Department of

Mathematics


Seminar Calendar
for events the day of Thursday, February 8, 2018.

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Thursday, February 8, 2018

11:00 am in 241 Altgeld Hall,Thursday, February 8, 2018

The theta operator and eta-quotients

Byungchan Kim (Seoul Technical University )

Abstract: Eta-quotients are a important explicit class of modular forms, and the theta-operator is a natural operator on modular forms. We investigate which eta-quotients are preserved by the theta-operator. This is motivated by a particular partition congruence. This talk is based on joint work with P.-C. Toh and with D. Choi and S. Lim.

2:00 pm in 241 Altgeld Hall,Thursday, February 8, 2018

Multiplicative functions which are additive on polygonal numbers

Byungchan Kim (SeoulTech)

Abstract: Spiro showed that a multiplicative function which is additive on prime numbers should be the identity function. After Spiro's work, there are many variations. One direction is to investigate multiplicative functions which are additive on polygonal numbers. It is known that a multiplicative function which is additive on triangular numbers should be the identity while there are non-identity functions which is multiplicative and is additive on square numbers. In this talk, we investigate multiplicative functions which are additive on several polygonal numbers and present some open questions. This talk is based on joint works with J.-Y. Kim, C.G. Lee and P.-S. Park.

2:00 pm in 243 Altgeld Hall,Thursday, February 8, 2018

(Non)-Removability of the Sierpiński Gasket

Dimitrios Ntalampekos (UCLA)

Abstract: Removability of sets for quasiconformal maps and Sobolev functions has applications in Complex Dynamics, in Conformal Welding, and in other problems that require``gluing" of functions to obtain a new function of the same class. We, therefore, seek geometric conditions on sets which guarantee their removability. In this talk, I will discuss some very recent results on the (non)-removability of the Sierpiński gasket. A first result is that the Sierpiński gasket is removable for continuous functions of the class $W^{1,p}$ for $p>2$. The method used applies to more general fractals that resemble the Sierpiński gasket, such as Apollonian gaskets and generalized Sierpiński gasket Julia sets. Then, I will sketch a proof that the Sierpiński gasket is non-removable for quasiconformal maps and thus for $W^{1,p}$ functions, for $1\leq p\leq 2$. The argument involves the construction of a non-Euclidean sphere, and then the use of the Bonk-Kleiner theorem to embed it quasisymmetrically to the plane.

3:00 pm in 345 Altgeld Hall,Thursday, February 8, 2018

Polynomial Families from Combinatorial $K$-Theory

Cara Monical (UIUC)

Abstract: Set-valued tableaux play an important role in combinatorial $K$-theory. Separately, semistandard skyline fillings are a combinatorial model for Demazure atoms and key polynomials. We unify these two concepts by defining a set-valued extension of semistandard skyline fillings. We use this combinatorial model to give $K$-theoretic extensions of Demazure atoms, key polynomials, quasisymmetric Grothendieck functions, and quasikey polynomials. We then connect these new bases to existing $K$-theoretic analogues. This is joint work with O. Pechenik and D. Searles.