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Series: Geometry Topology Seminar

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. Our proof relies on the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

Monday, February 18, 2019 - 13:55 ,
Location: Skiles 005 ,
Rongjie Lai ,
Rensselaer Polytechnic Institute ,
lair@rpi.edu ,
Organizer: Wenjing Liao

Abstract: The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. Instead of directly reconstruct the incomplete distance matrix, we consider a low-rank matrix completion method to reconstruct the associated Gram matrix with respect to a suitable basis. Computationally, simple and fast algorithms are designed to solve the proposed problem. Theoretically, the well known restricted isometry property (RIP) can not be satisfied in the scenario. Instead, a dual basis approach is considered to theoretically analyze the reconstruction problem. Furthermore, by introducing a new condition on the basis called the correlation condition, our theoretical analysis can be also extended to a more general setting to handle low-rank matrix completion problems under any given non-orthogonal basis. This new condition is polynomial time checkable and holds for many cases of deterministic basis where RIP might not hold or is NP-hard to verify. If time permits, I will also discuss a combination of low-rank matrix completion with geometric PDEs on point clouds to understanding manifold-structured data represented as incomplete inter-point distance data. This talk is based on:1. A. Tasissa, R. Lai, “Low-rank Matrix Completion in a General Non-orthogonal Basis”, arXiv:1812.05786 2018. 2. A. Tasissa, R. Lai, “Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion”, accepted, IEEE. Transaction on Information Theory, 2018. 3. R. Lai, J. Li, “Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance”, SIAM Journal on Scientific Computing, 39(5), pp. 2231-2256, 2017.

Series: Algebra Seminar

We survey dissertation work of my academic sister Sarah Mayes-Tang (2013 Ph.D.). As time allows, we aim towards two objectives. First, in terms of combinatorial algebraic geometry we weave a narrative from linear star configurations in projective spaces to matroid configurations therein, the latter being a recent development investigated by the quartet of Geramita -- Harbourne -- Migliore -- Nagel. Second, we pitch a prospectus for further work in follow-up to both Sarah's work and the matroid configuration investigation.

Series: CDSNS Colloquium

We
will discuss the regularity of the conjugacy between an Anosov
automorphism L of a torus and its small perturbation. We assume that L
has no more than two eigenvalues of the same modulus and that L^4 is
irreducible over rationals. We consider a volume-preserving
C^1-small perturbation f of L. We show that if the Lyapunov exponents of
f with respect to the volume are the same as the Lyapunov exponents of
L, then f is C^1+ conjugate to L. Further, we establish a similar result
for irreducible partially hyperbolic automorphisms
with two-dimensional center bundle. This is joint work with Andrey
Gogolev and Victoria Sadovskaya

Series: CDSNS Colloquium

We consider a hyperbolic dynamical system (X,f) and a Holder continuous
cocycle A over (X,f) with values in GL(d,R), or more generally in the
group of invertible bounded linear operators on a Banach space. We
discuss approximation of the Lyapunov exponents
of A in terms of its periodic data, i.e. its return values along the
periodic orbits of f. For a GL(d,R)-valued cocycle A, its Lyapunov
exponents with respect to any ergodic f-invariant measure can be
approximated by its Lyapunov exponents at periodic orbits
of f. In the infinite-dimensional case, the upper and lower Lyapunov
exponents of A can be approximated in terms of the norms of the return
values of A at periodic points of f. Similar results are obtained in the
non-uniformly hyperbolic setting, i.e. for hyperbolic
invariant measures. This is joint work with B. Kalinin.

Saturday, February 16, 2019 - 21:30 ,
Location: Skiles 005 ,
Various speakers ,
GT, Emory, UGA and GSU ,
Organizer: Sung Ha Kang

The Georgia Scientific Computing Symposium is a forum for professors,
postdocs, graduate students and other researchers in Georgia to meet in
an informal setting, to exchange ideas, and to highlight local
scientific computing research. The symposium has been held every year
since 2009 and is open to the entire research community.
This year, the symposium will be held on Saturday, February 16, 2019, at Georgia Institute of Technology. Please see
<a href="http://gtmap.gatech.edu/events/2019-georgia-scientific-computing-symposium" title="http://gtmap.gatech.edu/events/2019-georgia-scientific-computing-symposium">http://gtmap.gatech.edu/events/2019-georgia-scientific-computing-symposium</a>
for more information

Series: Combinatorics Seminar

We provide a framework for testing the possibility of large cascades in random networks. Our results extend previous studies on contagion in random graphs to inhomogeneous directed graphs with a given degree sequence and arbitrary distribution of weights. This allows us to study systemic risk in financial networks, where we introduce a criterion for the resilience of a large network to the failure (insolvency) of a small group of institutions and quantify how contagion amplifies small shocks to the network.

Friday, February 15, 2019 - 03:05 ,
Location: Skiles 246 ,
Yian Yao ,
GT Math ,
Organizer: Jiaqi Yang

I will present a proof of an abstract Nash-Moser Implicit Function Theorem. This theorem can cope with derivatives which are not boundly invertible from one space to itself. The main technique is to combine Newton steps - which loses derivatives with some smoothing that restores them.

Series: Stochastics Seminar

In this talk we construct a
net around the unit sphere with strong properties. We show that with
exponentially high probability, the value of |Ax| on the sphere can be
approximated well using this net, where A is a random matrix with
independent columns. We apply it to study the smallest singular value of
random matrices under very mild assumptions, and obtain sharp small
ball behavior.
As a partial case, we estimate (essentially optimally) the
smallest singular value for matrices of arbitrary aspect ratio with
i.i.d. mean zero variance one entries. Further, in the square case we
show an estimate that holds only under simply the assumptions of
independent entries with bounded concentration functions, and with
appropriately bounded expected Hilbert-Schmidt norm. A key aspect of our
results is the absence of structural requirements such as mean zero and
equal variance of the entries.

Thursday, February 14, 2019 - 11:00 ,
Location: Skiles 006 ,
Dhruv Mubayi ,
University of Illinois at Chicago ,
Organizer: Prasad Tetali

After a brief introduction to classical hypergraph Ramsey numbers, I will focus on the following problem. What is the minimum t such that there exist arbitrarily large k-uniform hypergraphs whose independence number is at most polylogarithmic in the number of vertices and every s vertices span at most t edges? Erdos and Hajnal conjectured (1972) that this minimum can be calculated precisely using a recursive formula and Erdos offered $500 for a proof. For k=3, this has been settled for many values of s, but it was not known for larger k. Here we settle the conjecture for all k at least 4. Our method also answers a question of Bhatt and Rodl about the maximum upper density of quasirandom hypergraphs. This is joint work with Alexander Razborov.