We study the dynamics of a single vortex ring of small cross-section in the three-dimensional incompressible Euler equations. For a broad class of initial vorticities concentrated near a vortex ring, we prove that the solution remains sharply localized around a moving core for all times and propagates along its axis with the classical logarithmic speed predicted by the vortex filament conjecture. Moreover, we show that such vortex rings are dynamically unstable under arbitrarily small perturbations: suitable smooth perturbations lead to linear-in-time filamentation in the axial direction. These results provide a quantitative description of the coexistence of long-time coherence and instability mechanisms for vortex rings in inviscid flows.

Hirzebruch proved a beautiful inequality for complex line arrangements in CP^2, giving strong bounds on the their combinatorics. In the quest for a topological proof of this inequality, Paolo Aceto and I studied odd and even line arrangements (which I will define in the talk). We proved Hirzebruch-like inequalities for these arrangements, and drew some corollaries about configurations of lines. Time (and audience) permitting, I will also discuss some more speculative ideas and generalisations of our results.

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

In this talk, we discuss the initial–boundary value problem for one-dimensional hyperbolic conservation laws on the half-line, focusing on linear systems and scalar conservation laws. We begin with a discussion of the theory of the Cauchy problem. We then turn to the half-line setting, where we introduce two formulations of boundary conditions: one based on the vanishing viscosity method and the other based on the Riemann problem. We show that these two formulations are equivalent for linear systems and scalar conservation laws. Finally, we present remarks on boundary conditions for general hyperbolic systems of conservation laws. Reference: Dubois, F., and LeFloch, P. Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71, 1 (1988), 93–122.

In this talk, I will begin by presenting some classic constructions of smooth non-orientable 4-manifolds arising from certain Brieskorn homology 3-spheres. I will then explain how to construct new examples, including infinitely many smooth fake copies of *RP4#*CP2. In addition, I will describe a method for generating a collection of Brieskorn homology 3-spheres that can be realized via integer surgery on knots in the 3-sphere. This is joint work with Jae Choon Cha and Oguz Savk.

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

Hamiltonian dynamics is a fundamental mathematical framework for describing classical mechanics, and it can be formulated in terms of vector fields on manifolds. While studying the three-body problem, a central example in Hamiltonian dynamics, Poincaré highlighted the crucial role of periodic orbits. This theme remains central in modern symplectic geometry. In this talk, we introduce the relationship between Hamiltonian dynamics and symplectic geometry, and survey classical and modern approaches to the study of periodic orbits. We also explain how minimal period orbits can be understood from a symplectic-geometric perspective and present an approach to establishing the existence of Birkhoff sections of minimal area using these ideas.

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

In this seminar, we study the Vlasov–Maxwell system, a fundamental collisionless kinetic model for plasmas, posed in a three-dimensional half-space with boundaries. We begin with a brief warm-up by revisiting the one-dimensional Vlasov–Poisson system in the absence of magnetic fields, focusing on Penrose’s classical 1960 spectral criterion for linear stability and instability. We then turn to the full Vlasov–Maxwell system and discuss the major analytical difficulties introduced by electromagnetic coupling, boundary effects, and nonlinear interactions. In particular, we highlight the role of an effective gravitational force directed toward the boundary and its interplay with boundary temperature conditions. This viewpoint naturally leads us to formulate a conjectural linear instability criterion associated with boundary-induced confinement effects. Within this framework, we construct global-in-time classical solutions to the nonlinear Vlasov–Maxwell system beyond the vacuum scattering regime. Our approach combines the construction of stationary boundary equilibria with a proof of their asymptotic stability in the $L^\infty$ setting under small perturbations. This work provides a new framework for analyzing long-time plasma dynamics in bounded domains with interacting magnetic fields. To our knowledge, it yields the first construction of asymptotically stable non-vacuum steady states for the full three-dimensional nonlinear Vlasov–Maxwell system. This is joint work with Chanwoo Kim.

