Sanchit Chaturvedi : Phase mixing in astrophysical plasmas with an external Kepler potential
- Applied Math and Analysis ( 6 Views )In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in towards the object or escape to infinity. We focus on this regime and prove first a linear phase mixing result in 3D outside symmetry with exact Kepler potential. Then we also prove a long-time nonlinear phase mixing result in spherical symmetry. The mechanism is phenomenologically similar to Landau damping on a torus but mathematically the situation is quite a lot more complex. This is based on an upcoming joint work with Jonathan Luk at Stanford.
Chen-Yun Lin : An embedding theorem: differential geometry behind massive data analysis
- Geometry and Topology ( 130 Views )High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis. In this talk, I will present a theoretical analysis of the effectiveness of the VDM. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.
Jie Wang : The primitive cohomology of the theta divisor of an abelian fivefold
- Algebraic Geometry ( 114 Views )The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this talk, I will explain how one can use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold. This is joint work with Izadi and Tam\'as.
Daniel Forger : A mechanism for robust daily timekeeping
- Mathematical Biology ( 97 Views )Circadian clocks persist with a constant period (~24-hour) even after a significant change of the expression level of clock genes. To study the biochemical mechanisms of timekeeping, we develop the most accurate mathematical model of mammalian intracellular timekeeping, as well as a simplified model amenable to mathematical analysis. This modeling work raises interesting questions about existence and uniqueness of models given knowledge of their solutions. Although much is known about cellular circadian timekeeping, little is known about how these rhythms are sustained with a constant period. Here, we show how a universal motif of circadian timekeeping, where repressors bind activators rather than directly binding to DNA, can generate oscillations when activators and repressors are in stoichiometric balance. Furthermore, we find that, even in the presence of large changes in gene expression levels, an additional slow negative feedback loop keeps this stoichiometry in balance and maintains oscillations with a fixed period. These results explain why the network structure found naturally in circadian clocks can generate ~24-hour oscillations in many conditions.
Suliana Manley : Super-resolution imaging and single-molecule tracking, from viruses to chromatin
- Nonlinear and Complex Systems ( 102 Views )We apply super-resolution imaging and single-molecule tracking to gain insight into how proteins assemble to form organized structures in cells. We describe several new tools that were developed to study diverse systems, from viruses to chromatin. The HIV structural protein Gag assembles to form spherical particles of radius ~70 nm. During the assembly process, the number of Gag proteins increases over several orders of magnitude, from a few at nucleation to thousands at completion. We demonstrated an approach that permits quantitative morphological and molecular counting analysis of hundreds of HIV-Gag clusters at the cellular plasma membrane, thus elucidating how different fluorescent labels can change the assembly of virions. Higher-order chromatin structure determines the degree of local DNA condensation, which in turn influences gene accessibility and therefore the expression of particular genes. We present two complementary approaches to address this limitation: super-resolution imaging of directly labeled DNA, and singlemolecule high density tracking of proteins participating in DNA packaging. For STORM imaging of DNA, we stained cells with the DNA-specific dye Picogreen, and obtained a ~5-fold improvement in resolution, resolving the sub-diffraction organization of chromatin structures in living cells. For single molecule tracking (sptPALM), we used small chemical tags to target synthetic dyes to specific protein targets, and visualized their dynamics3. The combination of DNA and protein superresolution imaging and single particle tracking will allow us to study chromatin organization in living cells, and rearrangements in response to exogenous signals.
Asif Zaman : Moments of other random multiplicative functions
- Number Theory ( 154 Views )Random multiplicative functions naturally serve as models for number theoretic objects such as the Mobius function. After fixing a particular model, there are many interesting questions one can ask. For example, what is the distribution of their partial sums? Harper has recently made remarkable progress for partial sums of certain random multiplicative functions with values that lie on the complex unit circle. He settled the correct order of magnitude for their low moments and surprisingly established that one expects better than square-root cancellation in their partial sums. I will discuss an extension of Harper's analysis to a wider class of multiplicative functions such as those modeling the coefficients of automorphic $L$-functions.
Chenglong Yu : Moduli of symmetric cubic fourfolds and nodal sextic curves
- Algebraic Geometry ( 202 Views )Period map is a powerful tool to study geometric objects related to K3 surfaces and cubic 4-folds. In this talk, we focus on moduli of cubic 4-folds and sextic curves with specified symmetries and singularities. We identify the geometric (GIT) compactifications with the Hodge theoretic (Looijenga, mostly Baily-Borel) compactifications of locally symmetric varieties. As a corollary, the algebra of GIT invariants is identified with the algebra of automorphic forms on the corresponding period domains. One of the key inputs is the functorial property of semi-toric compactifications of locally symmetric varieties. Our work generalizes results of Matsumoto-Sasaki-Yoshida, Allcock-Carlson-Toledo, Looijenga-Swierstra and Laza-Pearlstein-Zhang. This is joint work with Zhiwei Zheng.
