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Series: Analysis Seminar

In
this talk, I will discuss some polynomials that are best approximants
(in some sense!) to reciprocals of functions in some analytic function
spaces of the unit disk. I will examine the extremal
problem of finding a zero of minimal modulus, and will show how that
extremal problem is related to the spectrum of a certain Jacobi matrix
and real orthogonal polynomials on the real line.

Wednesday, November 29, 2017 - 13:55 ,
Location: Skiles 006 ,
Anubhav Mukherjee ,
Georgia Tech ,
Organizer: Jennifer Hom

I'll try to describe some known facts about 3 manifolds. And in the end I want to give some idea about Geometrization Conjecture/theorem.

Wednesday, November 29, 2017 - 13:55 ,
Location: Skiles 006 ,
Anubhav Mukherjee ,
Georgia Tech ,
Organizer: Jennifer Hom
I'll try to describe some known facts about 3 manifolds. And in the end I want to give some idea about Geometrization Conjecture/theorem.

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: Research Horizons Seminar

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: PDE Seminar

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

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.