Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, February 14, 2017.

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Tuesday, February 14, 2017

9:30 am in Illini Union Ballroom,Tuesday, February 14, 2017

SAS Day 2017

Abstract: SAS Day 2017 The U of I WebStore, Department of Statistics, SAS, and State Farm invite you to SAS Day 2017. Explore SAS technologies and real-world applications with leaders in the industry. Presentation topics will include:
• Data Visualization and Exploration as the Entry Point to Big Data Analytics
• SAS Studio Applications of Data Science Methods and Visualizations in Insurance Ratemaking
• Program, Drag-and-Drop, and Integrate Python using SAS Machine Learning Technology
• Advanced Statistical Research Enhancements: A Collaboration between SAS and State Farm
Students should be sure to enter the poster competition. You’ll not only gain recognition for your research, but you’ll be eligible to win up to $250. The entry deadline is Friday, January 27th so don’t delay. For more information or to register for the competition, visit https://webstore.illinois.edu/home/SASDay/SAS_Poster.aspx.

1:00 pm in 347 Altgeld Hall,Tuesday, February 14, 2017

Gevrey smoothing of weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules

Tobias Ried (Karlsruhe Institute of Technology)

Abstract: We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with finite mass, energy and entropy, that is, $f_0 \in L^1_2({\mathbb R}^d) \cap L \log L({\mathbb R}^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity. This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers. (Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)

3:00 pm in 241 Altgeld Hall,Tuesday, February 14, 2017

Families in posets minimizing the number of comparable pairs

Sarka Petrickova (Illinois Math)

Abstract: Given a poset $P$, we say a family $\mathcal{F} \subseteq P$ is centered if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the centeredness property if for any $M$, amongst all families of size $M$ in $P$, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice $\{0,1\}^n$ has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset $\{0,1,\ldots,k\}^n$ (where $(A_1,\dots,A_n)\le (B_1,\dots,B_n)$ if $A_i\le B_i$ for each $i\in [n]$) also has the centeredness property, provided $n$ is sufficiently large compared to $k$. We show that this conjecture is false for all $k\geq 2$ and investigate the range of $M$ for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of $\mathbf{F}_q^n$ has the centeredness property. This is joint work with Jozsef Balogh and Adam Zsolt Wagner.

4:00 pm in 131 English Building,Tuesday, February 14, 2017

Banach Lattices

Chris Gartland   [email] (UIUC Math)

Abstract: Many classical spaces such as $C(K)$ and $L^p(\mu)$ carry not only a normed linear structure, but also a lattice structure which behaves well with respect to the norm and vector space operations. The abstraction of this structure gives rise to objects known as Banach lattices, and classical theorems from point-set topology and measure theory can be proved in this purely abstract setting. We'll define the category of Banach lattices and concentrate on the specific subcategories of M-spaces and abstract $L^p$-spaces. We'll outline the Kakutani representation theorems for the spaces, in the former case establishing a duality between the category of M-spaces and the category of compact Hausdorff spaces. Nutter Butters will be provided.