Kalle Karu


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Graduate Student Supervision

Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

Derived algebraic cobordism (2022)

We construct and study a theory of bivariant cobordism of derived schemes. Our theory provides a vast generalization of the algebraic bordism theory of characteristic 0 algebraic schemes, constructed earlier by Levine and Morel, and a (partial) non-?¹-invariant refinement of the motivic cohomology theory MGL in Morel--Voevodsky's stable motivic homotopy theory. Our main result is that bivariant cobordism satisfies the projective bundle formula. As applications of this, we construct cobordism Chern classes of vector bundles, and establish a strong connection between the cobordism cohomology rings and the Grothendieck ring of vector bundles. We also provide several universal properties for our theory. Additionally, our algebraic cobordism is also used to construct a candidate for the elusive theory of Chow cohomology of schemes.

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Negative curves in blowups of weighted projective planes (2020)

We study the Mori dream space property for blowups at a general point of weighted projec-tive planes or, more generally, of toric surfaces with Picard number one. Such a variety isa Mori dream space if and only if it contains two irreducible disjoint curves; one of themnecessarily having non-positive self-intersection. We call such a curve a “negative curve”. Asignificant part of this thesis is dedicated to the study of such negative curves, as they largelygovern the Mori dream space property for these varieties.Our study begins by constructing two one-parameter families of negative curves andsubsequently a larger two-parameter class of negative curves having the previous twofamilies as boundary cases.Once such a variety is known to contain a negative curve, we determine if it contains adisjoint curve by using different procedures. For example, prime characteristic and coho-mological methods. Furthermore, we introduce an independent technique that applies to abroader class of cases. As a result, for each of the negative curves constructed we provideexamples and non-examples of Mori dream spaces containing the curve.

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Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

On the anisotropy theorem of Papadakis and Petrotou (2023)

We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen's g-conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic 2 follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field Q. These results provide another self-contained proof of the g-conjecture for homology spheres in characteristic 2.

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Moduli Space of Sheaves on Fans (2011)

A conjecture of H. Hopf states that if x(M²n) is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic x(M²n), should satify (-1)n x(M²n)≥ 0. Ruth Charney and Michael Davis investigated the conjecture in the context of piecewise Euclidean manifolds having "nonpositive curvature" in the sense of Gromov's CAT(0) inequality. In that context the conjecture can be reduced to a local version which predicts the sign of a "local Euler characteristic" at each vertex. They stated precisely various conjectures in their paper which we are interested in one of them stated as Conjecture D (see [1]) which is equivalent to the Hopf Conjecture for piecewise Euclidean manifolds cellulated by cubes.The goal of this thesis is to study the Charney - Davis Conjecture stated as Conjecture (D) by using sheaves on fans.

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