Mathematics

Ultracommutative monoids

Charles Rezk (UIUC Math)

Abstract: This talk is a sequel to one I gave in the spring about Stefan Schwede's orthogonal spaces and their relation to global equivariant homotopy theory. This time I'll talk about an interesting symmetric monoidal structure on this category, whose monoids may be called "ultracommutative".

A fundamental dichotomy for definably complete expansions of ordered fields

Philipp Hieronymi (UIUC Math)

Abstract: An expansion of a definably complete field either defines a discrete subring, or the image of a definable discrete set under a definable map is nowhere dense. As an application we show a definable version of Lebesgue's differentiation theorem.

Recent work on the regularity problem of some variation of 2D Boussinesq equations

Lizheng Tao (UIUC Math)

Abstract: In this talk, I will present my previous work for the 2D Boussinesq equations. Regularity results are achieved for a slightly super-critical variation of the standard equations. A modified Besov space will be introduced to handle the logarithmically super-critical operator.

Quasi-Regular Mappings of Lens Spaces

Anton Lukyanenko (UIUC Math)

Abstract: A quasi-regular QR mapping between metric manifolds is a branched cover with bounded dilatation, e.g. $f(z)=z^2$. In a joint work with K. Fassler and K. Peltonen, we define QR mappings of sub-Riemannian manifolds and show that:
1) Every lens space admits a uniformly QR (UQR) mapping $f$.
2) Every UQR mapping leaves invariant a measurable conformal structure.
The first result uses an explicit "conformal trap" construction, while the second builds on similar results by Sullivan-Tukia and a connection to higher-rank symmetric spaces.

Fleming-Viot particle system driven by a random walk on naturals

Nevena Maric (U Missouri St Louis)

Abstract: Random walk on naturals with negative drift and absorption at 0, when conditioned on survival, has uncountably many invariant measures (quasi-stationary distributions, qsd). We study a Fleming-Viot (FV) particle system driven by this process. In this particle system there are N particles where each particle evolves as the random walk described above. As soon as one particle is absorbed, it reappears, choosing a new position according to the empirical measure at that time. Between the absorptions, the particles move independently of each other. Our focus is in the relation of empirical measure of the FV process with qsds of the random walk. Firstly, mean normalized densities of the FV unique stationary measure converge to the minimal qsd, as N goes to infinity. Moreover, every other qsd of the random walk corresponds to a metastable state of the FV particle system.

Driving finite elements with a goal

Luke Olson (UIUC Computer Science)

Abstract: The finite element method provides a rich theoretical landscape and flexible computational framework for the numerical approximation of partial differential equations. In particular, least-squares based finite element methods are a natural framework for building simulations with optimal numerical properties and many numerical advantages (symmetric positive-definiteness, ellipticity, a posteriori estimation of the error). Yet, as with most approximation methods, adapting the solution to a particular feature or goal in the problem (e.g. conservation or a highly accurate features) is not immediate. In this talk, we give an overview of the least-squares finite element method to motivate a work efficient approximation method. And we show a systematic approach to incorporating goal-oriented decisions in the approximation process. Some knowledge of partial differential equations and numerics is helpful, but this is intended for a general audience in applied mathematics.

Tight co-degree condition for the existence of loose Hamilton cycles in $3$-uniform hypergraphs

Theodore Molla   [email] (UIUC Math)

Abstract: Recently many results analogous to Dirac's Theorem have been proved for hypergraphs. A {$(k,l)$-cycle is a hypergraph in which the vertices can be arranged in a cycle so that every edge contains $k$ consecutive vertices and every pair of consecutive edges intersect in exactly $l$ vertices. Call a $(k, k-1)$-cycle a tight cycle and a $(k,1)$-cycle a loose cycle. Let $H$ be a $3$-uniform hypergraph on $n$ vertices with minimum co-degree $\delta(H)$. Rödl, Ruciński and Szemerédi proved that $\delta(H) \ge (1/2 + o(1))n$ implies that $H$ contains a tight Hamilton cycle and Kühn and Osthus showed that $\delta(H) \ge (1/4 + o(1))n$ is sufficient for $H$ to contain a loose Hamilton cycle. Both results are from 2006. In 2011 Rödl, Ruciński and Szemerédi improved their previous result by showing that, for sufficiently large $n$, $\delta(H) \ge \left \lfloor n/2 \right \rfloor$ implies the existence of a tight Hamilton cycle. We will sketch a proof of an analogous result for loose cycles, that is we will show that every sufficiently large $3$-uniform hypergraph on $n \in 2 \mathbb{Z}$ vertices with minimum co-degree at least $n/4$ contains a loose Hamilton cycle. This result is best possible and uses the probabilistic absorbing technique.

An Introduction to Tropical Geometry

Nathan Fieldsteel (UIUC Math)

Abstract: This will be the first talk in a two-part series in which we will give a broad overview of the relatively young field of tropical geometry, aiming to introduce the central objects of study while providing motivation, examples and connections to other fields. Professors are welcome to attend.