Monday,
November 3 |
- 15:30–16:30 Algebra and Number Theory Seminar, Kemeny 343
 
- The Hasse principle for families of generalized Fermat equations
- Christopher Keyes, King's College London
- Integral solutions to Diophantine equations of the form Ax^p + By^q = Cz^r have been of interest to number theorists since antiquity. A relatively recent development is to interpret these geometrically as integral points on a suitable stacky curve. This allows one to formulate a version of the Hasse principle and access techniques familiar from the theory of rational points on algebraic curves. In this talk, we explore how to use a descent approach to decide whether certain generalized Fermat equations satisfy or fail the Hasse principle. We will also consider statistics for the existence of solutions in these families. This work is joint with Duque-Rosero, Kobin, Roy, Sankar, and Wang.
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Tuesday,
November 4 |
- 11:00–12:00 Combinatorics Seminar, Kemeny 307
- Distributive lattices for proving log-concavity and -convexity
- Bruce Sagan, Michigan State University
- Let $L$ be a distributive lattice. It follows easily from the FKG inequality that if $I$ and $J$ are lower order ideals in $L$ then $|I|\cdot |J| \le |L|\cdot |I\cap J|$ where vertical bars denote cardinality. The inequality is reversed if $I$ is a lower order ideal and $J$ is an upper order ideal. We show how these inequalities can be used to prove log-concavity and log-convexity results for various sequences including binomial coefficients, Catalan numbers, Fibonacci numbers, Stirling numbers of the second kind, order polynomials, and more. In the process we define some new and interesting distributive lattices. This is joint work with Jinting Liang.
- 15:30–16:30 Functional Analysis Seminar, Kemeny 343
- Free-Fermion Structures in Quantum Information
- James D. Whitfield, Dartmouth College
- This seminar will explore foundational aspects of free-fermion systems and their mathematical structures, focusing on the role of symplectic transformations in diagonalizing fermionic operators. We will introduce the algebraic framework of fermionic operators, Gaussian fermionic states, and their covariance matrices. Central to the discussion is the Bloch-Messiah theorem, which underpins the symplectic diagonalization of skew-symmetric matrices arising in these systems. The interplay between orthogonal and symplectic unitaries, as well as the construction of symplectic diagonalizing transformations, will be examined with some proofs, some examples, and some numerical implementations. Applications to quantum dynamics, particularly in the context of efficiently simulating free-fermion evolutions, will be highlighted, along with connections to combinatorial matrix functions such as determinants and pfaffians. This talk aims to provide a rigorous yet accessible bridge between fermionic antisymmetry and quantum information science.
- Poster
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Thursday,
November 6 |
- 10:00 Topology Seminar, Kemeny 307
- On Zeeman's conjecture for non-standard polyhedra
- Ivan Dynnikov, Moscow State University and Steklov Institute for Mathematics, Moscow
- Zeeman's conjecture states that if P is a contractible compact
two-polyhedron, then PxI is collapsible. In 1983, Gillman and Rolfsen
demonstrated that the conjecture, when restricted to standard spines of
three-manifolds, is equivalent to the Poincaré conjecture, which was
settled by Perelman in 2003. In 1987, Matveev proved that Zeeman's
conjecture, when restricted to standard polyhedra that are not spines of
three-manifolds, is equivalent to the weakened Andrews--Curtis
conjecture. In the talk, I will explain how to reduce the general case
of Zeeman's conjecture to the one of a standard polyhedron.
- 15:15 Math Colloquium, Kemeny 007
- Measuring $K_M$: Experimental Uncertainty, Ill-Posedness, and Bias
- Santiago Schnell, Dartmouth College
- Understanding enzyme kinetics—often summarised by the Michaelis constant $K_M$—is central to biochemistry and drug discovery. Yet a recent meta-analysis reveals striking dispersion in reported $K_M$ values, with errors as large as 10.8-fold across studies. I will argue that two mechanisms dominate this variability. First, parameter estimation in standard kinetic models can be ill-posed or ill-conditioned under common experimental regimes, so that small observational noise produces large errors in $K_M$ (and related parameters). Second, systematic biases introduced by experimental design and data-analysis practices—restricted substrate ranges, inadequate replication, ad hoc linearisation, and mis-specified error models—further inflate uncertainty. I will delineate conditions under which $K_M$ is practically or structurally non-identifiable (including singular-perturbation limits of the Michaelis--Menten mechanism), illustrate how routine workflows amplify estimation error, and outline remedies: principled design over informative substrate ranges, replication strategies, inverse-problem regularisation or Bayesian inference, and stricter reporting standards. These interventions provide a path to more reliable and reproducible enzyme-kinetic measurements.
References
P. Halling, P. Fitzpatrick, F. M. Raushel, J. Rohwer, S. Schnell, U. Wittig, R. Wohlgemuth and C. Kettner (2018). An empirical analysis of enzyme function reporting for experimental reproducibility: missing/incomplete information in published papers. Biophysical Chemistry 242, 22--27.
J. Eilertsen, S. Schnell, S. Walcher (2022). On the anti-quasi-steady-state conditions of enzyme kinetics. Mathematical Biosciences 350, 108870.
J. Eilertsen, S. Schnell, S. Walcher (2023). Natural parameter conditions for singular perturbations of chemical and biochemical reaction networks. Bulletin of Mathematical Biology 85, 48.
J. Eilertsen, S. Schnell, S. Walcher (2024). Rigorous estimates for the quasi-steady state approximation of the Michaelis--Menten reaction mechanism at low enzyme concentrations. Nonlinear Analysis: Real World Applications 120, e2219683120.
- Poster
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