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Series: Research Horizons Seminar

In
this talk, we consider the structure of a real $n \times n$ matrix in
the form of $A=JL$, where $J$ is anti-symmetric and $L$ is symmetric.
Such a matrix comes from a linear Hamiltonian ODE system with $J$ from
the symplectic structure and the Hamiltonian
energy given by the quadratic form $\frac 12\langle Lx, x\rangle$. We
will discuss the distribution of the eigenvalues of $A$, the
relationship between the canonical form of $A$ and the structure of the
quadratic form $L$, Pontryagin invariant subspace theorem,
etc. Finally, some extension to infinite dimensions will be mentioned.

Series: PDE Seminar

Geometric tangential analysis refers to a constructive systematic approach based on the concept that a problem which enjoys greater regularity can be “tangentially" accessed by certain classes of PDEs. By means of iterative arguments, the method then imports regularity, properly corrected through the path used to access the tangential equation, to the original class. The roots of this idea likely go back to the foundation of De Giorgi’s geometric measure theory of minimal surfaces, and accordingly, it is present in the development of the contemporary theory of free boundary problems. This set of ideas also plays a decisive role in Caffarelli’s work on fully non-linear elliptic PDEs, and subsequently in his studies on Monge-Ampere equations from the 1990’s. In recent years, however, geometric tangential methods have been significantly enhanced, amplifying their range of applications and providing a more user-friendly platform for advancing these endeavors. In this talk, I will discuss some fundamental ideas supporting (modern) geometric tangential methods and will exemplify their power through select examples.

Series: Math Physics Seminar

In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of a nematic liquid crystal phase in it. In such a phase, dimers spontaneously align, but there is no long range translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. I will discuss a recent proof of this conjecture. This is joint work with Elliott H. Lieb.

Series: Math Physics Seminar

Abstract: A number of quantities in quantum many-body systems show
remarkable universality properties, in the sense of exact independence
from microscopic details. I will present some rigorous result
establishing universality in presence of many body interaction in
Graphene and in Topological Insulators, both for the bulk and edge
transport. The proof uses Renormalization Group methods and a
combination of lattice and emerging Ward Identities.

Series: SIAM Student Seminar

In this talk, we provide a deterministic algorithm for robotic path finding in unknown environment and an associated graph generator use only potential information. Also we will generalize the algorithm into a path planning algorithm for certain type of optimal control problems under some assumptions and will state some approximation methods if certain assumption no longer holds in some cases. And we hope to prove more theoretical results for those algorithms to guarantee the success.

Series: Algebra Seminar

Let X be a curve over a p-adic field K with semi-stable reduction and let $\omega$ be a meromorphic differential on X. There are two p-adic integrals one may associated to this data. One is the Vologodsky (abelian, Zarhin, Colmez) integral, which is a global function on the K-points of X defined up to a constant. The other is the collection of Coleman integrals on the subdomains reducing to the various components of the smooth locus. In this talk I will prove the following Theorem, joint with Sarah Zerbes: The Vologodsky integral is given on each subdomain by a Coleman integrals, and these integrals are related by the condition that their differences on the connecting annuli form a harmonic 1-cocyle on the edges of the dual graph of the special fiber.I will further explain the implications to the behavior of the Vologodsky integral on the connecting annuli, which has been observed independently and used, by Stoll, Katz-Rabinoff-Zureick-Brown, in works on global bounds on the number of rational points on curves, and an interesting product on 1-forms used in the proof of the Theorem as well as in work on p-adic height pairings. Time permitting I will explain the motivation for this result, which is relevant for the interesting question of generalizing the result to iterated integrals.

Monday, November 27, 2017 - 14:00 ,
Location: Skiles 005 ,
Zhiliang Xu ,
Applied and Computational Mathematics and Statistics Dept, U of Notre Dame ,
zxu2@nd.edu ,
Organizer: Yingjie Liu

In
this talk, we will present new central and central DG schemes for
solving ideal magnetohydrodynamic (MHD) equations while preserving
globally divergence-free magnetic field on triangular grids. These
schemes incorporate the constrained transport
(CT) scheme of Evans and Hawley with central schemes and central DG
methods on overlapping cells which have no need for solving Riemann
problems across cell edges where there are discontinuities of the
numerical solution. The schemes are formally second-order
accurate with major development on the reconstruction of globally
divergence-free magnetic field on polygonal dual mesh. Moreover, the
computational cost is reduced by solving the complete set of governing
equations on the primal grid while only solving the
magnetic induction equation on the polygonal dual mesh.

Series: Combinatorics Seminar

Official School Holiday: Thanksgiving Break

Series: PDE Seminar

The aim of talk is threefold. First, we solve the cubic nonlinear Schr\"odinger equation on the real line with initial data a sum of Dirac deltas. Secondly, we show a Talbot effect for the same equation. Finally, we prove an intermittency phenomena for a class of singular solutions of the binormal flow, that is used as a model for the vortex filaments dynamics in 3-D fluids and superfluids. If time permits some questions concerning the transfer of energy and momentum will be also considered.

Series: Algebra Seminar

Real-valued smooth differential forms on Berkovich analytic spaces were introduced by Chambert-Loir and Ducros. They show many fundamental properties analogous to smooth real differential forms on complex manifolds, which are used for example in Arakelov geometry. In particular, these forms define a real valued bigraded cohomology theory for Berkovich analytic space, called tropical Dolbeault cohomology. I will explain the definition and properties of these forms and their link to tropical geometry. I will then talk about results regarding the tropical Dolbeault cohomology of varietes and in particular curves. In particular, I will look at finite dimensionality and Poincar\'e duality.