Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, December 6, 2016.

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events for the
events containing

More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, December 6, 2016

1:00 pm in 347 Altgeld Hall,Tuesday, December 6, 2016

#### Maximal operators related to curves in the plane

###### Joris Roos (University of Bonn)

Abstract: The talk will consist of two parts. In the first part, the discussion will focus on an analogue of Carleson's operator associated with integration along a monomial curve. In that context it is natural to ask whether the methods of time-frequency analysis carry over to an anisotropic setting. We answer that question and also provide certain partial bounds for the Carleson operator along monomial curves using entirely different methods. In the second part, I will present some results for maximal operators and Hilbert transforms along variable curves. Apart from the intrinsic interest in these operators, another motivation stems from Zygmund's conjecture on differentiation along Lipschitz vector fields. In particular, we can prove a curved variant of the conjecture.

1:00 pm in 345 Altgeld Hall,Tuesday, December 6, 2016

#### Tame structure via bounds on multiplicative character sums over finite fields

###### Chieu Minh Tran (UIUC Math)

Abstract: We study the model theory of the structure $(F; <)$ where $F$ is the algebraic closure of the field of $p$ elements and $<$ is an ordering on $F^\times$ induced by an injective group homomorphism $\chi: F^\times \to \mathbb{C}^\times$. Various notions of model theoretic tameness of the structure turn out to be consequences of number theoretic behaviors of the character map $\chi$. The results obtained are the first steps in bringing number theoretic phenomena of this type into model theory, following a suggestion by van den Dries, Hrushovski and Kowalski.

2:00 pm in 241 Altgeld Hall,Tuesday, December 6, 2016

#### The eta function

###### Scott Ahlgren   [email] (UIUC)

Abstract: The Dedekind eta function is one of the fundamental functions of number theory. It is a basic building block for many types of modular forms, and it plays a central role in a range of number-theoretic problems. I will give a bunch of examples of objects which can be built using the eta function, and of the applications which are related to its properties.

2:00 pm in 241 Altgeld Hall,Tuesday, December 6, 2016

#### The eta function

###### Scott Ahlgren   [email] (UIUC)

Abstract: The Dedekind eta function is one of the fundamental functions of number theory. It is a basic building block for many types of modular forms, and it plays a central role in a range of number-theoretic problems. I will give a bunch of examples of objects which can be built using the eta function, and of the applications which are related to its properties.

2:00 pm in 243 Altgeld Hall,Tuesday, December 6, 2016

#### Different types of convexity for two-by-two matrices

###### Terry Harris   [email] (UIUC Math)

Abstract: I will introduce some generalized notions of convexity and explain how they arise from the calculus of variations. Then I will talk specifically about rank-one convexity for two-by-two matrices.

3:00 pm in 241 Altgeld Hall,Tuesday, December 6, 2016

#### The complexity of the Colorful Carathéodory Theorem and other total search problems

###### Spencer Gordon (Illinois Computer Science)

Abstract: The Colorful Carathéodory Theorem states that given sets $P_1,\dotsc,P_{d+1} \subseteq \mathbb{R}^d$ such that the origin is in the convex hull of each of the $P_i$, there exists a $(d+1)$-point "colorful" set $S \subseteq P_1 \cup P_2 \cup \dotsb \cup P_{d+1}$ with $|S \cap P_i| = 1$ for each $i$ such that the origin is in the convex hull of $S$. The proof of this theorem gives an algorithm for finding such a colorful set, but it can take time exponential in $d$. We discuss the complexity of finding a colorful set containing the origin and situate this problem in complexity hierarchy of total search problems.

3:00 pm in 243 Altgeld Hall,Tuesday, December 6, 2016