Dick Hain : Hecke actions on loops and periods of iterated itegrals of modular forms
- Number Theory ( 291 Views )Hecke operators act on many invariants associated to modular curves and their generalizations. For example, they act on modular forms and on cohomology groups of modular curves. In each of these cases, they generate a semi-simple, commutative algebra. In the first part of this talk, I will recall (in friendly, elementary, geometric terms) what Hecke operators are and how they act on the standard invariants. I will then show that they also act on loops in modular curves (aka, conjugacy classes in modular groups). In this case, the Hecke operators generate a non-commutative subalgebra of the vector space generated by the conjugacy classes, which leads to a very natural non-commutative generalization of the classical Hecke algebra. In the second part of the talk will discuss why one might want do construct such a Hecke action. As a prelude to this, I will explain why this Hecke action commutes with the natural action of the absolute Galois group after taking profinite completions. And, in the unlikely event that I have sufficient time, I will also explain how (after taking the appropriate completion) this Hecke action is also compatible with Hodge theory.
Marco Gualtieri : Generalized Kahler geometry and T-duality
- String Theory ( 9 Views )I will introduce generalized Kahler geometry and explain its relation to supersymmetric bihermitian geometry. I will then explain the behaviour of this geometry under T-duality transformations (which I will also define). Finally I will discuss topological D-branes in this geometry.
Leonid Berlyand : PDE/ODE models of motility in active biosystems
- Mathematical Biology ( 106 Views )In the first part of the talk we present a review of our work on PDE models of swimming bacteria. First we introduce a stochastic PDE model for a dilute suspension of self-propelled bacteria and obtain an explicit asymptotic formula for the effective viscosity (E.V.) that explains the mechanisms of the drastic reduction of E.V.. Next, we introduce a model for semi-dilute suspensions with pairwise interactions and excluded volume constraints. We compute E.V. analytically (based on a kinetic theory approach) and numerically. Comparison with the dilute case leads to a phenomenon of stochasticity arising from a deterministic system. We develop a ODE/PDE model that captures the phase transition, an appearance of correlations and large scale structures due to interbacterial interactions. Collaborators: S. Ryan, B. Haines, (PSU students); I. Aronson, A. Sokolov, D. Karpeev (Argonne); In the second part of the talk we discuss a system of two parabolic PDEs arising in modeling of motility of eukaryotic cells on substrates. The two key properties of this system are (i) presence of gradients in the coupling terms (gradient coupling) and (ii) mass (volume) preservation constraints. We derive the equation of the motion of the cell boundary, which is the mean curvature motion perturbed by a novel nonlinear term and prove that the sharp interface property of initial conditions is preserved in time. This novel term leads to surprising features of the motion of the interface such as discontinuities of the interface velocity and hysteresis. This is joint work with V. Rybalko and M. Potomkin.
Jie Shen : Phase-field models for multiphase complex fluids: modeling, numerical analysis and simulations
- Applied Math and Analysis ( 98 Views )I shall present some recent work on phase-field model for multiphase incompressible flows. We shall pay particular attention to situations with large density ratios as they lead to formidable challenges in both analysis and simulation. I shall present efficient and accurate numerical schemes for solving this coupled nonlinear system, in many case prove that they are energy stable, and show ample numerical results which not only demonstrate the effectiveness of the numerical schemes, but also validate the flexibility and robustness of the phase-field model.
Jayce Getz : Trace formulae
- Graduate/Faculty Seminar ( 110 Views )All right, brain. You don't like me and I don't like you, but let's just do this and I can get back to killing you with trace formulae. -Homer Simpson (misquoted) We will discuss trace formulae starting with the Poisson summation formula and working towards the case of compact locally symmetric spaces. No background is assumed. Oh, and I'll bring beverages (both the big kid and little kid kind).
