On The Fourier Coefficients of High-Dimensional Random Geometric Graphs
Guy Bresler (MIT, EECS)
- The random geometric graph RGG(n,d,p) on the d-dimensional sphere is formed by sampling n i.i.d. vectors uniformly and placing an edge between pairs of vertices i and j closer than some threshold chosen so that the marginal edge probability is p. We study the low-degree Fourier coefficients of this distribution. Our main conceptual contribution is a novel strategy for bounding Fourier coefficients which we believe is more widely applicable to studying latent space distributions: we prove that the dependence among edges can be localized to few fragile edges and then show how randomness in the remaining edges acts as a noise operator. We apply the resulting bounds to: 1) Settle the low-degree polynomial complexity of distinguishing spherical and Gaussian random geometric graphs from Erdos-Renyi both in the case of observing a complete set of edges and in the non-adaptively chosen mask 2) Exhibit a statistical-computational gap for distinguishing 𝖱𝖦𝖦 and the planted coloring model [KVWX23] in a regime when 𝖱𝖦𝖦 is distinguishable from Erdos-Renyi. Joint work with Kiril Bangachev. https://arxiv.org/abs/2402.12589
Braids in planar open books and fillable surgeries.
Eylem Zeliha Yildiz (Duke U, Mathematics)
- We'll give a useful description of braids in $\underset{n}{\#}(S^1\times S^2)$ using surgery diagrams, which will allow us to address families of knots in lens spaces that admit fillable positive contact surgery. We also demonstrate that smooth $16$ surgery to the knot $P(-2,3,7)$ bounds a rational ball, which admits a Stein handlebody. This answers a question left open by Thomas Mark and Bülent Tosun.
Math ∩ Computer Science: the relationship between programs and proofs
Stavan Jain (Duke University)
- While ChatGPT and other generative language models have been shown to be useful in explaining concepts and ideas in mathematics, their performance is exceedingly underwhelming when it comes to proving theorems. These forms of Artificial Intelligence have no notion of truth and instead produce output based off probability distributions computed from imperfect data. However, computer systems grounded in logical consistency through a novel interpretation of programs and proofs have recently indicated that they have the potential to transform the way that mathematicians and computer scientists prove theorems.
This talk will unravel the relationship between computer programs and mathematical proofs. The first half will build up the mathematical machinery and historical context to explain the conceptual bridge that connects programs and proofs [the Curry-Howard Isomorphism]. The second part will explore the potential implications of (computer) proof assistants on the mathematical community and propound the idea that students, educators, and researchers alike might benefit from this emerging paradigm.
There will be a tea before the talk (@ 4:00 PM), and pizza after (@ 5:30 PM).
Small scale creation of the Lagrangian flow in 2d perfect fluids
Ayman Said (University of Cambridge)
- In this talk I will present a recent result showing that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm of the associated Lagrangian flow blows up in infinite time like $t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact $L^2$ decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade which I will then show to be the quantitative phenomenon behind a microlocal generalized Lyapunov function constructed by Shnirelman.
TBA
Robin Zhang (MIT, Mathematics)
KPZ equation from driven lattice gases
Kevin Yang (Harvard University)
- We will discuss a family of exclusion processes in one spatial dimension, where the random walk particles feel a speed-changed drift coming from an external “force field”. We show that their height functions have large-N limit given by the KPZ equation with homogenized transport operator coming from the speed-change. A similar result is shown for another type of driven lattice gas, where the particles are in contact with two external reservoirs. In this case, we derive an “open” KPZ equation limit and answer a question of Corwin ‘22.
Spatial Jump Process Models for Estimating Antibody-Antigen Interactions
Samuel Isaacson (Boston University, Mathematics and Statistics)
- Surface Plasmon Resonance (SPR) assays are a standard approach for quantifying kinetic parameters in antibody-antigen binding reactions. Classical SPR approaches ignore the bivalent structure of antibodies, and use simplified ODE models to estimate effective reaction rates for such interactions. In this work we develop a new SPR protocol, coupling a model that explicitly accounts for the bivalent nature of such interactions and the limited spatial distance over which such interactions can occur, to a SPR assay that provides more features in the generated data. Our approach allows the estimation of bivalent binding kinetics and the spatial extent over which antibodies and antigens can interact, while also providing substantially more robust fits to experimental data compared to classical bivalent ODE models. I will present our new modeling and parameter estimation approach, and demonstrate how it is being used to study interactions between antibodies and spike protein. I will also explain how we make the overall parameter estimation problem computationally feasible via the construction of a surrogate approximation to the (computationally-expensive) particle model. The latter enables fitting of model parameters via standard optimization approaches.
TBA
Theodore Drivas (Stony Brook University)
Save The Date
Mathematics Graduation/ Diplomas Ceremony
Duke University
US-Memorial Day
Gergen Lectures Seminar 1 Title TBD
Noga Alon (Princeton University, Mathematics)
Gergen Lectures Seminar 2
Noga Alon (Princeton University, Mathematics)
Gergen Lecture Reception
Noga Alon (Princeton University, Mathematics)
Gergen Lecture Seminar 3 Title TBD
Noga Alon (Princeton University, Mathematics)
Department of Mathematics
All Hands Meeting
Duke University
US-Labor Day
Duke University
Founder’s Weekend
Department of Mathematics
Fall Picnic 2024
TBA
Mansoor Haider (North Carolina State University, Mathematics)
- TBA
TBA
Jeseth Delgado Vela (Duke University, Civil and Environmental Engineering)
- TBA
TBA
Dominic LaBella and Christina Huang (Duke University School of Medicine, Radiation Oncology)
- TBA
Department of Mathematics
US-Thanksgiving Holiday
Department of Mathematics
US-Christmas Holiday
Department of Mathematics
US-New Years Holiday