This Week in Mathematics


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Entries for this week: 3

Tuesday April 21, 2026

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.

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