Clarence W. Rowley : Low-order models for control of fluids
- Nonlinear and Complex Systems ( 132 Views )The ability to effectively control a fluid would enable many exciting technological advances, such as the design of quieter, more efficient aircraft. Model-based feedback control is a particularly attractive approach, but the equations governing the fluid, although known, are typically too complex to apply standard tools for dynamical systems analysis or control synthesis. This talk addresses model reduction techniques, which are used to simplify existing models, to obtain low-order models tractable enough to be used for analysis and control, while retaining the essential physics. In particular, we will discuss two techniques: balanced truncation and Koopman modes. Balanced truncation is a well-known technique for model reduction of linear systems, with provable error bounds, but it is not computationally tractable for very large systems. We present an approximate version, called Balanced POD, that is computationally tractable, and produces much better models than traditional Proper Orthogonal Decomposition (POD), at least for the examples studied. Koopman modes are based on spectral analysis of the Koopman operator, an infinite-dimensional linear operator that describes the full nonlinear dynamics of a nonlinear system, and we show how the associated modes can elucidate coherent structures in examples including a jet in crossflow and the wake of a flat plate.
Jiequn Han : Deep Learning-Based Numerical Methods for High-Dimensional Parabolic PDEs and Forward-Backward SDEs
- Applied Math and Analysis ( 108 Views )Developing algorithms for solving high-dimensional partial differential equations (PDEs) and forward-backward stochastic differential equations (FBSDEs) has been an exceedingly difficult task for a long time, due to the notorious difficulty known as the curse of dimensionality. In this talk we introduce a new type of algorithms, called "deep BSDE method", to solve general high-dimensional parabolic PDEs and FBSDEs. Starting from the BSDE formulation, we approximate the unknown Z-component by neural networks and design a least-squares objective function for parameter optimization. Numerical results of a variety of examples demonstrate that the proposed algorithm is quite effective in high-dimensions, in terms of both accuracy and speed. We furthermore provide a theoretical error analysis to illustrate the validity and property of the designed objective function.
Sergei Gukov : Warped Compactifications on Calabi-Yau Fourfolds
- String Theory ( 10 Views )Motivated by phenomenological applications we consider (warped) compactifications of type IIA string theory on Calabi-Yau fourfolds in the presence of Ramond-Ramond fluxes. The main goal of the talk is a description of low-energy physics, including a new superspace formulation of two-dimensional N = (2,2) dilaton supergravity coupled to matter, and computation of the effective superpotentials induced by fluxes.
Thomas Ivey : Cable knot solutions of the vortex filament flow
- Geometry and Topology ( 134 Views )The simplest model of vortex filament motion in an ideal fluid leads to an integrable nonlinear evolution equation, known as the localized induction approximation or the vortex filament flow, closely related to the cubic focusing nonlinear Schroedinger equation. For closed finite-gap solutions of this flow, certain geometric and topological features of the evolving curves appear to be correlated with the algebro-geometric data used to construct them. In this talk, I will briefly discuss this construction, and some low-genus examples (in particular, Kirchhoff elastic rod centerlines) where this correlation is well understood. I will mainly discuss recent joint work with Annalisa Calini, describing how to generate a family of closed finite-gap solutions of increasingly higher genus via a sequence of deformations of the multiply covered circle. We prove that every step in this sequence corresponds to constructing a cable on previous filament; moreover, the cable's knot type (which is invariant under the evolution) can be read off from the deformation sequence.
Dick Hain : What is an algebraic group?
- Algebraic Geometry ( 112 Views )Algebraic groups are important in algebraic and arithmetic geometry. This talk will be a general introduction to them. I will discuss some basic example (elliptic curves, GLn, ...) and then introduce linear algebraic groups and affine algebraic groups. There will be lots of examples, which will help explain why they are important.
M. Haluk Sengun : Torsion Homology of Hyperbolic 3-Manifolds
- Number Theory ( 107 Views )Hyperbolic 3-manifolds have been studied intensely by topologists since the mid-1970's. When the fundamental group arises from a certain number theoretic construction (in this case, the manifold is called "arithmetic"), the manifold acquires extra features that lead to important connections with number theory. Accordingly, arithmetic hyperbolic 3-manifolds have been studied by number theorists (perhaps not as intensely as the topologists) with different motivations. Very recently, number theorists have started to study the torsion in the homology of arithmetic hyperbolic 3-manifolds. The aim of the first half of this introductory talk, where we will touch upon notions like "arithmeticity", "Hecke operators", will be to illustrate the importance of torsion from the perspective of number theory. In the second half, I will present new joint work with N.Bergeron and A.Venkatesh which relates the topological complexity of homology cycles to the asymptotic growth of torsion in the homology. I will especially focus on the interesting use of the celebrated "Cheeger-Mueller Theorem" from global analysis.