Wenrui Hao Data-driven modeling is essential for deciphering complex biological systems, yet its utility is often constrained by two fundamental hurdles: the inability to guarantee parameter identifiability and the high computational cost of learning nonlinear dynamics. This talk introduces a unified computational framework designed to overcome these challenges, bridging theoretical rigor with scalable machine learning. The first component of the framework establishes a computational foundation for practical identifiability. By leveraging the Fisher Information Matrix and its theoretical links to coordinate identifiability, we propose an efficient method for identifiability assessment. We further introduce regularization-based strategies to manage non-identifiable parameters, thereby enhancing model reliability and facilitating robust uncertainty quantification. To address the discovery of nonlinear dynamics, we present the Laplacian Eigenfunction-Based Neural Operator (LE-NO). This operator learning framework is specifically engineered for modeling reaction–diffusion equations. By projecting nonlinear operators onto Laplacian eigenfunctions, LE-NO achieves superior computational efficiency and generalization across varying boundary conditions, effectively bypassing the limitations of large-scale architectures and data scarcity. Finally, we demonstrate the framework’s utility in the context of Alzheimer’s disease modeling. We show that this integrated approach ensures reliable parameter inference while capturing the intricate nonlinear dynamics of disease progression, providing a critical step toward the development of high-fidelity digital twins for neurodegenerative pathology.

Wenrui Hao Data-driven modeling is essential for deciphering complex biological systems, yet its utility is often constrained by two fundamental hurdles: the inability to guarantee parameter identifiability and the high computational cost of learning nonlinear dynamics. This talk introduces a unified computational framework designed to overcome these challenges, bridging theoretical rigor with scalable machine learning. The first component of the framework establishes a computational foundation for practical identifiability. By leveraging the Fisher Information Matrix and its theoretical links to coordinate identifiability, we propose an efficient method for identifiability assessment. We further introduce regularization-based strategies to manage non-identifiable parameters, thereby enhancing model reliability and facilitating robust uncertainty quantification. To address the discovery of nonlinear dynamics, we present the Laplacian Eigenfunction-Based Neural Operator (LE-NO). This operator learning framework is specifically engineered for modeling reaction–diffusion equations. By projecting nonlinear operators onto Laplacian eigenfunctions, LE-NO achieves superior computational efficiency and generalization across varying boundary conditions, effectively bypassing the limitations of large-scale architectures and data scarcity. Finally, we demonstrate the framework’s utility in the context of Alzheimer’s disease modeling. We show that this integrated approach ensures reliable parameter inference while capturing the intricate nonlinear dynamics of disease progression, providing a critical step toward the development of high-fidelity digital twins for neurodegenerative pathology.

Wenrui Hao Data-driven modeling is essential for deciphering complex biological systems, yet its utility is often constrained by two fundamental hurdles: the inability to guarantee parameter identifiability and the high computational cost of learning nonlinear dynamics. This talk introduces a unified computational framework designed to overcome these challenges, bridging theoretical rigor with scalable machine learning. The first component of the framework establishes a computational foundation for practical identifiability. By leveraging the Fisher Information Matrix and its theoretical links to coordinate identifiability, we propose an efficient method for identifiability assessment. We further introduce regularization-based strategies to manage non-identifiable parameters, thereby enhancing model reliability and facilitating robust uncertainty quantification. To address the discovery of nonlinear dynamics, we present the Laplacian Eigenfunction-Based Neural Operator (LE-NO). This operator learning framework is specifically engineered for modeling reaction–diffusion equations. By projecting nonlinear operators onto Laplacian eigenfunctions, LE-NO achieves superior computational efficiency and generalization across varying boundary conditions, effectively bypassing the limitations of large-scale architectures and data scarcity. Finally, we demonstrate the framework’s utility in the context of Alzheimer’s disease modeling. We show that this integrated approach ensures reliable parameter inference while capturing the intricate nonlinear dynamics of disease progression, providing a critical step toward the development of high-fidelity digital twins for neurodegenerative pathology.

Generative modeling has emerged as a powerful tool for molecular design and structure prediction, offering the ability for molecular discovery. However, challenges such as synthetic feasibility, novelty, diversity of generated molecules, and generalization ability of predictions remain critical for real-world applications, particularly in drug discovery. In this presentation, we introduce an overview of state-of-the-art generative models, including graph-based methods, generative flow networks, and diffusion methods, all aimed at addressing these challenges. First, we will show how generative modeling can facilitate the structural prediction of protein-ligand complexes and its expansion. Second, we focus on strategies that improve the synthesizability of generated molecules by incorporating chemical reaction templates, enabling the generation of novel compounds that are not only drug-like but also synthetically accessible. Third, large language models fine-tuned with drug-related data can be used to elucidating complex relationships between drugs, proteins, and diseases. Through case studies in drug design and broader molecular applications, we demonstrate how these generative modeling can help accelerate drug discovery, offering a pathway to more practical and innovative solutions across molecular discovery domains.

