Welcome to RepTile
A conference on representation theory and tilings
UGA, February 2224, 2019

Friday3?
Coverdell S175
Eduard Duryev & Leonid Monin
Eigengons
abstract±
How can one visualize algebraic curves given by equations? Consider a polygon in the complex plane with parallel equal sides identified by translations. Complex structure of the plane descends to give a Riemann surface, hence an algebraic curve. But how one identifies which algebraic curve it is? Usually this question is impossible to answer.
In our talk we will give a series of examples of such polygons together with the equations of algebraic curves that they define.

Friday4:30?
Coverdell S175
Peter Smillie & Philip Engel
Penrose tilings of compact surfaces, Part 1
abstract±
I'll give a few approaches to constructing tilings of compact surfaces by rhombuses glued according to Penrose's matching rules. I'll then give some boundsa logarithmic lower bound and a polynomial upper bound  for the growth of the function F_g(n) which gives the number of Penrose tilings of a genus g surface with n tiles.

Friday 6:30?
555 Morton
Dinner at Phil's Place
Bring food and wine
abstract±
We will discuss amazing things at this dinner. It will be awesome, which is why you should come.

Saturday910
Coverdell S175
Coffee & Bagels
Brought to you from ABC
abstract±
(Athens Bagel Company)

Saturday 10?
CoverdellS175
Leonid Monin
Counting geodesics on hyperbolic surfaces
abstract±

Saturday 11:15?
CoverdellS175
Eduard Duryev
Squaretiled surfaces and Hurwitz spaces of branched covers
abstract±

Saturday 12:30?
1427 Lumpkin
Lunch
Cali & Tito's
abstract±
You put food in your mouth and chew.

Saturday 2:30?
CoverdellS175
Peter Smillie
Penrose tilings of compact surfaces, Part 2
abstract±

Saturday 4?
CoverdellS175
Ben Dozier
Affine measures and short saddle connections
abstract±
A stratum of translation surfaces comes equipped with a natural Lebesgueclass measure, called the MasurVeech measure. MasurVeech volumes are intimately connected to counting squaretiled surfaces in a given stratum. It is also interesting to study general affine measures (supported on affine invariant submanifolds). By work of EskinMirzakhaniMohammadi, many dynamical questions can be reduced to the study of these affine measures. It is natural to ask about the interaction between measures of certain subsets of surfaces and the geometric properties of those surfaces.
I will discuss a proof in progress of a conjecture on the volume, with respect to any affine measure, of the locus of surfaces that have multiple independent short saddle connections. A key tool is the new compactification of strata due to BainbridgeChenGendronGrushevskyMoller, which gives a good picture of how a translation surface can degenerate. The techniques also give a new proof of the following very recent result of DucManh Nguyen (possibly of interest to reptilers): the natural measure on a stratum of higher order differentials is finite.

Saturday 6:30?
DownTown
Conference Dinner
The Place
abstract±
The place we're eating is called "The Place."

Sunday 9?
CoverdellS175
Pastries & Coffee
TBA
abstract±
TBA

Sunday 10?
CoverdellS175
Vanya Telpukhovskiy
On normalizations of Thurston measure on the space of measured laminations
abstract±
The space of measured laminations ML(Sigma) associated to a topological surface Sigma of genus g with n punctures is an integral piecewise linear manifold of real dimension 6g6+2n. There is also a natural symplectic structure on ML(Sigma) defined by Thurston. The integral and symplectic structures define a pair of measures on ML(Sigma) which are known to be proportional. The projective class of these measures on ML(Sigma) is called the Thurston measure. I will compute the ratio between two normalizations of the Thurston measure and talk about its relation to counting problems for quadratic differentials, in particular squaretiled surfaces (if time allows). This work is joint with Leonid Monin.

Sunday 11:30?
CoverdellS175
David Gay
Tiling questions motivated by the search for interesting trisection diagrams
abstract±
TBA

Sunday 1:30?
CoverdellS175
Math Discussion Time
A Time to Discuss Math
abstract±
When the math will be discussed.