Department of

Mathematics


Seminar Calendar
for events the day of Thursday, September 5, 2019.

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Thursday, September 5, 2019

12:00 pm in 243 Altgeld Hall ,Thursday, September 5, 2019

Counting incompressible surfaces in 3-manifolds

Nathan Dunfield   [email] (Illinois)

Abstract: Counting embedded curves on a hyperbolic surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting incompressible surfaces in a hyperbolic 3-manifold, with the key difference that now the surfaces themselves have intrinsic topology. As are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Hakenís normal surface theory and facts about branched surfaces, we can characterize not just the rate of growth but show it is (essentially) a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples. This is joint work with Stavros Garoufalidis and Hyam Rubinstein.

3:00 pm in 347 Altgeld Hall,Thursday, September 5, 2019

Polytopes, polynomials and recent results in 1989 mathematics

Bruce Reznick   [email] (University of Illinois at Urbana-Champaign)

Abstract: Hilbertís 17th Problem discusses the possibility of writing polynomials in several variables which only take non-negative values as a sum of squares of polynomials. One approach is to substitute squared monomials into the arithmetic-geometric inequality. Sometimes this is a sum of squares, sometimes it isnít, and I proved 30 years ago that this depends on a property of the polytope whose vertices are the exponents of the monomials in the substitution. Whatís new here is an additional then-unproved claim in that paper and its elementary, but non-obvious proof. This talk lies somewhere in the intersection of combinatorics, computational algebraic geometry and number theory and is designed to be accessible to first year graduate students.