Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, August 27, 2015.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, August 27, 2015

11:00 am in 241 Altgeld Hall,Thursday, August 27, 2015

#### Organizational meeting

Abstract: Please attend if you are a grad student, postdoc or faculty member interested in the number theory seminar

1:00 pm in 243 Altgeld Hall,Thursday, August 27, 2015

#### A tale of two norms

###### Nathan Dunfield (U of I)

Abstract: Abstract: The first cohomology of a hyperbolic 3-manifold has two natural norms: the Thurston norm, which measure topological complexity of surfaces representing the dual homology class, and the harmonic norm, which is just the L^2 norm on the corresponding space of harmonic 1-forms. Bergeron-Sengun-Venkatesh recently showed that these two norms are closely related, at least when the injectivity radius is bounded below. Their work was motivated by the connection of the harmonic norm to the Ray-Singer analytic torsion and issues of torsion growth in homology of towers of finite covers. After carefully introducing both norms, I will discuss new results that refine and clarify the precise relationship between them; a key tool here will be a third norm based on least-area surfaces. This is joint work with Jeff Brock and will feature some pretty pictures that are joint work with Anil Hirani. View talk at https://youtu.be/29cE5c1F04k

2:00 pm in 241 Altgeld Hall,Thursday, August 27, 2015

#### Arithmetic properties of sporadic Ap\'ery-like numbers

###### Amita Malik (UIUC Math)

Abstract: In 1982, Gessel showed that the Ap\'ery numbers associated to the irrationality of $\zeta(3)$ satisfy Lucas congruences. In this talk, we discuss the corresponding congruences for all known sporadic Ap\'ery-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol-van Straten and Rowland-Yassawi to establish these congruences. However, for the sequences often labeled $s_{18}$ and $\eta$, we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. If time permits, we will also discuss primes which do not divide any term of a given Ap\'ery-like sequence. This is joint work with Armin Straub.

4:00 pm in 245 Altgeld Hall,Thursday, August 27, 2015

#### Harmonic analysis and additive combinatorics on fractal sets

###### Izabella Laba (University of British Columbia)

Abstract: A recurring theme in Euclidean harmonic analysis is the connection between Fourier-analytic properties of measures and geometric characteristics of their supports. The best known classical results of this type concern estimates on singular and oscillatory integrals associated with surface measures on smooth manifolds. The behaviour of such integrals depends on geometric considerations such as dimension and curvature. In this lecture, we will discuss analogous results for fractal measures. It turns out that the right analogue of curvature in this context is provided by the additive-combinatorial notion of pseudorandomness, roughly meaning the absence of arithmetic structure. We will review several results of this type, including restriction estimates, maximal and differentiation theorems, and Szemeredi-type results.