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Series: Graph Theory Seminar

Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2.
We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint
connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1,
b2}⊆V(G2). In
this talk, we will complete a sketch of our arguments for characterizing when (G, a0, a1, a2, b1, b2) is feasible. Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

Series: Job Candidate Talk

Random effects models are commonly used to measure genetic
variance-covariance matrices of quantitative phenotypic traits. The
population eigenvalues of these matrices describe the evolutionary
response to selection. However, they may be difficult to estimate from
limited samples when the number of traits is large. In this talk, I will
present several results describing the eigenvalues of classical MANOVA
estimators of these matrices, including dispersion of the bulk
eigenvalue distribution, bias and aliasing of large "spike" eigenvalues,
and distributional limits of eigenvalues at the spectral edges. I will
then discuss a new procedure that uses these results to obtain better
estimates of the large population eigenvalues when there are many
traits, and a Tracy-Widom test for detecting true principal components
in these models. The theoretical results extend proof techniques in
random matrix theory and free probability, which I will also briefly
describe.This is joint work with Iain Johnstone, Yi Sun, Mark Blows, and Emma Hine.

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

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: 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

TBA

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.