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Monday, October 22, 2018 - 13:55 ,
Location: Skiles 005 ,
Professor Hans-Werner van Wyk ,
Auburn University ,
Organizer: Martin Short

The fractional Laplacian is a non-local spatial operator describing anomalous diffusion processes, which have been observed abundantly in nature. Despite its many similarities with the classical Laplacian in unbounded domains, its definition on bounded regions is more problematic. So is its numerical discretization. Difficulties arise as a result of the integral kernel's singularity at the origin as well as its unbounded support. In this talk, we discuss a novel finite difference method to discretize the fractional Laplacian in hypersingular integral form. By introducing a splitting parameter, we first formulate the fractional Laplacian as the weighted integral of a function with a weaker singularity, and then approximate it by a weighted trapezoidal rule. Our method generalizes the standard finite difference approximation of the classical Laplacian and exhibits the same quadratic convergence rate, for any fractional power in (0, 2), under sufficient regularity conditions. We present theoretical error bounds and demonstrate our method by applying it to the fractional Poisson equation. The accompanying numerical examples verify our results, as well as give additional insight into the convergence behavior of our method.

Series: Geometry Topology Seminar

A
fundamental result in equivariant homotopy theory due to Elmendorf
states that the homotopy theory of G-spaces, with w.e.s measured on all
ﬁxed points, is equivalent to the homotopy theory of G-coeﬃcient systems
in spaces, with w.e.s measured at each level
of the system. Furthermore, Elmendorf’s result is rather robust:
analogue results can be shown to hold for, among others, the categories
of (topological) categories and operads. However, it has been known for
some time that in the G-operad case such a result
does not capture the ”correct” notion of weak equivalence, a fact made
particularly clear in work of Blumberg and Hill discussing a whole
lattice of ”commutative operads with only some norms” that are not
distinguished at all by the notion of w.e. suggested
above. In this talk I will talk about part of a joint project which aims
at providing a more diagrammatic understanding of Blumberg and Hill’s
work using a notion of G-trees, which are a generalization of the trees
of Cisinski-Moerdijk-Weiss. More speciﬁcally,
I will describe a new algebraic structure, which we dub a ”genuine
equivariant operad”, which naturally arises from the study of G-trees
and which allows us to state the ”correct” analogue of Elmendorf’s
theorem for G-operads.

Series: Research Horizons Seminar

A knot is a simple closed
curve in the 3-space. Knots appeared as one of the first objects of
study in topology. At first knot theory was rather isolated in
mathematics.
Lately due to newly discovered invariants and newly established
connections to other branches of mathematics, knot theory has become an
attractive and fertile area where many interesting, intriguing ideas
collide. In this talk we discuss a new class of knot
invariants coming out of the Jones polynomial and an algebra of
surfaces based on knots (skein algebra) which has connections to many
important objects including hyperbolic structures of surfaces and
quantum groups. The talk is elementary.

Series: High Dimensional Seminar

We will discuss several open problems concerning unique determination of convex bodies in the n-dimensional Euclidean space given some information about their projections or sectionson all sub-spaces of dimension n-1. We will also present some related results.

Series: Analysis Seminar

In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved.Their fifth problem asks the following.Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). (proportional to the volume of the cone spanned by the secion and the support point). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid?We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach Mazur distance.This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.

Wednesday, October 24, 2018 - 14:00 ,
Location: Skiles 006 ,
Surena Hozoori ,
Georgia Institute of Technology ,
shozoori3@gatech.edu ,
Organizer: Surena Hozoori

Boothby Wang fibrations are historically important examples of contact manifolds and it turns out that we can equip these contact manifolds with extra structures, namely K-contact structures. Based on the study of the relation of these examples and the regularity properties of the corresponding Reeb vector fields, works of Boothby, Wang, Thomas and Rukimbira gives a classification of K-contact structures.

Series: Math Physics Seminar

We consider a class of singular ordinary differential equations,
describing systems subject to a quasi-periodic forcing term and
in the presence of large dissipation, and study the existence of
quasi-periodic solutions with the same frequency vector as the forcing term.
Let A be the inverse of the dissipation coefficient.
More or less strong non-resonance conditions on the frequency
assure different regularity in the dependence on the parameter A:
by requiring a non-degeneracy condition on the forcing term,
smoothness and analyticity, and even Borel-summability,
follow if suitable Diophantine conditions are assumed,
while, without assuming any condition, in general no more than a continuous dependence on A is obtained.
We investigate the possibility of weakening the non-degeneracy
condition and still obtaining a solution for arbitrary frequencies.

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.

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: 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: 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.

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: 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).

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: 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: 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.

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

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.