Curtis Porter : CRash CouRse in CR Geometry
- Graduate/Faculty Seminar,Uploaded Videos ( 2252 Views )CR geometry studies real hypersurfaces in complex vector spaces and their generalizations, CR manifolds. In many cases of interest to complex analysis and PDE, CR manifolds can be considered ``curved versions" of homogeneous spaces according to Elie Cartan’s generalization of Klein’s Erlangen program. Which homogeneous space is the ``flat model" of a CR manifold depends on the Levi form, a tensor named after a mathematician who used it to characterize boundaries of pseudoconvex domains. As in the analytic setting, the Levi form plays a central role in the geometry of CR manifolds, which we explore in relation to their homogeneous models.
Oliver Tough : The Fleming-Viot Particle System with McKean-Vlasov dynamics
- Probability,Uploaded Videos ( 1332 Views )Quasi-Stationary Distributions (QSDs) describe the long-time behaviour of killed Markov processes. The Fleming-Viot particle system provides a particle representation for the QSD of a Markov process killed upon contact with the boundary of its domain. Whereas previous work has dealt with killed Markov processes, we consider killed McKean-Vlasov processes. We show that the Fleming-Viot particle system with McKean-Vlasov dynamics provides a particle representation for the corresponding QSDs. Joint work with James Nolen.
Yiming Zhong : Fast algorithm for Radiative transport
- Graduate/Faculty Seminar,Uploaded Videos ( 991 Views )This talk consists of two aspects about solving the radiative transport through the integral formulation. The radiative transport equation has been numerically studied for many years, the equation is difficult to solve due to its high dimensionality and its hyperbolic nature, in recent decades, the computers are equipped with larger memories so it is possible to deal with the full-discretization in phase space, however, the numerical efficiency is quite limited because of many issues, such as iterative scheme, preconditioning, discretization, etc. In this talk, we first discuss about the special case of isotropic scattering and its integral formulation, then walk through the corresponding fast algorithm for it. In the second part, we try to trivially extend the method to anisotropic case, and talk about the method’s limitation and some perspectives in both theory and numerics.
James Keener : Flexing your Protein muscles: How to Pull with a Burning Rope
- Mathematical Biology ( 727 Views )The segregation of chromosomes during cell division is accomplished by kinetochore machinery that uses depolymerizing microtubules to pull the chromosomes to opposite poles of the dividing cell. While much is known about molecular motors that pull by walking or push by polymerizing, the mechanism of how a pulling force can be achieved by depolymerization is still unresolved. In this talk, I will describe a new model for the depolymerization motor that is used by eukaryotic cells to segregate chromosomes during mitosis. In the process we will explore the use of Huxley-type models (population models) of protein binding and unbinding to study load-velocity curves of several different motor-like proteins.
Alfio Fabio La Rosa : Translation functors and the trace formula
- Number Theory ( 490 Views )I will propose a way to combine the theory of translation functors with the trace formula to study automorphic representations of connected semisimple anisotropic algebraic groups over the rational numbers whose Archimedean component is a limit of discrete series. I will explain the main ideas of the derivation of a trace formula which, modulo a conjecture on the decomposition of the tensor product of a limit of discrete series with a finite-dimensional representation into basic representations, allows to isolate the non-Archimedean parts of a finite family of C-algebraic automorphic representations containing the ones whose Archimedean component is a given limit of discrete series.
Jacob Bedrossian : Positive Lyapunov exponents for 2d Galerkin-Navier-Stokes with stochastic forcing
- Applied Math and Analysis ( 436 Views )In this talk we discuss our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander's condition. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry.
Oguz Savk : Bridging the gaps between homology planes and Mazur manifolds.
- Geometry and Topology,Uploaded Videos ( 315 Views )We call a non-trivial homology 3-sphere a Kirby-Ramanujam sphere if it bounds a homology plane, an algebraic complex smooth surface with the same homology groups of the complex plane. In this talk, we present several infinite families of Kirby-Ramanujam spheres bounding Mazur type 4-manifolds, compact contractible smooth 4-manifolds built with only 0-, 1-, and 2-handles. Such an interplay between complex surfaces and 4-manifolds was first observed by Ramanujam and Kirby around nineteen-eighties. This is upcoming joint work with Rodolfo Aguilar Aguilar.