Louigi Addario-Berry : Probabilistic aspects of minimum spanning trees
- Probability ( 150 Views )Abstract: One of the most dynamic areas of probability theory is the study of the behaviour of discrete optimization problems on random inputs. My talk will focus on the probabilistic analysis of one of the first and foundational combinatorial optimization problems: the minimum spanning tree problem. The structure of a random minimum spanning tree (MST) of a graph G turns out to be intimately linked to the behaviour of critical and near-critical percolation on G. I will describe this connection and some of my results, alone and with coauthors, on the structure, scaling limits, and volume growth of random MSTs. It turns out that, on high-dimensional graphs, random minimum spanning trees are expected to be three-dimensional when viewed intrinsically, and six-dimensional when viewed as embedded objects.
Jeff Harvey : D-Branes and Asymmetric Orbifolds
- String Theory ( 11 Views )D-branes are usually treated as classical geometric objects in string theory. They can be defined more abstractly in conformal field theory using the boundary state formalism. Some general issues involving this formalism will be discussed and illustrated with examples involving D-branes on both symmetric and asymmetric orbifolds.
Shi Jin : An Eulerian surface hopping method for the Schr\{o}dinger equation with conical crossings
- Applied Math and Analysis ( 104 Views )In a nucleonic propagation through conical crossings of electronic energy levels, the codimension two conical crossings are the simplest energy level crossings, which affect the Born-Oppenheimer approximation in the zeroth order term. The purpose of this paper is to develop the surface hopping method for the Schr\"{o}dinger equation with conical crossings in the Eulerian formulation. The approach is based on the semiclassical approximation governed by the Liouville equations, which are valid away from the conical crossing manifold. At the crossing manifold, electrons hop to another energy level with the probability determined by the Landau-Zener formula. This hopping mechanics is formulated as an interface condition, which is then built into the numerical flux for solving the underlying Liouville equation for each energy level. While a Lagrangian particle method requires the increase in time of the particle numbers, or a large number of statistical samples in a Monte Carlo setting, the advantage of an Eulerian method is that it relies on fixed number of partial differential equations with a uniform in time computational accuracy. We prove the positivity and $l^{1}$-stability and illustrate by several numerical examples the validity and accuracy of the proposed method.
Ina Petkova : Knot Floer homology and the gl(1|1) link invariant
- Geometry and Topology ( 113 Views )The Reshetikhin-Turaev construction for the standard representation of the quantum group gl(1|1) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. After a brief review of this construction, I will give an introduction to tangle Floer homology a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant. This is joint work with Alexander Ellis and Vera Vertesi.
Dennis Frank-Ito : The Future of Computational Fluid Dynamics Modeling in Assessing Upper Airway Respiratory Physiology
- Mathematical Biology ( 94 Views )The complexity of the human airway coupled with advances in computational technology have led to the growing interest in the use of computational fluid dynamics (CFD) techniques to simulate airway physiology in order to collect objective data due to inter-individual anatomy and pre- & post-surgical changes. Outcomes from airway surgery are sometimes difficult to predict a priori, and it is not known whether topical medications are reaching target sites within the human air passage. This talk will give an overview of how CFD is used to explore these issues, as well as demonstrate the potential ability of this methodology in pre-surgical planning.
Boyce E. Griffith : Multiphysics and multiscale modeling of cardiac dynamics
- Applied Math and Analysis ( 123 Views )The heart is a coupled electro-fluid-mechanical system. The contractions of the cardiac muscle are stimulated and coordinated by the electrophysiology of the heart; these contractions in turn affect the electrical function of the heart by altering the macroscopic conductivity of the tissue and by influencing stretch-activated transmembrane ion channels. In this talk, I will present mathematical models and adaptive numerical methods for describing cardiac mechanics, fluid dynamics, and electrophysiology, as well as applications of these models and methods to cardiac fluid-structure and electro-mechanical interaction. I will also describe novel models of cardiac electrophysiology that go beyond traditional macroscopic (tissue-scale) descriptions of cardiac electrical impulse propagation by explicitly incorporating details of the cellular microstructure into the model equations. Standard models of cardiac electrophysiology, such as the monodomain or bidomain equations, account for this cellular microstructure in only a homogenized or averaged sense, and we have demonstrated that such homogenized models yield incorrect results in certain pathophysiological parameter regimes. To obtain accurate model predictions in these parameter regimes without resorting to a fully cellular model, we have developed an adaptive multiscale model of cardiac conduction that uses detailed cellular models only where needed, while resorting to the more efficient macroscale equations where those equations suffice. Applications of these methods will be presented to simulating cardiac and cardiovascular dynamics in whole heart models, as well as in detailed models of cardiac valves and novel models of aortic dissection. Necessary physiological details will be introduced as needed.