Seminars from until

This talk offers a leisurely-paced and informal introduction to some classical results at the intersection of mathematics and machine learning theory. We will explore the subject through three central lenses: approximation, optimization, and generalization. Particular attention will be given to universal approximation theorems, which illustrate the expressive power of neural networks. The focus is on foundational ideas and mathematical intuition, I will also highlight some limitations of these classical tools. The goal is not to be exhaustive, but to offer a broad perspective and present a few selected proofs related to expressivity along the way.

References

1. A. Pinkus. Approximation theory of the MLP model in neural networks. Acta Numerica, 1999.
2. J. Berner, P. Grohs, G. Kutyniok and P. Petersen. The Modern Mathematics of Deep Learning, in: Mathematical Aspects of Deep Learning, CUP, 2023.

This talk offers a leisurely-paced and informal introduction to some classical results at the intersection of mathematics and machine learning theory. We will explore the subject through three central lenses: approximation, optimization, and generalization. Particular attention will be given to universal approximation theorems, which illustrate the expressive power of neural networks. The focus is on foundational ideas and mathematical intuition, I will also highlight some limitations of these classical tools. The goal is not to be exhaustive, but to offer a broad perspective and present a few selected proofs related to expressivity along the way.

References

1. A. Pinkus. Approximation theory of the MLP model in neural networks. Acta Numerica, 1999.
2. J. Berner, P. Grohs, G. Kutyniok and P. Petersen. The Modern Mathematics of Deep Learning, in: Mathematical Aspects of Deep Learning, CUP, 2023.

I will explain how to think of infinitely wide neural networks at both initialization and during training. This means, its initial value and how it evolves along its training. At initialization, I will show that such neural networks are equivalent to a Gaussian process. During training, I will show that their evolution is equivalent to an autonomous linear flow in the space of functions. This is related to a phenomenon called (the lack of) feature learning and I intend to at least mention what that is.

Based on:

1. Luís Carvalho, João Lopes Costa, José Mourão, Gonçalo Oliveira. Wide neural networks: From non-gaussian random fields at initialization to the NTK geometry of training, arXiv:2304.03385.
2. L. Carvalho, J. L. Costa, J. Mourão, G. Oliveira. The positivity of the Neural Tangent Kernel, to appear in SIMODS (SIAM Journal on Mathematics of Data Science), arXiv:2404.12928.

I will explain how to think of infinitely wide neural networks at both initialization and during training. This means, its initial value and how it evolves along its training. At initialization, I will show that such neural networks are equivalent to a Gaussian process. During training, I will show that their evolution is equivalent to an autonomous linear flow in the space of functions. This is related to a phenomenon called (the lack of) feature learning and I intend to at least mention what that is.

Based on:

1. Luís Carvalho, João Lopes Costa, José Mourão, Gonçalo Oliveira. Wide neural networks: From non-gaussian random fields at initialization to the NTK geometry of training, arXiv:2304.03385.
2. L. Carvalho, J. L. Costa, J. Mourão, G. Oliveira. The positivity of the Neural Tangent Kernel, to appear in SIMODS (SIAM Journal on Mathematics of Data Science), arXiv:2404.12928.

Survival models with cure fractions, known as long-term survival models, are widely used in epidemiology to account for both immune and susceptible patients regarding a failure event. In such studies, it is also necessary to estimate unobservable heterogeneity caused by unmeasured prognostic factors. Moreover, the hazard function may exhibit a non-monotonic shape, specifically, an unimodal hazard function. In this article, we propose a long-term survival model based on a defective version of the Dagum distribution, incorporating a power variance function frailty term to account for unobservable heterogeneity. This model accommodates survival data with cure fractions and non-monotonic hazard functions. The distribution is reparameterized in terms of the cure fraction, with covariates linked via a logit link, allowing for direct interpretation of covariate effects on the cure fractionan uncommon feature in defective approaches. We present maximum likelihood estimation for model parameters, assess performance through Monte Carlo simulations, and illustrate the models applicability using two health-related datasets: severe COVID-19 in pregnant and postpartum women and patients with malignant skin neoplasms.

