Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, September 10, 2020.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, September 10, 2020

3:30 pm in Zoom Meeting (email jobal@illinois.edu for info),Thursday, September 10, 2020

#### Fair representation

###### Noga Alon (Princeton University and Tel Aviv)

Abstract: A substantial number of results and conjectures deal with the existence of a set of prescribed type which contains a fair share from each member of a finite collection of objects in a space. Examples include the Ham-Sandwich Theorem in Measure Theory, the Hobby-Rice Theorem in Approximation Theory, the Necklace Theorem and the Ryser Conjecture in Discrete Mathematics, and more. The techniques in the study of these results combine combinatorial, topological, geometric and algebraic tools. I will describe the topic, focusing on several recent results.

6:00 pm in Zoom,Thursday, September 10, 2020

#### Kohnert polynomials and their stable limits

###### Dominic Searles   [email] (University of Otago)

Abstract: A. Kohnert introduced an algorithmic process beginning with left-justified box diagrams in the first quadrant of the plane, and used this to give a combinatorial formula for the (type A) Demazure characters, an important basis of polynomials arising from representation theory. We consider the polynomials arising from applying Kohnert's algorithm to any starting diagram, which we call Kohnert polynomials. We establish some structural results concerning Kohnert polynomials, including that their stable limits are quasisymmetric. As an application, we introduce a basis of Kohnert polynomials which stabilises to a new basis of quasisymmetric functions that contain the Schur functions. We additionally give a representation-theoretic interpretation of this new basis. (Includes joint work with S. Assaf.) Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.