Please see the following for more related talks:

- Friday, March 31, 2017, 3:15pm, 119 Physics, Algebraic Geometry Seminar
*Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups*

Seth Baldwin (UNC - Chapel Hill)- The cohomology rings of flag varieties have long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.

- Friday, April 7, 2017, 3:15pm, 119 Physics, Algebraic Geometry Seminar
*TBA*

Prakash Belkale (UNC) - Friday, April 14, 2017, 3:15pm, 119 Physics, Algebraic Geometry Seminar
*Fake projective spaces and fake tori*

Olivier Debarre (Ecole Normale Supérieure and Duke)- We discuss compact complex manifolds which ``look like'' complex projective
spaces or complex tori.
Hirzebruch and Kodaira proved in 1957 that when
*n*is odd, any compact Kähler manifold*X*which is homeomorphic to**P**^{n}is isomorphic to**P**^{n}. This holds for all*n*by Aubin and Yau's proofs of the Calabi conjecture. One may conjecture that it should be sufficient to assume that the integral cohomology rings*H*^{*}(*X*,**Z**) and*H*^{*}(**P**^{n},**Z**) are isomorphic.Catanese observed that complex tori are characterized among compact Kähler manifolds

*X*by the fact that their integral cohomology rings are exterior algebras on*H*^{1}(*X*,**Z**) and asked whether this remains true under the weaker assumption that the rational cohomology ring is an exterior algebra on*H*^{1}(*X*,**Q**). (We call the corresponding compact Kähler manifolds ``rational cohomology tori".) We give a negative answer to Catanese's question by producing explicit examples. We also prove some structure theorems for rational cohomology tori. This is work in collaboration with Z. Jiang, M. Lahoz, and W. F. Sawin.

- We discuss compact complex manifolds which ``look like'' complex projective
spaces or complex tori.
Hirzebruch and Kodaira proved in 1957 that when
- Monday, May 1, 2017, 3:15pm, 119 Physics, Geometry Seminar
*TBA*

Matt Kerr (Washington U in St. Louis) - Friday, September 22, 2017, 3:15pm, 119 Physics, Algebraic Geometry Seminar
*TBA*

Rita Pardini (U Pisa)

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