Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day, informal meeting of mathematicians from the University of Illinois, Purdue University and nearby universities, with interests in algebra, geometry and combinatorics (widely interpreted).

Location: Department of Mathematics, Altgeld Hall, University of Illinois at Urbana-Champaign

Speakers  and schedule:

       Coffee and pastries          8h30   

Abstract. The B. and M. Shapiro conjecture has been originally 

formulated as a problem of real algebraic geometry: 

Let P_1(x),...,P_k(x) be linearly independent univariate 

polynomials with complex coefficients, such that all zeros 

of their Wronski determinant are real. Then the subspace of C[x] 

spanned by P_1,...,P_k is real. 

For k=2, it can be reformulated as a problem about rational 

functions: If all critical points of a rational function f(x) 

are real, then f(R) is a circle on the Riemann sphere. 

In this case, it was proved by Eremenko and Gabrielov using 

methods of complex analysis. The only known proof for 

the general case, by Mukhin, Tarasov and Varchenko, is based 

on the Bethe ansatz for the Gaudin quantum integrable model. 

I'll give a review of relations of this problem to different 

areas of mathematics: complex analysis, differential equations, 

Schubert calculus, combinatorics of Young tableaux, arrangements 

of hyperplanes, etc.

Abstract. The classical Noether-Lefschetz theorem asserts that, on a generic surface S of degree at least four in P3, every curve is the intersection of S with a hypersurface. The surfaces that do have additional curves are parametrized by countably many algebraic varieties, the so-called Noether Lefschetz loci. When P3 is replaced by a Calabi-Yau threefold, each of these loci is typically a finite set of points. Surprising modularity properties for the cardinalities of these sets are predicted by physicists. In the talk, I will describe how to get a sensible "count" of the Noether-Lefschetz loci (including degenerate cases, points at infinity, etc.) using Hodge theory. This is a joint project with Davesh Maulik.

Abstract: We study the Teichmuller curves parameterizing covers of
elliptic curves with a unique branch point, with a focus on the interplay
between algebraic combinatorics, algebraic geometry and complex dynamics.

Abstract: Given a smooth complex projective variety,

let me say that it is Picard maximal if it has the maximal

number of algebraic cycles with respect to the Hodge bound.

Algebraic surfaces with this property have been studied

quite extensively, but as far as I know, this hasn't been looked at

in higher dimensions. I'll give some constructions in

higher dimensions involving fibred products of surfaces.

Part of the motivation for looking at such examples is that

they automatically satisfy certain well known conjectures,

but mostly I think these are worthy of study for their

own sake.

Homewood Suites  (holding rooms for AlGeCom)
1417 S. Neil, Champaign, IL   217-352-9960

Other hotels in the area:
Hampton Inn
1200 W. University, Urbana, IL  217-337-1100  
They have a shuttle to and from campus.

Eastland Suites
1907 N. Cunningham, Urbana, IL  217-367-8331    
They have a shuttle to and from campus.

Hawthorne Suites
101 Trade Center, Champaign, IL  217-398-3400

I-Hotel
1900 S. First, Champaign, IL 217-819-5002   
They have a shuttle to and from campus.

Hilton Garden Inn
1501 S. Neil, Champaign, IL  217-352-9970

Holiday Inn
1001 Killarney Rd, Urbana, IL  217-328-7900

Holiday Inn Express
1002 Killarney Rd., Urbana, IL 217-328-0328