WAGS Spring 2017
April 8-9, 2017 
University of British Columbia

Speakers
 
 Jarod Alper (Washington) Slice theorems for stacks and applications
 Melody Chan (Brown) Brill-Noether varietes and tableaux
 Daniel Halpern-Leistner     (Columbia) Equivariant Geometry and Calabi-Yau Manifolds
 Sam Payne (Yale) A tropical motivic Fubini theorem with applications to Donaldson-Thomas theory
 Giulia Sacca (Stony Brook)  Degenerations of hyperkähler manifolds
 Bernd Sturmfels (Berkeley/MPI Leipzig)
 Algebraic Geometry of Gaussian Mixtures
 Jim Bryan (UBC) Quantum Entanglement and Geometric Invariant Theory

Tentative Schedule

 Saturday     Sunday 
 9:30-11:00 Mendelbaldeko lecture: Sturmfels 9:00-10:00 Payne
 11:30-12:30 Chan 10:30-11:30 Halpern-Leistner
 12:30-2:30 Lunch 11:45-12:45 Bryan
 2:30-3:30 Sacca  
 4:00-5:00 Alper 

Abstracts
Jarod Alper: Slice theorems for stacks and applications
We will begin by discussing the following theorem proven in joint work with Jack Hall and David Rydh:  every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is etale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. After briefly discussing extensions of this result to arbitrary fields and base schemes, we will focus on applications.  First, we will show how this result allows us to extend classical theorems concerning algebraic groups.  Second, we will apply this theorem to construct projective moduli spaces of objects (such as semistable vector bundles over a smooth projective curve) which may have infinite automorphism groups.

Jim Bryan: Quantum Entanglement and Geometric Invariant Theory
A fundamental question in quantum mechanics is whether a multipartite quantum  mechanical system has a completely entangled state. I will explain how this question is equivalent to the following very natural question in geometric invariant theory: Let H_1,…,H_n be complex vector spaces of dimensions d_1,…,d_n, let V be the tensor product of the H’s, and let G = SL(d_1)x…xSL(d_n), then is the GIT quotient P(V)//G non-empty? We give a complete answer to this question. The answer turns out to exhibit surprising arithmetic complexity. (This is joint work with Mark Van Raamsdonk and Zinovy Reichstein). 

Melody Chan: Brill-Noether varieties and tableaux
Brill-Noether theory on curves is the classical study of linear series on curves: essentially, maps of curves to projective space. On a smooth compact curve X of genus g, the Brill-Noether variety G^r_d(X) parametrizes linear series on X of rank r and degree d. I will discuss joint work with Alberto Lopez Martin, Nathan Pflueger, and Montserrat Teixidor i Bigas, in which we use combinatorics related to Buch’s set-valued tableaux, along with Osserman’s machinery of degenerations to Eisenbud-Harris schemes of limit linear series, to study the geometry of G^r_d(X).

Daniel Halpern-Leistner: Equivariant geometry and Calabi-Yau manifolds
Developments in high energy physics, specifically the theory of mirror symmetry, have led to deep conjectures regarding the geometry of a special class of complex manifolds called Calabi-Yau manifolds. One of the most intriguing of these conjectures states that various geometric invariants, some classical and some more homological in nature, agree for any two Calabi-Yau manifolds which are birationally equivalent to one another. I will discuss how new methods in equivariant geometry have shed light on this conjecture over the past few years, leading to the first substantial progress for compact Calabi-Yau manifolds of dimension greater than three. The key technique is the new theory of "Theta-stratifications" and "Theta-stability" -- a generalization of geometric invariant theory -- which allows one to bring ideas from equivariant Morse theory into the setting of algebraic geometry.

Sam Payne: A tropical motivic Fubini theorem with applications to Donaldson-Thomas theory
I will present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan’s theory of motivic volumes of semi-algebraic sets.  As time permits, I will discuss applications of this method, which include the solution to a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson-Thomas theory. This is based on joint work with Johannes Nicaise.

