As the continuous limit of many discretized algorithms, PDEs can provide a qualitative description of algorithm’s behavior and give principled theoretical insight into many mysteries in machine learning. In this talk, I will give a theoretical interpretation of several machine learning algorithms using Fokker-Planck (FP) equations. In the first one, we provide a mathematically rigorous explanation of why resampling outperforms reweighting in correcting biased data when stochastic gradient-type algorithms are used in training. In the second one, we propose a new method to alleviate the double sampling problem in model-free reinforcement learning, where the FP equation is used to do error analysis for the algorithm. In the last one, inspired by an interactive particle system whose mean-field limit is a non-linear FP equation, we develop an efficient gradient-free method that finds the global minimum exponentially fast.

I will present joint work with Jacek Jendrej (CRNS, Sorbonne Paris Nord) on equivariant wave maps with values in the two-sphere. We prove that every finite energy equivariant wave map resolves, as time passes, into a superposition of decoupled harmonic maps and radiation, settling the soliton resolution conjecture for this equation.  It was proved in works of Côte, and Jia and Kenig, that such a decomposition holds along a sequence of times. We show the resolution holds continuously-in-time via a “no-return” lemma based on the virial identity. The proof combines a collision analysis of solutions near a multi-soliton configuration with concentration compactness techniques. As a byproduct of our analysis we also prove that there are no elastic collisions between pure multi-solitons. 

Let $C_2$ denote the cyclic group of order 2. In this talk, we’ll explore some recent computations done in $RO(C_2)$-graded cohomology with constant integral coefficients for $C_2$-surfaces. We’ll also explore some interesting patterns in these computations, and discuss how these might generalize to $C_2$-manifolds of higher dimension.

The ability of cells to exert forces and move about in their environment is essential to the survival of single-celled and multicellular organisms. Cell movement requires the coordination a number of sub-processes involving biochemical signaling and mechanical force generation. How such coordination can occur is a major area of study. In this talk I will discuss two lines of research that address different aspects of cell motion and biological transport.

In the first part, I will discuss a model and numerical simulation method to study how cell membranes change shape in response to forces that arise from the cell cortex. The cell cortex is a thin layer of cytoskeletal material that lies beneath the cell membrane in many cells. While models of biological membranes have existed for some time, the cell cortex is much more complicated, and detailed models do not yet exist. Thus, as a first step toward understanding the effect of the cell cortex, I will discuss how forces that mimic those generated by the cell cortex affect cell shape, leading to biologically realistic results.

In the second part, I will discuss how cell-level behaviors impact tissue and organ-scale properties through the use of multi-state continuous-time random walk models. These models can be posed in a general framework that includes details such as spatially heterogeneous binding, stochastic internal state changes, and various modes of spatial transport. Macro-scale equations with coefficients that depend on the local details are then obtained to describe transport on a tissue or organ-scale. Both lines of research have extensions that will be discussed throughout the talk.

Approximately an eternity ago, I gave a talk about Fréchet derivatives of maps between normed vector spaces and an infinite-dimensional Taylor's Theorem. I also promised this talk would be part of a series of infinite-dimensional calculus talks. I shall finally partially deliver on this promise by discussing "vector-valued integrals": what they are, when they exist, and -- time permitting -- some applications.

We study the problem of decision-making in the setting of a scarcity of shared resources when the preferences of agents are unknown a priori and must be learned from data. Taking the two-sided matching market as a running example, we focus on the decentralized setting, where agents do not share their learned preferences with a central authority. Our approach is based on the representation of preferences in a reproducing kernel Hilbert space, and a learning algorithm for preferences that accounts for uncertainty due to the competition among the agents in the market. Under regularity conditions, we show that our estimator of preferences converges at a minimax optimal rate. Given this result, we derive optimal strategies that maximize agents’ expected payoffs and we calibrate the uncertain state by taking opportunity costs into account. We also derive an incentive-compatibility property and show that the outcome from the learned strategies has a stability property. Finally, we prove a fairness property that asserts that there exists no justified envy according to the learned strategies.

This is a joint work with Michael I. Jordan.

In 1986, Nishikawa conjectured that a closed Riemannian manifold with positive curvature operator of the second kind is diffeomorphic to a spherical space form and a closed Riemannian manifold with nonnegative curvature operator of the second kind is diffeomorphic to a Riemannian locally symmetric space. Recently, the positive case of Nishikawa's conjecture was proved by Cao-Gursky-Tran and the nonnegative case was settled by myself. In this talk, I will first talk about curvature operators of the second kind and then present a proof of Nishikawa's conjecture under weaker assumptions.

For more than a century and a half it has been widely-believed that the physics of diffraction imposes certain fundamental limits on the resolution of an optical system. However our understanding of what exactly can and cannot be resolved has never risen above heuristic arguments which, even worse, appear contradictory.

In this work we remedy this gap by studying the diffraction limit as a statistical inverse problem and, based on connections to provable algorithms for learning mixture models, we rigorously prove upper and lower bounds on how many photons we need (and how precisely we need to record their locations) to resolve closely-spaced point sources. Moreover we show the emergence of a phase transition, which helps explain why the diffraction limit can be broken in some domains but not in others.

This is based on joint work with Sitan Chen.

A famous theorem of Giordano, Putnam and Skau (1995) states that two minimal homeomorphisms of a Cantor space X are orbit equivalent (i.e, the equivalence relations induced by the two associated actions are isomorphic) as soon as they have the same invariant Borel probability measures. I will explain a new "elementary" approach to prove this theorem, based on a strengthening of a result of Krieger (1980). I will not assume prior familiarity with Cantor dynamics. This is joint work with S. Robert (Lyon).

Fluctuations of synaptic-weights, among many other physical, biological and ecological quantities, are driven by coincident events originating from two 'parent' processes. We propose a multiplicative shot-noise model that can capture the behavior of a broad range of such natural phenomena, and analytically derive an approximation that accurately predicts its statistics. We apply our results to study the effects of a multiplicative synaptic plasticity rule that was recently extracted from measurements in physiological conditions. Using mean-field theory analysis and network simulations we investigate how this rule shapes the connectivity and dynamics of recurrent spiking neural networks. We show that the multiplicative plasticity rule, without fine-tuning, gives a stable, unimodal synaptic-weight distribution with a large fraction of strong synapses. The strong synapses remain stable over long times but do not `run away'. Our results suggest that the multiplicative plasticity rule offers a new route to understand the tradeoff between flexibility and stability in neural circuits and other dynamic networks. Joint work with Bin Wang.

https://mathweb.ucsd.edu/~bli/research/mathbiosci/MBBseminar/

Take $E$ to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, we will outline a proof that 100% of the quadratic twists of $E$ have rank at most one. To do this, we will find the distribution of $2^k$-Selmer ranks in this family for every positive $k$. We will also show how are techniques may be applied to find the distribution of $2^k$-class groups of quadratic fields.

The pre-talk will focus on the definition of Selmer groups. We will also give some context for the study of the arithmetic statistics of these groups.