Abstract: Neural network quantization is a branch of neural network compression that aims to replace high-precision parameters (such as 32-bit or 64-bit weights) with lower-bit representations while maintaining the model’s architecture and performance. In this talk, I will begin by introducing the basic concepts of neural network quantization and highlighting several interesting quantization methods. I will then present my paper Frame Quantization of Neural Networks—a joint work with Professor Wojciech Czaja—recently published in the Journal of Fourier Analysis and Applications. Finally, I will discuss several open questions related to neural network quantization and low-bit representations of neural networks.
Abstract: Mean-Field Games (MFGs) study the Nash equilibrium of non-cooperative games involving a continuum of players. They have broad applications and deep connections to areas such as sampling, optimal transport and economics, etc. In this talk, I will present our recent works in both forward and inverse problems in MFGs, with insights gained through numerical analysis and computational methods. In this talk, I will begin by presenting a convergence analysis of a learning algorithm for MFGs. Our results highlight the central role of the best response in understanding both the game dynamics and the algorithm behavior. Then, I will present a simulation-free method to generalize the algorithm to high-dimension with application to generative models. In the end, I will introduce a simple and efficient iterative strategy for solving a class of inverse MFG problems. This approach shows that measurements of the Nash equilibrium state can be remarkably effective in inferring unknown ambient potentials, such as obstacles.
Abstract: Since their introduction by Francois Labourie, Anosov subgroups of real semisimple Lie groups have come to be accepted as the "correct" generalization of convex cocompact subgroups in rank one. Convex cocompact subgroups of rank one groups may be characterized as holonomy groups of negatively curved locally symmetric Riemannian manifolds whose geodesic flows are Axiom A in the sense of Smale. The naive generalization of this to the higher rank Anosov subgroup situation is false, but we will explain the construction of another locally homogeneous Axiom A flow (often with no interpretation as a geodesic flow) associated to any Anosov subgroup. Time permitting, we will discuss applications of this construction to the meromorphic continuation of certain zeta functions and questions of exponential mixing. All results are joint work with Benjamin Delarue and Daniel Monclair.
Abstract: We discuss modeling, numerical analysis and computation of liquid crystal networks (LCNs). These materials couple a nematic liquid crystal with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Thin bodies of LCNs are thus natural candidates for soft robotics applications. We start from the classical 3D trace energy formula and derive a reduced 2D membrane energy as the formal asymptotic limit of vanishing thickness, including both stretching and bending energies, and characterize the zero energy deformations. We design a sound numerical method and discuss its Gamma convergence. We present computations showing the geometric effects that arise from liquid crystal defects as well as computations of non-isometric origami within and beyond theory. This work is joint with the former students L. Bouck and S. Yang, and the current student G. Benavides.
Abstract: Inflation is a method to select a model from within a large hierarchy of models with the goal of finding the simplest model that economically simulates a small set of objective functions with near optimal fidelity. It begins with an extremely large model, the parameters of which each have an associated intrinsic uncertainty. A hierarchy of smaller models is constructed from a family of reductions of this large model. Starting from a small model in the hierarchy, adjoint sensitivity analysis is used to select a slightly larger model from within the hierarchy that improves the fidelity of a small set of objective functions. This process is repeated until the improved fidelity of these functions becomes comparable to their intrinsic fidelity. This model inflation method will be illustrated on a family of quadratic population models.
Abstract: Class field theory is the study of the abelian Galois extensions of local and global fields, and can be used to define Galois extensions whose Galois groups encode information about the failure of unique factorization in a number field.
Abstract: We show that, up to biholomorphism, a given noncompact complex manifold admits at most one shrinking gradient Kähler-Ricci soliton with Ricci curvature vanishing at infinity. Time permitting, we will also discuss how the technique for proving the uniqueness of the soliton vector field can be applied to other settings, such as AC Calabi-Yau manifolds.