This talk provides an overview of Photoacoustic Tomography (PAT) from both the imaging and mathematical perspectives, and then develops a unified integral-transform viewpoint via a generalized spherical mean operator. In PAT, a short optical pulse induces an initial acoustic pressure distribution \(f(\mathbf x)\), which evolves according to a wave equation. The measured time-dependent acoustic data on an acquisition surface \(\Gamma\) form the forward map, and the central inverse problem is to reconstruct \(f\) from boundary observations. Key mathematical issues include uniqueness, and explicit reconstruction formulas, all of which depend sensitively on the measurement geometry and observation time.

In this talk, we study the non-cutoff Boltzmann equation with moderately soft potentials, a classical kinetic model. The uniqueness of large weak solutions is challenging due to the nonlinearity and limited regularity. To overcome these difficulties, we utilize dilated dyadic decompositions in phase space $(v,\xi,\eta)$ to capture hypoellipticity and reduce the fractional derivative structure $(-\Delta_v)^s$ of the Boltzmann collision operator to a zeroth-order form. Within this framework, we establish the uniqueness of large-data weak solutions under the assumption of finite $L^2$--$L^r$ energy, namely that $\|\mu^{-\frac{1}{2}}(F-\mu)\|_{L^\infty_t L^{r}_{x,v}}+\|\mu^{-\frac{1}{2}}(F-\mu)\|_{L^\infty_t L^2_{x,v}}$ is bounded for some sufficiently large $r>0$. The challenges arising from large solutions are handled via a negative-order hypoelliptic estimate, which yields additional integrability in $(t,x)$.

Stochastic modeling and analysis can help answer pressing medical questions. In this talk, I will attempt to justify this claim by describing recent work on two problems in medicine. The first problem concerns ovarian tissue cryopreservation, which is a proven tool to preserve ovarian follicles prior to gonadotoxic treatments. Can this procedure be applied to healthy women to delay or eliminate menopause? How can it be optimized? The second problem concerns medication nonadherence. What should you do if you miss a dose of medication? How can physicians design dosing regimens that are robust to missed/late doses? I will describe (a) how stochastics theory offers insights into these questions and (b) the mathematical questions that emerge from this investigation.

Any reasonable exotic phenomena in simply-connected 4-manifolds are unstable. It is an open question if there is an universal upper bound to the number of stabilizations needed. The case of 1 stabilization was proven in works of Lin and Guth-K., but whether we need more than two stabilizations has been open because it is significantly harder. In this talk, we discuss my recent proof with Park and Taniguchi that two stabilizations are indeed not enough for exotic diffeomorphisms.

A freshman can calculate that the probability of picking $k$ blue balls after sampling $n$ balls from a bin of $K$ blue balls and $N-K$ red balls is $$\frac{\dbinom{n}{k} \dbinom{N-n}{K-k}}{\dbinom{N}{K}}$$ if one samples without replacement, while it is $$\frac{\dbinom{n}{k} (\frac{K}{N})^k(\frac{N-K}{N})^{n-k}$$ if one samples with replacement. We demonstrate that comparing probabilities of sampling with replacement vs. without replacement leads to De Finetti's Theorem, the Aldous-Hoover Theorem, and even a weak form of Szemeredi's Regularity Lemma which plays a crucial role in the study of graphons. This comparison also leads to a strong version of a representation for DAG-exchangeable arrays (Jung, Lee, Staton, Yang (2021)) which generalize Aldous-Hoover arrays as well as Hierarchical Exchangeable arrays (Austin-Panchenko (2014)).

We establish the stability of a pair of Hill's spherical vortices moving away from each other in 3D incompressible axisymmetric Euler equations without swirl. Each vortex in the pair propagates away from its odd-symmetric counterpart, while keeping its vortex profile close to Hill's vortex. This is achieved by analyzing the evolution of the interaction energy of the pair and combining it with the compactness of energy-maximizing sequences in the variational problem concerning Hill's vortex. The key strategy is to confirm that, if the interaction energy is initially small enough, the kinetic energy of each vortex in the pair remains so close to that of a single Hill's vortex for all time that each vortex profile stays close to the energy maximizer: Hill's vortex. An estimate of the propagating speed of each vortex in the pair is also obtained by tracking the center of mass of each vortex. This estimate is optimal in the sense that the power exponent of the epsilon (the small perturbation measured in the "L^1+L^2+impulse" norm) appearing in the error bound cannot be improved. This talk is based on the paper [Y.-J. Sim, Nonlinearity, 2026].