Giullia Sacca:  Degenerations of hyperkähler manifolds
The problem of understanding semistable degenerations of K3 surfaces has been greatly studied and is completely understood. The aim of this talk is to present joint work in progress with J. Kollár, R. Laza, and C. Voisin giving partial generalizations to higher dimensional hyperkähler (HK) manifolds. I will also present some applications, including a generalization of theorem of Huybrechts to possibly singular symplectic varieties and shortcuts to showing that certain HK manifolds are of a given deformation type.

Bernd Sturmfels:  Algebraic Geometry of Gaussian Mixtures
Mixtures of Gaussians are ubiquitous in data science. We give an introduction to the geometry of these statistical models, with focus on the projective varieties represented by their moments. Recent work with Carlos Amendola and Kristian Ranestad characterizes circumstances under which these moment varieties have the expected dimensions.



Related Colloquium Talk
Bernd Stumfels will also be speaking in the Math Colloquium on Friday April 7th from 3-4pm. More information is available here.

About WAGS
WAGS is a twice-yearly meeting of algebraic geometers in the western half of the United States and Canada that traces its origins back to the Utah-UCLA Algebraic Geometry Seminar started in 1989.

Long term planning for WAGS is currently being organized by Aaron Bertram, Sebastian Casalaina-Martin, Renzo Cavalieri, Sándor Kovács, Ravi Vakil, and Bianca Viray.

For more information about WAGS, please visit www.wagsymposium.org

Mendebaldeko Lectures are named after the Basque algebraic geometer Hirune Mendebaldeko.
She was a Basque pacifist, a contemporary of Nicholas Bourbaki, whom she met in Paris while studying algebraic geometry. They were rumored to be carrying on a secret affair, with not infrequent trysts in the Pyrenees. Whenever they appeared together in public, however, there was no hint of a relationship. Also see  https://plus.google.com/+lievenlebruyn/posts/bKqgwkmZZqQ

Diversity

We hope to broaden the community of algebraic geometers. All are welcome to attend. We especially encourage participation from women and members of groups traditionally under-represented in mathematics.

Location

The meeting will be held in room ESB 1012 at the University of British Columbia. See here for a map to the ESB (Earth Sciences Building).  

Registration and Financial Assistance

WAGS is partially supported by the National Science Foundation and by the Department of Mathematics; if you need funding, please apply when you register.

We ask that all participants fill out the first form; you must fill out both forms in order to be reimbursed:
Important for reimbursement:Please help us keeping organized by mailing your reimbursement claims within 2 weeks of the end of WAGS. We may run into trouble allocating funds for late submissions!

Accommodation

On campus, there are many suitable hotels:

There are many reasonably priced off campus accommodations such as Quilchena House and the famous Sylvia Hotel in the West End. See for example here.


Transportation

You can travel by airplane or car:

By plane:  Fly into Vancouver International Airport (YVR). Public transport (Skytrain) is available from the airport to Vancouver and UBC. To reach UBC from the airport, take the Skytrain to the Broadway/City Hall stop, then ride the 99 B-Line bus (westbound) until the end of the line. The taxi fare is about $35.00. Vancouver does not have Uber. 

By car:  Take Interstate 5 to the Peace Arch border crossing, then continue north on Highway 99. Once in Vancouver, turn left on W Broadway, and follow the road all the way to UBC. Visitor parking information can be found here.

Important:
To enter Canada, US citizens need a valid passport or the equivalent. If you are flying to Vancouver and you are not a US citizen, you may need to obtain an eTA (electronic travel visa): to see, go to http://www.cic.gc.ca/english/visit/visas.asp 
The procedure only takes a few minutes online but it costs CAD $7. If you are driving, you don't need an eTA.

Local information

There are a number of fantastic restaurants on or near the UBC campus. They include (in no particular order):

Bierrcraft in Westbrook village:  http://biercraft.com/wesbrook-at-ubc/
Green Leaf Sushi (3416 W Broadway)
Maria's Taverna: (2324 W 4th Ave)

You may view their locations on this map:

Most of the restaurants are near W 10th and Broadway, and are accessible by the 99 B-line bus.

Local organizers

Local organization is by Jim Bryan, Jim Carrell, Sabin Cautis, Nathan Ilten, and Kalle Karu.  Send questions to spring2017@wagsymposium.org.