- You are here:
- GT Home
- Home
- News & Events

Friday, October 26, 2018 - 15:05 ,
Location: Skiles 156 ,
Jiaqi Yang ,
GT Math ,
Organizer: Jiaqi Yang

We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace, and some non-resonance conditions are satisfied. Then the map leaves invariant a smooth (as smooth as the map) manifold, which is unique among C^L invariant manifolds. Here, L only depends on the spectrum of the linearization. This is based on a work of Prof. Rafael de la Llave.

Series: Combinatorics Seminar

We discuss some recent developments on the critical behavior of percolation on finite random networks. In a seminal paper, Aldous (1997) identified the scaling limit for the component sizes in the critical window of phase transition for the Erdos-Renyi random graph (ERRG). Subsequently, there has been a surge in the literature, revealing several interesting scaling limits of these critical components, namely, the component size, diameter, or the component itself when viewed as a metric space. Fascinatingly, when the third moment of the asymptotic degree distribution is finite, many random graph models has been shown to exhibit a universality phenomenon in the sense that their scaling exponents and limit laws are the same as the ERRG. In contrast, when the asymptotic degree distribution is heavy-tailed (having an infinite third moment), the limit law turns out to be fundamentally different from the ERRG case and in particular, becomes sensitive to the precise asymptotics of the highest degree vertices. In this talk, we will focus on random graphs with a prescribed degree sequence. We start by discussing recent scaling limit results, and explore the universality classes that arise from heavy-tailed networks. Of particular interest is a new universality class that arises when the asymptotic degree distribution has an infinite second moment. Not only it gives rise to a completely new universality class, it also exhibits several surprising features that have never been observed in any other universality class so far. This is based on joint works with Shankar Bhamidi, Remco van der Hofstad, Johan van Leeuwaarden and Sanchayan Sen.

Series: Algebra Seminar

An
algorithm to compute chi-y genera of generic complete intersections in
algebraic tori has already been known since the work of Danilov and
Khovanskii in 1978, yet a closed formula has been given only very
recently
by Di Rocco, Haase, and Nill. In my talk, I will show how this formula
simplifies considerably after an extension of scalars. I will give an
algebraic explanation for this phenomenon using the Grothendieck rings
of vector bundles on toric varieties. We will
then see how the tropical Chern character gives rise to a refined
tropicalization, which retains the good properties of the usual,
unrefined tropicalization.

Series: ACO Student Seminar

We study the fundamental problem of high-dimensional mean
estimation in a robust model where a constant fraction of the samples
are adversarially corrupted. Recent work gave the first polynomial time
algorithms for this problem with dimension-independent
error guarantees for several families of structured distributions.
In this work, we give the first nearly-linear time algorithms for
high-dimensional robust mean estimation. Specifically, we focus on
distributions with (i) known covariance and sub-gaussian tails, and (ii)
unknown bounded covariance. Given $N$ samples
on $R^d$, an $\epsilon$-fraction of which may be arbitrarily corrupted,
our algorithms run in time $\tilde{O}(Nd)/poly(\epsilon)$ and
approximate the true mean within the information-theoretically optimal
error, up to constant factors. Previous robust algorithms
with comparable error guarantees have running times $\tilde{\Omega}(N
d^2)$.
Our algorithms rely on a natural family of SDPs parameterized by
our current guess $\nu$ for the unknown mean $\mu^\star$. We give a
win-win analysis establishing the following: either a near-optimal
solution to the primal SDP yields a good candidate for
$\mu^\star$ -- independent of our current guess $\nu$ -- or the dual SDP
yields a new guess $\nu'$ whose distance from $\mu^\star$ is smaller by
a constant factor. We exploit the special structure of the
corresponding SDPs to show that they are approximately
solvable in nearly-linear time. Our approach is quite general, and we
believe it can also be applied to obtain nearly-linear time algorithms
for other high-dimensional robust learning problems.
This is a joint work with Ilias Diakonikolas and Rong Ge.

Series: Stochastics Seminar

We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA as a finite time growth process and questions about the number of arms, growth and dimension. I will present some conjectures and open problems. This is joint work with Ron Rosenthal (Technion) and Yuan Zhang (Pekin University).

Thursday, October 25, 2018 - 13:30 ,
Location: Skiles 006 ,
Daniel Minahan ,
Georgia Tech ,
Organizer: Trevor Gunn

We will discuss some basic concepts in étale cohomology and compare them
to the more explicit constructions in both algebraic geometry
and algebraic topology.

Series: Graph Theory Seminar

We present an algebraic framework which simultaneously
generalizes the notion of linear subspaces, matroids, valuated matroids,
oriented matroids, and regular matroids. To do this, we first
introduce algebraic objects which we call
pastures; they generalize both hyperfields in the sense
of Krasner and partial fields in the sense of Semple and Whittle. We
then define matroids over pastures; in fact, there are at least two
natural notions of matroid in this general context,
which we call weak and strong matroids. We present ``cryptomorphic'’ descriptions of each kind of matroid. To a (classical) rank-$r$ matroid $M$ on $E$, we can associate a
universal pasture (resp. weak universal pasture)
$k_M$ (resp. $k_M^w$). We show that morphisms from the universal
pasture (resp. weak universal pasture) of $M$ to a pasture $F$ are
canonically in bijection with strong (resp. weak) representations
of $M$ over $F$. Similarly, the sub-pasture $k_M^f$ of $k_M^w$
generated by ``cross-ratios'', which we call the
foundation of $M$, parametrizes rescaling classes of
weak $F$-matroid structures on $M$. As a sample application of these
considerations, we give a new proof of the fact that a matroid is
regular if and only if it is both binary and orientable.

Series: School of Mathematics Colloquium

In these lectures we discuss some statistical problems with an interesting combinatorial structure behind. We start by reviewing the "hidden clique" problem, a simple prototypical example with a surprisingly rich structure. We also discuss various "combinatorial" testing problems and their connections to high-dimensional random geometric graphs. Time permitting, we study the problem of estimating the mean of a random variable.

Series: Other Talks

Thanks are due to our colleague, Vladimir Koltchinskii, for arranging this visit. Please write to Vladimir if you would like to meet with Professor Gabor Lugosi during his visit, or for additional information.

In these lectures we discuss some statistical problems with an interesting combinatorial structure behind. We start by reviewing the "hidden clique" problem, a simple prototypical example with a surprisingly rich structure. We also discuss various "combinatorial" testing problems and their connections to high-dimensional random geometric graphs. Time permitting, we study the problem of estimating the mean of a random variable.

Series: Graph Theory Working Seminar

Erdős and Nešetřil conjectured in 1985 that every graph with maximum degree
Δ can be strong edge colored using at most (5/4)Δ^2 colors. A (Δ_a, Δ_
b)-bipartite graphs is an bipartite
graph such that its components A,B has maximum degree Δ_a, Δ_ b
respectively. R.A. Brualdi and J.J.
Quinn Massey (1993)
conjectured that the strong chromatic index of (Δ_a, Δ_ b)-bipartite
graphs is bounded by Δ_a*Δ_ b. In
this talk, we focus on a recent result affirming the conjecture for (3, Δ)-bipartite
graphs.