Biochemical reaction networks and gene regulatory networks in cells are prototypical examples of complex systems, characterized by highly nonlinear and stochastic, multilevel dynamical interactions. Gaining a deep understanding of the stochastic dynamics and thermodynamic principles governing biochemical reaction networks not only helps elucidate the intrinsic mechanisms underlying cell fate decisions and the onset and progression of diseases, but also provides new theoretical paradigms for the study of complex systems. This line of research has become one of the forefront interdisciplinary areas, bridging mathematics, physics, biology, chemistry, statistics, and intelligent science. In this talk, I will present our recent research progress in this area, with the hope of stimulating further discussion and inspiring new ideas.

We consider the computation of internal solutions for a time domain plasma wave equation with unknown coefficients from the data obtained by sampling its transfer function at the boundary. The computation is performed by transforming the background snapshots for a known background coefficient using the Cholesky decomposition of the data-driven Gramian. We show that this approximation is asymptotically close to the projection of the internal solution onto the subspace of background snapshots. This allows us to derive a generally applicable bound for the error in the approximation of internal fields from boundary data only for a time domain plasma wave equation with an unknown potential q. We use this to show convergence for general unknown $q$ in one dimension. We show numerical experiments and applications to SAR imaging in higher dimensions.

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Meeting ID: 830 2123 3470 Passcode: 080691

We discuss sharp local smoothing estimates for curve averages. The proof introduces a new method for estimating oscillatory integrals based on wave packet analysis and a high–low decomposition. We outline the main ideas of the local smoothing estimates for curve averages in three dimensions, focusing on the treatment of the relevant oscillatory integrals.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

Bubbling is a form of singularity formation that commonly arises in critical partial differential equations. In the context of dispersive equations, this phenomenon is closely related to what is commonly known as soliton resolution: the asymptotic decomposition of solutions into a sum of several solitons (each modulated by time dependent parameters) and a radiation term. This talk focuses on classifying the asymptotic behavior of these modulation parameters, thereby providing a more refined understanding of bubbling dynamics. Modulation analysis serves as both the key methodology and the guiding philosophy. I will present several results in this direction.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We show how specific families of positive definite kernels serve as powerful tools in analyses of iteration algorithms for multiple layer feedforward Neural Network models. Our focus is on particular kernels that adapt well to learning algorithms for data-sets/features which display intrinsic self-similarities at feedforward iterations of scaling.

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Meeting ID: 856 9123 8129 Passcode: 712180

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

Abstract: In this talk, we discuss finite-time blow-up dynamics for the nonlinear heat equation (NLH). We explain the notion of finite-time blow-up, introduce Type I and Type II blow-ups, and discuss the difference between these two behaviors. Restricting to radially symmetric solutions, we review known blow-up results and give a heuristic explanation of when only Type I blow-up is possible and when Type II blow-up may occur. Finally, we describe possible Type II blow-up scenarios through their formal mechanisms. Reference: [1] Hiroshi Matano, Frank Merle. On Nonexistence of type II blowup for a supercritical nonlinear heat equation. Communications on Pure and Applied Mathematics, 2004, 57. 1494 - 1541. [2] Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41(10): 4847-4885

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

Given two knots in a 3-manifold M (e.g. in Euclidean 3-space) the Gordian distance is defined as the minimal number of crossing changes needed to turn one knot into the other. I will discuss the relationship between the Gordian distance in Euclidean 3-space and Gordian distance in general 3-dimensional manifolds.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

Abstract: In this seminar, we study the logistic diffusion equation, a reaction–diffusion model, and its equilibria. We first establish existence and regularity of positive solutions to the parabolic problem. We then use the comparison principle to show that, as time tends to infinity, the solution converges to a steady state solving the corresponding elliptic equation. We recall why the existence of solutions to this elliptic problem is not easily obtained by standard variational methods. Finally, we discuss how stability depends on the resource term and how the solution behavior changes with the diffusion rate. References: [1] Cantrell, R.S., Cosner, C. Spatial ecology via reaction-diffusion equation. Wiley series in mathematical and computational biology, John Wiley & Sons Ltd (2003)

Quantum computing offers new possibilities for scientific computing by enabling operations on exponentially large state spaces. In this lecture, we discuss how nonlinear partial differential equations (PDEs) can be connected to quantum algorithms through mathematical linearization frameworks. After a brief introduction to the fundamentals of quantum computation and quantum numerical linear algebra, we present Koopman and Koopman–von Neumann (KvN) formulations that embed nonlinear dynamics into linear operators. We then outline how these ideas, combined with Carleman linearization and relaxation-based methods, can lead to quantum-ready formulations of nonlinear PDE solvers.