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Works in Lun-Yi's Show: "Demonstrations: Collaborations with Mathematicians"
Shift Collaborative Studio, through Saturday, June 2
noon-5 p.m. Fridays and Saturdays, and by appointment
306 S. Washington St., #105, Seattle WA 98107
www.shiftstudio.org


Listen to an interview with Lun-Yi Tsai on Seattle's NPR Station, KUOW.

 

"Shafarevich's Conjecture"

This is a collaboration with Prof. Sándor Kovács of the University of Washington.

All mathematicians are asked at some point: "Hey man, what was your thesis about?" To which the response is some absurd oversimplification if not an all out deformation of their work. After all, how is it possible to sum up in a couple sentences what took several years to accomplish? But now if you ask Prof. Sándor Kovács what his thesis was about he'll no longer give you a nebulous answer about black holes and the universe, instead he'll just point to this drawing from our collaboration.

The story goes back to a famous conjecture given by the Russian mathematician Shafarevich at the International Congress of Mathematicians in 1962. The conjecture in its terse mathematical language predicted that "for a fixed base and a fixed genus there are only finitely many non-isotrivial families of smooth projective curves of the given genus over the given base." Sándor proposed this theorem to me in his office drawing a few figures on the whiteboard of how he saw this thing. The conjecture is particularly close to him as he has been able to prove several generalizations of it in the past few years. The work piqued my interest especially because of certain metaphysical ideas that I have been entertaining regarding covering spaces and human existence. This seemed to be a generalization of those musings and I definitely wanted to explore the visualizations to clarify my own ideas regarding what in my mind essentially boils down to parametrizations. Our collaboration has been very rewarding and has inspired me to work more with algebraic theorems; after all, according to Sándor, "Algebra = Geometry."

As I was drawing this, I would go from the math and start thinking about people at a cocktail party or at an art opening. And then these genus 4 surfaces came to life and became these clusters of folks sipping champagne and martinis and checking each other out while the jazz music playing in the background. So if it looks like they are swaying to the music it's because they are! They are also reminiscent of my father's cybernetic sculpture, which are deeply embedded in my subconscious.


"Shafarevich's Conjecture"
2007
Charcoal and graphite on paper
35 x 40 in

 

"Ricci Flow with Surgery"

This drawing was inspired by a talk given last December at the University of Washington by Prof. Richard Hamilton on the "Ricci Flow," which was used famously by Grigori Perelman in 2002 to prove the long-standing Poincaré conjecture. The appearance in the lecture of things like cigars and necks and pinches immediately lent themselves to visual representation and interpretation. Afterward, we had a conversation about mathematical history and the possibility of progress in mathematics. Mathematical history, Hamilton mused, is always written as things should have happened, but never how they actually did happen. He observed that even his part in the proof of the Poincaré conjecture was being rewritten. Then remarking mostly to himself, he said in hindsight the problem seemed so simple and wondered with amazement why no one, including himself, saw how reversing the direction of time was just the trick needed. He had reversed time in other situations, but not in that case! This is so typical of math: when a problem is unsolved it looks impossible, but when one knows the answer, it is completely obvious!

When I was a child, I used to have this recurring nightmare of the gigantic smooth surfaces crowding around me and smothering me with their smoothness and their massive proportions. You would think that such a thing would turn me away from manifolds, but curiously I feel a certain intimate familiarity with these objects.

It is important to note that the ideas of French mathematician Henri Poincaré are really behind many pieces in this series: it's in the two Whitehead Continuum drawings, it's in the three different Hopf Fibrations, and it's in this work.


"Ricci Flow with Surgery"
2007
charcoal and graphite on paper
26.5 x 40 in

 

"Whitehead Continuum I and II"

Both these drawings were intiated by discussions with Prof. Aravind Asok at the University of Washington. He suggested that I look at contractible spaces, which are spaces that you can shrink to a point. In dimensions one and two the only contractible spaces are just the one and two-dimensional "balls," but in dimension 3, things get more interesting. In the 1930s, Henry Whitehead, who was trying to prove the Poincaré Conjecture, which says things that are compact without holes are basically spheres. Whitehead gave a "proof" of this, but then discovered a mistake and created a counterexample known as the Whitehead manifold. So these two drawings are of an object born out of the mistake of a first-rate mathematician.

This happens often in math: good mathematicians making good mistakes, which lead to breakthroughs in the field. But as Goro Shimura noted: "It's hard to make good mistakes."


"Whitehead Continuum I"
2007
charcoal and graphite on paper
25 x 41 in

 

"Contraction Mapping and Picard's Theorem"

When asked for a cool theorem, Prof. Eric Todd Quinto reminded me of one he had taught me in first year undergraduate analysis (which is what you take after calculus) called the contraction mapping principle. This lemma says that if you have a kind of map (function) that contracts your (complete metric) space so that each time you apply it the points in the space get closer together, then there's going to be a unique point that is fixed. So all the points in the space are being shrunk to that point. This has powerful applications in many areas of math especially in the theory of differential equations, which has the result known as Picard's Theorem that uses a contraction mapping to show that there is a unique solution to a particular differential equation.

But mathematics aside, every time I now look at this drawing, I think of Christmas in New York. I think of Rockefeller Center and the ice skating rink. I think nostalgically of Central Park and the cold empty streets of my childhood.

"Contraction Mapping and Picard's Theorem"
2007
Charcoal and graphite on paper
25.5 x 40 in

 

"Wormhole Construction on Sigma T"
This drawing is a collaboration with Prof. Dan Pollack at the University of Washington.

Dan and his collaborators have found that between any two points in the universe, it is possible mathematically to construct a "wormhole" while maintaining the structure of spacetime according the Einstein's Theory of General Relativity. This collaboration was particularly stimulating because of the number of ideas I had to learn and the time Dan and I spent at the blackboard in his office ironing out the details. It's my first drawing involving the physics of spacetime.


"Wormhole Construction on Sigma T"
2007
Charcoal and graphite on paper
30.5 x 40 in