## Eylem Zeliha Yildiz : Braids in planar open books and fillable surgeries.

- Geometry and Topology ( 74 Views )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.

## Ayman Said : Small scale creation of the Lagrangian flow in 2d perfect fluids

- Class Department of Mathematic ( 45 Views )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.

## Robin Zhang : Harris–Venkatesh plus Stark

- Number Theory ( 16 Views )The class number formula describes the behavior of the Dedekind zeta function at s = 0. The Stark conjecture extends the class number formula, describing the behavior of Artin L-functions at s = 0 in terms of units. The Harris–Venkatesh conjecture, originally motivated by the conjectures of Venkatesh and Prasanna–Venkatesh on derived Hecke algebras, can be viewed as an analogue to the Stark conjecture modulo p. In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harris–Venkatesh and Stark for modular forms of weight 1, and describe the proof of this in the imaginary dihedral case. Time permitting, I will also describe some new questions and in-progress work modulo pn.

## Samuel Isaacson : Spatial Jump Process Models for Estimating Antibody-Antigen Interactions

- Mathematical Biology ( 47 Views )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.

## Theodore Drivas : The Feynman-Lagerstrom criterion for boundary layers

- Class Department of Mathematic ( 44 Views )We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single "eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. We will show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. This is joint work with S. Iyer and T. Nguyen.

## Amarjit Budhiraja : Invariant measures of the infinite Atlas model: domains of attraction, extremality, and equilibrium fluctuations.

- Probability ( 40 Views )The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model has a one parameter family {p(a), a > 0} of product form mutually singular stationary distributions. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure p(a) if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to p(a) weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of p(a) for each a. Results on extremality and ergodicity of p(a) will be presented. Finally, I will describe some recent results on fluctuations of the Atlas model from inhomogeneous stationary profiles. This is based on joint work with Sayan Banerjee and Peter Rudzis.

## Mark Stern : Introduction to p-harmonic forms, L^p Hodge theory, and L^p cohomology

- Geometry and Topology ( 57 Views )In this talk I will lay the foundations of the geometry of p-harmonic forms and L^p-Hodge theory. As an application, I will give strong evidence for (half of) a conjecture of Gromov on the L^p cohomology of negatively curved symmetric spaces.

## Kiran Kedlaya : Census-taking for curves over finite fields

- Number Theory ( 49 Views )With Yongyuan Huang and Jun Bo Lau, we recently completed a census of genus-6 curves over the field F_2, and are working on a similar census in genus 7. This uses Mukai's "flowcharts" for describing canonical curves in this genera. We discuss some of the key features of this classification; some aspects of computational group theory required to convert this classification into tractable computations; and some applications of the results, including relative class number problems for function fields, gonality of curves over finite fields (work of Faber-Grantham-Howe), and cohomology of modular curves (work of Canning-Larson and Bergstrom-Canning-Petersen-Schmitt).