This Week in Mathematics


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Entries for this week: 6

Tuesday April 21, 2026

Topology seminar
Metric reconstruction via optimal transport
- Henry Adams, University of Florida
Time: 3:05PM Room: LOV 232
Abstract/Desc: Let X be a sample of points from a metric space M. How do we recover the geometric and topological properties of M from X? One way is to build a metric thickening space P(X;r) of all probability measures on X whose "size" is at most r, where size could be measured as an L^p variance or as an L^p diameter, for 1 <= p <= infty. Can a Morse theory be developed to describe how P(X;r) changes as r increases? Particular cases of interest are when X is finite (which I will mention) or when X is a Riemannian manifold (which I will focus on).

Applied and Computational Mathematics
From Micro-physics to Stable Macro-models: Variational Learning for Non-Newtonian Fluids
- Dr. Huan Lei, Michigan State University
Time: 3:05 Room: 231
Abstract/Desc: Machine-learned PDE models provide a promising approach to predict in-sample dynamics, yet their numerical robustness and out-of-distribution generalization remain open challenges, especially when physical structure is not enforced. We introduce a general approach for learning stable, interpretable macroscale PDEs by constructing the energy variational structure directly from microscale physical laws. In this framework, we introduce a set of micro-macro encoders to model the unresolved micro-physics as generalized field variables, along with an extendable energy functional and variational form that strictly preserve the conservation laws and entropy production. We illustrate this approach through the non-Newtonian hydrodynamics of polymeric fluids, a canonical multiscale problem where conventional empirical closures often fail. The resulting model naturally inherits microscale structural-dependent, nonlinear interactions that challenge empirical closures. More importantly, the variational informed formulation guarantees frame-indifference objectivity, free energy decay, and positivity-preserving dynamics; various pre-existing energy-stable numerical schemes can be used to establish long-time simulations. In contrast, the direct PDE form-based learning leads to models that may fit training data but fail beyond it due to instability and loss of physical fidelity.

ATE
Networks of Bistable Mutual Activation Switches
- Philip Nyamele Asare,
Time: 1:00 PM Room: LOV 204-A
Abstract/Desc: Biological decision-making is often discrete: regulatory programs tend to settle into stable “ON/OFF” regimes that persist despite molecular noise. This proposal investigates how such categorical out- comes can emerge not from isolated motifs, but from networks of interacting bistable switches. Our central hypothesis is that coupling strength g governs a tradeoff between fate diversity and coordination: weak coupling sustains multiple mixed equilibria with sizable basins of attraction, whereas strong coupling drives consensus by expanding the basins of coordinated attractors until mixed basins are swallowed. We frame this claim in the language of dynamical landscapes, where cell-level outcomes correspond to attractors and robustness corresponds to basin volume. These predictions yield measurable signatures for cell decision-making.

Wednesday April 22, 2026

Biomathematics Seminar
Nanorobots for Targeted Drug Delivery in Biofilm Removal / Exotic patterns in Quasi-Ring Networks
- Rachel Veit-Holt / Toma Debnath, FSU
Time: 3:05 Room: Love 232
Abstract/Desc: Veit-Holt: Biofilms are structured communities of microorganisms embedded in an extracellular polymeric substance, or EPS, matrix. That matrix protects cells from antibiotics, mechanical removal, and other environmental stresses, which is why biofilms are especially difficult to eliminate in dental, medical, and industrial settings. This presentation examines how magnetic nanorobots may address that problem through targeted transport, mechanical disruption, and localized chemical action. In addition to the biological motivation, the talk introduces several simple mathematical ideas that clarify why these systems behave the way they do: continuum transport models for biofilm growth, reaction-diffusion limits on drug penetration, low-Reynolds-number propulsion, magnetic force balance, and a viscoelastic description of EPS resistance. Together, these ideas show why passive treatment often fails and why actively controlled nanoscale devices are promising for targeted biofilm removal. Debnath: Pattern formation in network-coupled dynamical systems arises in many physical, chemical, and biological contexts. In this work, we study reaction–diffusion dynamics on networks, with a focus on quasi-ring topologies. The dynamics are modeled using a two-species system, where diffusion is represented through the graph Laplacian. The stability of the homogeneous steady state is analyzed using the Master Stability Function (MSF) framework by decomposing perturbations into Laplacian eigenmodes. To capture the nonlinear evolution near instability, weakly nonlinear theory is employed, yielding amplitude equations that describe pattern growth and saturation. Numerical simulations of the Brusselator model are performed on both regular ring and perturbed quasi-ring networks. While the regular ring supports spatially extended patterns associated with delocalized Fourier modes, small perturbations in the quasi-ring break symmetry and induce localization of Laplacian eigenvectors. In particular, localized eigenvectors associated with short-wave instabilities interact with delocalized modes from long-wave instabilities, significantly shaping the resulting patterns. As a result, quasi-ring networks exhibit a range of dynamical behaviors, including synchronized oscillations, oscillation death, traveling waves, and mixed states with coexisting stationary and oscillatory regions. These findings highlight how small structural perturbations can strongly influence network spectra and lead to complex, chimera-like pattern formation.

Thursday April 23, 2026

ATE
Phase Dynamics in Linear and Nonlinear Multi-Track Models of Active Molecular Transport
- Gustav Jennetten,
Time: 2:00 PM Room: LOV 204-A
Abstract/Desc: Intracellular transport processes are fundamental to cell health. Breakdown in transport is linked to onset of several diseases. This is particularly true for neurons, for which deficiencies in active transport processes have been implicated in neurodegenerative diseases. However, the mechanism by which such deficient transport emerges is an open question. In this talk, we introduce a model of neuronal transport that explicitly takes into account the number of tracks that are available for active transport to occur. We ask how the number of available tracks impacts macroscopic flux of material through the neuron and answer this question by partitioning model behavior in parameter space. We also describe a biophysical mechanism that could drive transport breakdown.

Friday April 24, 2026

Mathematics Colloquium [url]
Designing dynamic measure transport for sampling and quantization
- Youssef M Marzouk, MIT
Time: 3:05 Room: Lov 101
Abstract/Desc: Sampling or otherwise summarizing complex probability distributions is a central task in applied mathematics, statistics, and machine learning. Many modern algorithms for this task introduce dynamics in the space of probability measures and design these dynamics to achieve good computational performance. We will discuss several aspects of this broad design endeavor. First is the problem of optimal scheduling of dynamic transport, i.e., with what speed should one proceed along a prescribed path of probability measures? Though many popular methods seek “straight line” trajectories, i.e., trajectories with zero acceleration in a Lagrangian frame, we show how a specific class of “curved” trajectories can improve approximation and learning. We then present extensions of this idea which seek not only schedules but paths that improve spatial regularity of the underlying velocity. Second, we discuss the problem of weighted quantization, i.e., summarizing a complex distribution with a small set of weighted samples. We study this problem from the perspective of minimizing maximum mean discrepancy via gradient flow in the Wasserstein–Fisher–Rao geometry. This perspective motivates a new fixed-point algorithm, called mean shift interacting particles (MSIP), which outperforms state-of-the-art methods for quantization. We describe how MSIP can be used not only to quantize an empirical distribution, but to sample given an unnormalized density.

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Department of Mathematics

This Week in Mathematics