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Series: PDE Seminar

The magnetohydrodynamic (MHD) equations govern the motion of electrically conducting fluids such as plasmas, liquid metals, and electrolytes. They consist of a coupled system of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. Besides their wide physical applicability, the MHD equations are also of great interest in mathematics. They share many similar features with the Navier-Stokes and the Euler equations. In the last few years there have been substantial developments on the global regularity problem concerning the magnetohydrodynamic (MHD) equations, especially when there is only partial or fractional dissipation. The talk presents recent results on the global well-posedness problem for the MHD equations with various partial or fractional dissipation.

Series: Geometry Topology Seminar

I'll introduce you to one of my favorite knotted objects: fibered,
homotopy-ribbon disk-knots. After giving a thorough overview of these
objects, I'll discuss joint work with Kyle Larson that brings some new
techniques to bear on their study. Then, I'll
present new work with Alex Zupan that introduces connections with Dehn
surgery and trisections. I'll finish by presenting a classification
result for fibered, homotopy-ribbon disk-knots bounded by square knots.

Series: Algebra Seminar

A classical theorem in modern homotopy theory states that functors from finite pointed sets to spaces satisfying certain conditions model infinite loop spaces (Segal 1974). This theorem offers a recognition principle for infinite loop spaces. An analogous theorem for Morel-Voevodsky's motivic homotopy theory has been sought for since its inception. In joint work with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, we provide such a theorem. The category of finite pointed sets is replaced by a category where the objects are smooth schemes and the maps are spans whose "left legs" are finite syntomic maps equipped with a K-theoretic trivialization of its contangent complex. I will explain what this means, how it is not so different from finite pointed sets and why it was a natural guess. In particular, I will explain some of the requisite algebraic geometry.Time permitting, I will also provide 1) an explicit model for the motivic sphere spectrum as a torsor over a Hilbert scheme and,2) a model for all motivic Eilenberg-Maclane spaces as simplicial ind-smooth schemes.

Monday, October 2, 2017 - 13:55 ,
Location: Skiles 005 ,
Weilin Li ,
University of Maryland, College Park ,
wl298@math.umd.edu ,
Organizer: Wenjing Liao

We formulate
super-resolution as an inverse problem in the space of measures, and
introduce a discrete and a continuous model. For the discrete model, the
problem is to accurately recover a sparse high dimensional vector from
its noisy low frequency Fourier coefficients. We determine a sharp bound
on the min-max recovery error, and this is an immediate consequence of a
sharp bound on the smallest singular value of restricted Fourier
matrices. For the continuous model, we study the total variation
minimization method. We borrow ideas from Beurling in order to determine
general conditions for the recovery of singular measures, even those
that do not satisfy a minimum separation condition. This presentation
includes joint work with John Benedetto and Wenjing Liao.

Series: Geometry Topology Seminar

We use Manolescu's Pin(2)-equivariant Floer homology to study homology cobordisms among Seifert spaces. In particular, we will show that the subgroup of the homology cobordism group generated by Seifert spaces admits a \mathbb{Z}^\infty summand. This is joint work with Irving Dai.

Friday, September 29, 2017 - 15:00 ,
Location: Skiles 154 ,
Sergio Mayorga ,
Georgia Tech ,
Organizer: Jiaqi Yang

We will look at a system of hamiltonian equations on the torus, with an initial condition in momentum and a terminal condition in position, that arises in mean field game theory. Existence of and uniqueness of solutions will be shown, and a few remarks will be made in regard to its connection to the minimization problem of a cost functional.

Series: Combinatorics Seminar

No Combinatorics Seminar, but many others of interest:
(a) on Friday [September 29th, 1pm-2pm in Skiles 005] He Guo, will give an ACO Student Seminar on "Packing nearly optimal Ramsey R(3,t) Graphs"
(b) on Thursday [September 28th, 11am-12am in Skiles 006] Tom Bohman will give an ACO colloquim talk on "Randomized Controlled Trials for Combinatorial Construction"
(c) on Saturday and Sunday [September 30th and October 1st] Atlanta Lecture Series in Combinatorics and Graph Theory XX takes place at Georgia Tech, with featured speaker Paul Seymour

Friday, September 29, 2017 - 13:55 ,
Location: Skiles 006 ,
Peter Lambert-Cole ,
Georgia Institute of Technology ,
Organizer: Peter Lambert-Cole

In this talk, I will present Arnold's famous ADE classification of simple singularities.

Series: ACO Student Seminar

In 1995 Kim famously proved the Ramsey bound $R(3,t) \ge c t^2/\log t$ by constructing an $n$-vertex graph that is triangle-free and has independence number at most $C \sqrt{n \log n}$. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph $K_n$ into a packing of such nearly optimal Ramsey $R(3,t)$ graphs. More precisely, for any $\epsilon>0$ we find an edge-disjoint collection $(G_i)_i$ of $n$-vertex graphs $G_i \subseteq K_n$ such that (a) each $G_i$ is triangle-free and has independence number at most $C_\epsilon \sqrt{n \log n}$, and (b) the union of all the $G_i$ contains at least $(1-\epsilon)\binom{n}{2}$ edges. Our algorithmic proof proceeds by sequentially choosing the graphs $G_i$ via a semi-random (i.e., Rödl nibble type) variation of the triangle-free process. As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabó (concerning a Ramsey-type parameter introduced by Burr, Erdös, Lovász in 1976). Namely, denoting by $s_r(H)$ the smallest minimum degree of $r$-Ramsey minimal graphs for $H$, we close the existing logarithmic gap for $H=K_3$ and establish that $s_r(K_3) = \Theta(r^2 \log r)$. Based on joint work with Lutz Warnke.

Series: Graph Theory Seminar

In this talk we will discuss some Tur\'an-type results on graphs with a given circumference.
Let $W_{n,k,c}$ be the graph obtained from a clique $K_{c-k+1}$
by adding $n-(c-k+1)$ isolated vertices each joined to the same $k$ vertices of the clique,
and let $f(n,k,c)=e(W_{n,k,c})$.
Kopylov proved in 1977 that for $c\max\{f(n,3,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)\}$,
then either $G$ is a subgraph of $W_{n,2,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$,
or $c$ is odd and $G$ is a subgraph of a member of two well-characterized families
which we define as $\mathcal{X}_{n,c}$ and $\mathcal{Y}_{n,c}$.
We extend and refine their result by showing that if $G$ is a 2-connected graph on $n$
vertices with minimum degree at least $k$ and circumference $c$
such that $10\leq c\max\{f(n,k+1,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)\}$,
then one of the following holds:\\
(i) $G$ is a subgraph of $W_{n,k,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$, \\
(ii) $k=2$, $c$ is odd, and $G$ is a subgraph of a member of $\mathcal{X}_{n,c}\cup \mathcal{Y}_{n,c}$, or \\
(iii) $k\geq 3$ and $G$ is a subgraph of the union of a clique $K_{c-k+1}$ and some cliques $K_{k+1}$'s,
where any two cliques share the same two vertices.
This provides a unified generalization of the above result of F\"{u}redi et al. as well as
a recent result of Li et al. and independently, of F\"{u}redi et al. on non-Hamiltonian graphs.
Moreover, we prove a stability result on a classical theorem of Bondy on the circumference.
We use a novel approach, which combines several proof ideas including a closure operation and an edge-switching technique.