What we are calling Emden-Fowler-Hénon (EFH) type equations are elliptic partial differential equations that have the main structure as $-\Delta u=h(x)|u|^{p-1}u$, usually considering $h(x)$ a radial function and with $p$ a subcritical exponent in the sense of Sobolev embeddings, that is, $1 < p < 2^*-1$. Taking $h(x) \equiv 1$, the resulting equation is called Lane-Emden equation which is used in astrophysics to model stellar structures. If we consider $h(x) = |x|^\alpha$, we obtain the so called Hénon equation, proposed in [1] as a perturbation of Lane-Emden equation to study the stability of spherical stellar systems. In mathematics, the Hénon equation plays many important roles in the theory of qualitative analysis of PDE, for instance, by providing a nontrivial example to theorems and working as a guide to develop new tools to fortify the theory. Significant developments have been made by [2, 3, 4, 5] to understand the qualitative properties of the Hénon equation and to set a process to analyze EFH type equations. In this setting, we propose an EFH type equation considering $h(x) = (4|x|(1 − |x|))^\alpha$, which is a weight that presents a behavior unexplored in literature and, we believe, necessary to comprehend the full picture of EFH type equations. We are also interested in understanding how well the developed technics apply to this new configuration.

1. Hénon, M. (1974). Numerical Experiments on the Stability of Spherical Stellar Systems. Symposium - International Astronomical Union, 62, 259-259. https://doi.org/10.1017/s0074180900070662
2. Smets, D., Willem, M. & Su, J. Non-radial ground states for the Hénon equation. Communications In Contemporary Mathematics. 4, 467-480 (2002,8)
3. Cao, D., Peng, S. & Yan, S. Asymptotic behaviour of ground state solutions for the Hénon equation. IMA Journal Of Applied Mathematics. 74, 468-480 (2008,12)
4. Silva, W. & Santos, E. Asymptotic profile and Morse index of the radial solutions of the Hénon equation. Journal Of Differential Equations. 287 pp. 212-235 (2021,6)
5. Mercuri, C. & Santos, E. Quantitative symmetry breaking of groundstates for a class of weighted Emden-Fowler equations. Nonlinearity. 32, 4445-4464 (2019,10).

In the past decade, there has been extensive research on the nonlinear Schrödinger equation (NLS) on metric graphs, driven by both the physical and mathematical communities. Metric graphs, in essence, are one-dimensional objects that can model network-like structures. The first goal of this talk, particularly for those who are new to metric graphs, is to provide an introduction to these structures and present the appropriate functional framework for studying the NLS equation on them.

It is well-established that both the metric (size) and topological (shape) properties of the graphs can impact the existence of solutions to the NLS. As a result, no general theory currently exists for analyzing the NLS equation on metric graphs. A common approach is to focus on specific classes of graphs. In this talk, we focus on two such graphs: the $\mathcal{T}$-graph and tadpole graphs. We then discuss, using techniques from the theory of ordinary differential equations (specifically, parts of the period function), how to approach questions related to the existence, uniqueness, and multiplicity of positive solutions on these graphs.

Time permitting, we will demonstrate how this careful analysis leads to a series of existence and uniqueness/multiplicity results for a class of graphs known as single-knot graphs.

This talk is based on joint work with Simão Correia and Hugo Tavares.

To develop a model for non-equilibrium statistical mechanics, the system is typically brought into contact with two thermodynamic reservoirs, known as boundary reservoirs. These reservoirs impose their own particle density at the system's boundary, thereby inducing a current. Over time, a non-equilibrium steady state emerges, characterized by a stationary current value.

Recently, there has been increasing interest in multi-component systems, where various particle species (sometimes referred to as colors) coexist. In such setups, interactions between diferent species are possible alongside the occupation of available sites.

This work focuses on the boundary-driven multi-species stirring process on a one-dimensional lattice. This process extends naturally from the symmetric exclusion process (SEP) when multiple particle species are considered. Its dynamics involve particles exchanging positions with holes or with particles of diferent colors, each occurring at a rate of 1. Additionally, the system interacts with boundary reservoirs that inject, remove, and exchange types of particles.

After defining the process's generator using an appropriate representation of the gl(N) Lie algebra, we establish the existence of an absorbing dual process defined on an extended chain, where the two boundary reservoir are replaced by absorbing extra sites. This dual process shares bulk dynamics with the original but includes extra sites that absorb particles over extended time periods.

This multi-species stirring process can be mapped onto a higher rank open XXX-Heisenberg spin chain, therefore we employ absorbing duality and the matrix product ansatz to derive closed-form expressions for the non-equilibrium steady-state multi-point correlations of the process. This result is reported in [1].

