Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, November 6, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, November 6, 2018

12:00 pm in Altgeld Hall 102,Tuesday, November 6, 2018

#### 2FA Open Hour

###### Math IT   [email] (Illinois Math)

Abstract: If you have any questions regarding the mandatory 2FA enrollment, require a 2FA token, or need help setting up the Duo Mobile app, please stop by our office, 102 Altgeld Hall, from noon to 1pm. If none of our early-November open hours work for you, please contact us at math-it@illinois.edu.

1:00 pm in 345 Altgeld Hall,Tuesday, November 6, 2018

#### Logarithmic Hyperseries

###### Elliot Kaplan (UIUC)

Abstract: In a joint work with Lou van den Dries and Joris van der Hoeven, we constructed the field of Logarithmic Hyperseries. This is a proper class-sized ordered differential field which is also equipped with a logarithm and a composition. In this talk, I will briefly detail the construction of this field and indicate how the logarithm, composition, and derivation interact with each other. I will also indicate where this field fits among transseries, logarithmic transseries, and the surreal numbers.

2:00 pm in 345 Altgeld Hall,Tuesday, November 6, 2018

#### One-point function estimates and natural parametrization for loop-erased random walk in three dimensions

###### Xinyi Li (University of Chicago)

Abstract: In this talk, I will talk about loop-erased random walk (LERW) in three dimensions. I will first give an asymptotic estimate on the probability that 3D LERW passes a given point (commonly referred to as the one-point function). I will then talk about how to apply this estimate to show that 3D LERW as a curve converges to its scaling limit in natural parametrization. If time permits, I will also talk about the asymptotics of non-intersection probabilities of 3D LERW with simple random walk. This is a joint work with Daisuke Shiraishi (Kyoto).

2:00 pm in 243 Altgeld Hall,Tuesday, November 6, 2018

#### Paths and Arctic Curves: the Tangent Method at Work

###### Philippe R. Di Francesco (Illinois Math)

Abstract: Tiling problems of finite domains of the plane with a fixed set of tiles can often be rephrased in terms of non-intersecting lattice paths. For large scaled domains, random tilings can exhibit a sharp separation between “frozen” regions tiled regularly and “liquid” regions tiled wildly. This is the arctic phenomenon. The separating curve is called "arctic curve”.

We present a new technique, called the tangent method, to derive the arctic curve using only boundary properties of the set of paths describing the tilings. We apply this technique to the celebrated domino tiling problem of the Aztec diamond, and to the rhombus tiling of certain domains with arbitrary boundary shape. We perform exact enumeration using the Gessel-Viennot theorem for non-intersecting lattice paths, and asymptotic analysis. This leads to compact expressions for arctic curves and their q-deformations in the presence of area-dependent weights.

(Based on joint works with M.F. Lapa and E. Guitter.)