Department of

Mathematics


Seminar Calendar
for events the day of Friday, February 25, 2022.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, February 25, 2022

1:00 pm in Zoom,Friday, February 25, 2022

Dimension distortion and applications

Efstathios Konstantinos Chrontsios Garitsis (UIUC Math)

Abstract: Since Hausdorff dimension was first introduced in 1918, many different notions of dimension have been defined and used throughout many areas of Mathematics. An interesting topic has always been the distortion of said dimensions of a given set under a specific class of mappings. More specifically, Gehring and V\"ais\"al\"a proved in 1973 a theorem concerning the distortion of Hausdorff dimension under quasiconformal maps, while Kaufman in 2000 proved the analogous result for Box-counting dimension. In this talk, an introduction to the different types of dimensions will be presented, along with the results of of Gehring, V\"ais\"al\"a and Kaufman. We will then proceed to discuss analogous theorems we proved for the Assouad dimension and spectrum, which describe how K-quasiconformal maps change these notions of a given subset of $\mathbb{R}^n$. We will conclude the talk by demonstrating how said theorems can be applied to fully classify polynomial spirals up to quasiconformal equivalence. Contact: ryanm12@illinois.edu for Zoom coordinates.

1:00 pm in Zoom,Friday, February 25, 2022

Dimension distortion and applications

Efstathios Konstantinos Chrontsios Garitsis (UIUC Math)

Abstract: Since Hausdorff dimension was first introduced in 1918, many different notions of dimension have been defined and used throughout many areas of Mathematics. An interesting topic has always been the distortion of said dimensions of a given set under a specific class of mappings. More specifically, Gehring and V\"ais\"al\"a proved in 1973 a theorem concerning the distortion of Hausdorff dimension under quasiconformal maps, while Kaufman in 2000 proved the analogous result for Box-counting dimension. In this talk, an introduction to the different types of dimensions will be presented, along with the results of of Gehring, V\"ais\"al\"a and Kaufman. We will then proceed to discuss analogous theorems we proved for the Assouad dimension and spectrum, which describe how K-quasiconformal maps change these notions of a given subset of $\mathbb{R}^n$. We will conclude the talk by demonstrating how said theorems can be applied to fully classify polynomial spirals up to quasiconformal equivalence. Contact: ryanm12@illinois.edu for Zoom coordinates.

3:00 pm in 341 Altgeld Hall,Friday, February 25, 2022

Introduction to Equivariant Homotopy Theory and RO(G)-graded Cohomology

Zach Halladay (UIUC Math)

Abstract: In its simplest form, equivariant homotopy theory is the study of homotopy theory with the addition of actions by a group G. By mixing representation theory into homotopy theory, we create additional structure and complexities to consider. To every representation of our group, we may take the one point compactification and the group will then act on the resulting sphere. In so called “genuine” G-spectra, these representation spheres are invertible, so in particular we may grade cohomology theories on (virtual) G-representations. In this introductory talk, I will go over some of the additional properties and structures of equivariant homotopy theory and if time permits illustrate these structures with a cohomology computation.

4:00 pm in 347 Altgeld Hall,Friday, February 25, 2022

The Jones Polynomial and its Big Brother, Khovanov Homology

Joseph Malionek (UIUC)

Abstract: In this talk, I will acquaint the audience with a link invariant called the Jones Polynomial and one of way of making it much more complicated (but potentially much stronger) called Khovanov Homology. I will go over some key properties of each of these invariants, some example calculations, some big results, and some open problems.