Erika Berenice Roldan Roa : Asymptotic behavior of the homology of random polyominoes
- Probability ( 197 Views )In this talk we study the rate of growth of the expectation of the number of holes (the rank of the first homology group) in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. This results can be extend, using the same techniques, to other polyforms and higher dimensions.
Qin Li : Stability of stationary inverse transport equation in diffusion scaling
- Applied Math and Analysis ( 149 Views )We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales char- acterized by the magnitude of a dimensionless parameterthe Knudsen number (Kn). In the diffusive scaling (Kn ≪ 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well- posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn → 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Knp (p = 1 or 2), and as a result lead to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.
Benjamin Dodson : Concentration compactness for the L^2 critical nonlinear Schrodinger equation
- Applied Math and Analysis ( 136 Views )The nonlinear Schrodinger equation
i u_{t} + D u = m |u|^{(4/d)}u | (1) |
As time permits the talk will also discuss the energy - critical problem in R^{d} \ W,
i u_{t} + D u = |u|^{4/(d - 2)} u | , u|_{Bdry(W)} = 0, (2) |
Junchi Li : New stochastic voting systems on fixed and random graphs
- Graduate/Faculty Seminar ( 121 Views )In this talk I will introduce two stochastic voting systems and results we proved. (i) Axelrod's model generalizes the voter model in which individuals have one of Q possible opinions about each of F issues and neighbors interact at a rate proportional to the fraction of opinions they share. We proved that on large two-dimensional torus if Q/F is small, then there is a giant component of individuals who share at least one opinion and consensus develops on this percolating cluster. (ii) The latent voter model allows a latent period after each site flips its opinion. We will present Shirshendu's result on a random r-regular graph with n vertices that as the rate of exponential latent period $\lambda \gg \log n$, dynamics converge to coexistence behavior with quasi-stationary density = 1/2 at $O(\lambda)$ times. Using different technologies one can generalize it to the varying degree case, a.k.a. the configuration models. Joint work with Rick Durrett and Shirshendu Chatterjee
Sayan Mukherjee : Geometric Perspectives on Supervised Dimension Reduction
- Applied Math and Analysis ( 117 Views )The statistical problem of supervised dimension reduction (SDR) is given observations of high-dimensional data as explanatory variables and univariate response variable, find a submanifold or subspace of the explanatory variables that predict the response. It is generally assumed that the data is concentrated on a low dimensional manifold in the high-dimensional space of explanatory variables.
The gradient of the manifold will be shown to be a central quantity in the problem of SDR. We will present a regularization algorithm for inferring the gradient geiven data. We will prove the rate of convergence of the gradient estimate to the gradient on the manifold of the true function to be of the order of the dimension of the manifold and not the much larger
The second part of the talk will rephrase the problem of SDR in a classical probabilistic (Bayesian) setting of mixture models of multivariate normals. An interesting result of this procedure is that the subspaces relevant to prediction are drawn from a posterior distribution on Grassmannian manifolds. For both methods efficacy on simulated and real data will be shown. ambient space.
Adam Levine : Heegaard Floer invariants for homology S^1 x S^3s
- Geometry and Topology ( 109 Views )Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of S^1 x S^3. Specifically, we show that for any smoothly embedded 3-manifold Y representing a generator of H_3(X), a suitable version of the Heegaard Floer d invariant of Y, defined using twisted coefficients, is a diffeomorphism invariant of X. We show how this invariant can be used to obstruct embeddings of certain types of 3-manifolds, including those obtained as a connected sum of a rational homology 3-sphere and any number of copies of S^1 x S^2. We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic R^4s. This is joint work with Danny Ruberman.
Robin Zhang : Harris–Venkatesh plus Stark
- Number Theory ( 64 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.