Next, scaling limits of the process are examined, particularly the behavior of the properly scaled empirical density of the process. First, hydrodynamic equations are derived, illustrating typical system behavior (in the spirit of the law of large numbers). Second, fluctuations from this hydrodynamic limit are investigated, revealing a set of Gaussian processes coupled through noise, resembling aspects of the central limit theorem. Finally, large deviation results are reported, describing the probability of rare trajectories deviating from typical behaviors. An additional outcome of this analysis is the identification of a system of hydrodynamic equations featuring a drift due to interaction with an external field. These scaling limit results are reported in [2] and [3].

References:

[1] F. Casini, R. Frassek, C. Giardinà, Duality for the multispecies stirring process with open boundaries. (2024) J. Phys. A: Math. Theor. 57 295001

[2] F. Casini, C. Giardinà, F. Redig, Density Fluctuations for the Multi-Species Stirring Process. (2024) J Theor Probab 37, 33173354.

[3] F. Casini, F. Redig, H. van Wiechen, A large deviation principle for the multispecies stirring process. (2024), Arxiv: 2410.20857.

In this work, we deal with the symmetric exclusion process with k slow bonds equally spaced in the torus with kn sites, where the strength of a slow bond is $\alpha n^{-\beta}$, where $\beta>1$. For k fixed, it was known (T. Franco, P. Gonçalves, A. Neumann, AIHP'13) that the hydrodynamic limit in the diffusive scaling of this process is given by the heat equation with Neumann boundary conditions, meaning that the system does not allow flux through a slow bond in the limit. In this joint work with Tiecheng Xu and Dirk Erhard, we obtain another three superdiffusive scalings for this system. If $k$ is fixed and the (time) scaling is $n^{\theta}$, where $2< \theta<1+\beta$, the system reaches equilibrium instantaneously at each box between two consecutive slow bonds, being constant in time and space. If k is fixed and $\theta=1+\beta$, the density is spatially constant inside each box, and evolves as the discrete heat equation. And if $\theta=1+\beta$ and $k$ goes to infinity, we recover the continuous heat equation (on the torus).

Cells make fate decisions in response to dynamic environments, and multicellular structures emerge from multiscale interplays among cells and genes in space and time. The recent single-cell genomics technology provides an unprecedented opportunity to profile cells for all their genes. While those measurements provide high-dimensional gene expression profiles for all cells, it requires fixing individual cells that lose many important spatiotemporal information. Is it possible to infer temporal relationships among cells from single or multiple snapshots? How to recover spatial interactions among cells, for example, cell-cell communication? In this talk I will present our newly developed computational tools to study cell fate in the context of single cells as a system. In particular, I will show dynamical models and machine-learning methods, with a focus on inference and analysis of transitional properties of cells and cell-cell communication using both high-dimensional single-cell and spatial transcriptomics, as well as multi-omics data for some cases. Through their applications to various complex systems in development, regeneration, and diseases, we show the discovery power of such methods in addition to identifying areas for further method development for spatiotemporal analysis of single-cell data.

We begin with a brief overview of the rapidly developing research area of active matter (a.k.a. active materials). These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools. We focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction. We introduce a minimal two-dimensional free-boundary PDE model that captures the evolution of the cell shape and nonlinear diffusion of myosin.

We first consider a linear diffusion model with two sources of nonlinearity: Keller-Segel cross-diffusion term and the free boundary that models moving/deformable cell membranes. Here we establish asymptotic linear stability and derive the explicit formula for the stability-determining eigenvalue.

Next, we consider the effect of nonlinear myosin diffusion, which results in the change of the bifurcation type from super- to subcritical, and we obtain an asymptotic representation of the bifurcation curve (for small velocities). This allows us to derive an explicit formula for the curvature at the bifurcation point that controls the bifurcation type. In the most recent work in progress with the Heidelberg biophysics group, we study the relation between various types of nonlinear diffusion and bistability.

Finally, we discuss novel mathematical features of this free boundary model with a focus on non-self-adjointness, which plays a key role in the spectral stability analysis. Our mathematics reveals the physical origins of the non-self-adjoint of the operators in this free boundary model.

Joint works with A. Safsten & V. Rybalko (Transactions of AMS 2023, and Phys. Rev. E 2022), with O. Krupchytskyi &T. Laux (Preprint 2024), and with A. Safsten & L. Truskinovsky ( Arxiv preprint 2024). This work has been supported by NSF grants DMS-2404546, DMS-2005262, and DMS-2404546.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.