Abstract: In the classification of first-order theories, many dividing lines are defined by forbidding specific configurations of definable sets (e.g. Stability = No Order Property, Simplicity = No Tree Property, etc). In this talk, we abstract the combinatorial nature of these properties; using the concept of patterns of consistency and inconsistency, we give a general framework for studying this kind of dividing lines. Taking this idea to its limit, we review a notion of maximal complexity (SM) in the context of patterns. Weakening SM, we define new dividing lines (PM and the PM^{(k)} hierarchy), provide examples separating these properties from previously defined dividing lines, and prove various results about them.
Abstract: Hamiltonian simulation becomes more challenging as the underlying unitary becomes more oscillatory. In such cases, an algorithm with commutator scaling and a weak dependence, such as logarithmic, on the derivatives of the Hamiltonian is desired. We introduce a family of new time-dependent Hamiltonian simulation algorithm based on the Magnus series expansion that exhibits both features. Importantly, when applied to unbounded Hamiltonian simulation in the interaction picture, we prove that the commutator in the p-order algorithm leads to a surprising 2p-order superconvergence, with an error preconstant independent of the number of spatial grids. The proof of superconvergence is based on semiclassical analysis that is of independent interest.
Abstract: A branchvariety of a projective k-scheme X is a geometrically reduced scheme Y equipped with a finite map to X. Alexeev and Knutson showed the existence of a proper moduli space of branchvarieties with fixed numerical invariants, but the projectivity of this space remained an open question. In this talk, we will discuss positivity results for certain line bundles on the moduli space of equidimensional branchvarieties. As a consequence, we establish that this moduli space is projective.
Abstract: One of the ways to generalize the classical concept of ergodic averages is as follows: instead of considering average values of some function $f$ along the whole orbit $(T^n(x))$, consider them only along a subsequence $(T^{a_n}(x))$. We focus on the case when $(a_n)=(P(n))$ for some polynomial $P$, and let $(X, T)$ be a uniquely ergodic topological system. We are then interested in the property that such ergodic averages along $(a_n)$ converge for every $x\in X$ to the integral of $f$. This is a very delicate property, and not many examples of such systems are known. We will see a new method for establishing this kind of property for $(a_n)=(n^2)$, by assuming a strong rigidity condition on $X$, in particular obtaining weakly mixing examples. Here by rigidity we mean the existence of a sequence $(T^{q_n})$ of iterates of $T$ converging to the identity map on $X$ — we will in particular require a uniform, quantitative form of this convergence. The method uses a lot of input from number theory about the distribution of square residues in arithmetic progressions, and interestingly it does not seem to yield results for polynomials of larger degree. We will discuss the reason for this, as well as examples of systems satisfying the required rigidity condition.
Abstract: In this talk, we study the large-time behavior of the pressureless Euler system with nonlocal alignment and interaction forces. For general interaction potentials and communication weights, we obtain quantitative convergence of classical solutions. In 1D, $(\lambda,\Lambda)$-convex potentials yield exponential decay for bounded weights and sharp algebraic rates for weakly singular ones. For the Coulomb-quadratic potential, we prove exponential convergence with bounded weights and polynomial bounds with singular ones. In multi-dimensions, $(\lambda,\Lambda)$-convex potentials give exponential or improved algebraic decay depending on the weight. In all cases, the density converges (up to translation) to the interaction energy minimizer, while velocities align to a constant. Our results highlight that convergence rates are determined solely by the local behavior of the communication kernel: bounded weights produce exponential decay, while weakly singular ones yield algebraic rates.
Abstract: This talk explores the deep connections between two foundational pillars of twentieth-century statistics—survey sampling and experimental design. Though these connections became somewhat esoteric in the late twentieth century, they have experienced a revival through the modern framework of finite-population causal inference. Central to this connection is the concept of potential outcomes (or counterfactuals), first introduced by Neyman in 1923 and later expanded and formalized by Rubin in the 1970s. Through illustrative examples, we will show how the classical results developed in the early twentieth century can be reinterpreted and extended to address contemporary challenges, particularly as randomized experiments gain renewed prominence across the social, behavioral, and biomedical sciences.
Abstract: Stochastic gradients are now commonly used in modern machine learning. But where do they come from and where else are they used? This talk will explore stochastic gradients (mostly from a simulation-based viewpoint) and discuss some applications and examples of stochastic gradients.