory-independent mathematical framework I will make use of point-free topological spaces (locales). I will discuss why this new proposal should be promising for circumventing the conclusions of Bell’s theorem and understanding quantum theory. I will conclude by presenting several open questions.
Special Lagrangians form an important class of minimal submanifolds in Calabi-Yau manifolds. In this talk, we will consider the Calabi-Yau $3$-folds with a K3-fibration and the size of the K3-fibres are small. Motivated by tropical geometry, Donaldson-Scaduto conjectured that special Lagrangian collapse to ``gradient cycles" when the K3-fibres collapse. This phenomenon is similar to holomorphic curves in Calabi-Yau manifolds with collapsing special Lagrangian fibrations converging to tropical curves. Similar to the realization problem in tropical geometry, one might expect to reconstruct special Lagrangians from gradient cycles. In this talk, I will report the first theorem of this kind based on a joint work with Shih-Kai Chiu.
Weak linearisation was defined years ago through a static characterisation of the intuitive notion of virtual redex, based on (legal) paths computed from the (syntactical) term tree. Weak-linear terms impose a linearity condition only on functions that are applied (consumed by reduction) and functions that are not applied (therefore persist in the term along any reduction) can be non-linear. This class of terms was shown to be strongly normalising with deciding typability in polynomial time. We revisit this notion through non-idempotent intersection types (also called quantitative types). By using an effective characterisation of minimal typings, based on the notion of tightness, we are able to distinguish between “consumed” and “persistent” term constructors, which allows us to define an expansion relation, between general lambda-terms and weak-linear lambda-terms, whilst preserving normal forms by reduction.
A crucial problem in geometric quantization is to understand the relationship among quantum spaces associated to different polarizations. Two types of polarizations on toric varieties, Kähler and real, have been studied extensively. This talk will focus on the quantum spaces associated with mixed polarizations and explore their relationships with those associated with Kähler polarizations on toric varieties.
A Ferrers diagram is a graphical way of representing an integer partition. A q-series is a series in which the ratio of the nth term to the next is a rational function of $q^n$. With reference to the origins of the subject in the work of Sylvester, I will present a short introduction to the use of Ferrers diagrams in giving combinatorial interpretations of q-series identities. I will then move on to more recent developments involving a sort of generalized partition, called an overpartition. Finally, I will describe some further generalizations and related open problems.
3d mirror symmetry is a mysterious duality for certian pairs of hyperkähler manifolds, or more generally complex symplectic manifolds/stacks. In this talk, we will describe its relationships with 2d mirror symmetry. This could be regarded as a 3d analog of the paper "Mirror Symmetry is T-Duality" by Strominger, Yau and Zaslow which described 2d mirror symmetry via 1d dualities.
