WAGS
/ Spring 2013
Western 
Federico Ardila 

Abstract: Tropical geometry is a valuable tool which translates questions in classical enumerative geometry into questions about polyhedral complexes. It is no suprise that the resulting combinatorial questions are usually rather subtle. I will discuss two instances where combinatorial techniques have provided (somewhat puzzling) new insight into classical problems: Severi degrees of toric surfaces, and double Hurwitz numbers.
The first of these two projects is joint with Florian Block (UC Berkeley). 
Noah Giansiracusa 

Abstract: I'll discuss joint work with Brent Doran (ETH) in which we
construct an "algebraic uniformization" of the moduli space
\bar{M}_{0,n} of stable, rational npointed curves: we present it as a
nonreductive GIT quotient of affine space by a nonlinearizable
action of a solvable group. This is accomplished by replacing the
universal torsor with an A^1homotopic model that encodes the global
projective geometry of the moduli space in a more accesible way. We
show in particular that the Cox ring of \bar{M}_{0,n} is a
G_ainvariant subring of a polynomial ring, which allows it to be
computed for any fixed n, in principle, by standard invariant theory
algorithms, and which shows a precise sense in which \bar{M}_{0,n} is
"one G_a away" from being a toric variety. 
Ravi Vakil 

Abstract:
I will give an introduction to the Grothendieck ring of varieties, giving a survey of a small portion
of interesting results in the field due to various people (Kapranov,
Cheah, LarsenLunts, LiuSebag, Bittner, Litt, ...). 
Chenyang Xu 

Abstract: Dual complex of a singularity is originally defined as the homotopy class of CW complexes which characterize how the exceptional divisors in a log resolutions intersect each other. Using MMP, for isolated singularities, we indeed find a canonical representative of the dual complex which is defined up to PL homeomorphism. The same method also answers the question that the dual complex of a klt singularity is always contractible. This is a joint work with de Fernex and Koll\'ar. 
Xinyi Yuan 

Abstract: In this talk, I will first introduce a refined Noether
inequality for fibered algebraic surfaces. Then we introduce an
arithmetic version of the inequality in the setting of Arakelov
theory. These are joint works with Tong Zhang.

Zhiwei Yun 

Abstract: The zeta function of a smooth curve over a finite field
packages information about the number of points on it in larger and
larger finite fields. It is a rational function and has a functional
equation. How do we define a zeta function for a singular curve that
captures not just the number of points but also information about its
singularities? Does this function enjoy similar properties of the
usual zeta function? What is its analog for number fields? We will
answer some of these questions in the talk. 
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