Geometry seminars

Benoit Assi examines limits on reliable theoretical predictions for present and future particle colliders. Low-order simulations of QCD radiation carry large uncertainties, hadronization is commonly modeled rather than derived, and effective theories contain more operators than measurements can constrain. Information theory and machine learning can incorporate improved calculations into simulations, quantify uncertainty, choose informative observables and advance hadronization theory. Geometry of effective-theory field space combines infinite operator families into finite physical quantities, while information measures identify the combinations experiments can resolve. The talk develops these complementary approaches to extracting the available information from LHC and future-collider data.

Gregorio Baldi surveys recent results on how the Hodge locus is distributed in mixed variations of Hodge structures, including the pure setting. The presentation also examines concrete applications of these results to moduli spaces.

Neural activity in many sensory systems is organized on low-dimensional manifolds by means of convex receptive fields. Neural codes in these areas are constrained by this organization, as not every neural code is compatible with convex receptive fields. The same codes are also constrained by the structure of the underlying neural network. In my talk I will attempt to provide answers to the following natural questions: (i) How do recurrent circuits generate codes that are compatible with the convexity of receptive fields? (ii) How can we utilize the constraints imposed by the convex receptive field to understand the underlying stimulus space. To answer question (i), we describe the combinatorics of the steady states and fixed points of recurrent networks that satisfy the Dale’s law. It turns out the combinatorics of the fixed points are completely determined by two distinct conditions: (a) the connectivity graph of the network and (b) a spectral condition on the synaptic matrix. We give a characterization of exactly which features of connectivity determine the combinatorics of the fixed points. We also find that a generic recurrent network that satisfies Dale's law outputs convex combinatorial codes. To address question (ii), I will describe methods based on ideas from topology and geometry that take advantage of the convex receptive field properties to infer the dimension of (non-linear) neural representations. I will illustrate the first method by inferring basic features of the neural representations in the mouse olfactory bulb.

Self-organized complex structures in nature, e.g., viral capsids, hierarchical biopolymers, and bacterial flagella, offer efficiency, adaptability, robustness, and multi-functionality. Can we program the self-assembly of three-dimensional (3D) complex structures using simple building blocks, and reach similar or higher level of sophistication in engineered materials? Here we present an analytic theory for the self-assembly of polyhedral nanoparticles (NPs) based on their crystal structures in non-Euclidean space. We show that the unavoidable geometrical frustration of these particle shapes, combined with competing attractive and repulsive interparticle interactions, lead to controllable self-assembly of structures of complex order. Applying this theory to tetrahedral NPs, we find high-yield and enantiopure self-assembly of helicoidal ribbons, exhibiting qualitative agreement with experimental observations. We expect that this theory will offer a general framework for the self-assembly of simple polyhedral building blocks into rich complex morphologies with new material capabilities such as tunable optical activity, essential for multiple emerging technologies.

The dream of programmable matter is to create materials whose physical properties (shape, moduli, response to perturbations, etc.) can be changed on the fly. For many years, my group has been thinking about how to program flat sheets that fold up into three dimensional shapes, most recently by exploiting the principles of origami design. Unfortunately, a combinatorial explosion of folding pathways makes robust folding particularly challenging. In this talk, I will discuss how this pluripotency arises from the topology of the configuration space. This suggests a broader understanding of a larger class of materials spanning from folding forms to spring networks to mechanical structures that perform computational logic.

This talk establishes a general reduction from nonlinear spectral gaps of metric spaces to data-dependent space partitions in the form of locality-sensitive hashing. The reduction yields an approach to high-dimensional approximate near-neighbor search and a data structure for any d-dimensional norm whose query algorithm makes a sublinear number of probes. Its approximation factor is O(log d), improving on the square-root-of-d factor available from the generic approach based on John’s ellipsoid. The presentation connects spectral properties, geometric partitions, and efficient approximate search. Joint work with Assaf Naor, Aleksandar Nikolov, Ilya Razenshteyn, and Erik Waingarten.