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Combinatorics with an analytic structure

Objective

"Combinatorics, and its interplay with geometry, has fascinated our ancestors as shown by early stone carvings in the Neolithic period. Modern combinatorics is motivated by the ubiquity of its structures in both pure and applied mathematics.
The work of Hochster and Stanley, who realized the relation of enumerative questions to commutative algebra and toric geometry made a vital contribution to the development of this subject. Their work was a central contribution to the classification of face numbers of simple polytopes, and the initial success lead to a wealth of research in which combinatorial problems were translated to algebra and geometry and then solved using deep results such as Saito's hard Lefschetz theorem. As a caveat, this also made branches of combinatorics reliant on algebra and geometry to provide new ideas.

In this proposal, I want to reverse this approach and extend our understanding of geometry and algebra guided by combinatorial methods. In this spirit I propose new combinatorial approaches to the interplay of curvature and topology, to isoperimetry, geometric analysis, and intersection theory, to name a few. In addition, while these subjects are interesting by themselves, they are also designed to advance classical topics, for example, the diameter of polyhedra (as in the Hirsch conjecture), arrangement theory (and the study of arrangement complements), Hodge theory (as in Grothendieck's standard conjectures), and realization problems of discrete objects (as in Connes embedding problem for type II factors).

This proposal is supported by the review of some already developed tools, such as relative Stanley--Reisner theory (which is equipped to deal with combinatorial isoperimetries), combinatorial Hodge theory (which extends the ``K\""ahler package'' to purely combinatorial settings), and discrete PDEs (which were used to construct counterexamples to old problems in discrete geometry)."

Host institution

KOBENHAVNS UNIVERSITET
Net EU contribution
€ 917 494,40
Address
Norregade 10
1165 Kobenhavn
Denmark

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Region
Danmark Hovedstaden Byen København
Activity type
Higher or Secondary Education Establishments
Non-EU contribution
€ 0,00

Beneficiaries (3)

KOBENHAVNS UNIVERSITET
Denmark
Net EU contribution
€ 917 494,40
Address
Norregade 10
1165 Kobenhavn

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Region
Danmark Hovedstaden Byen København
Activity type
Higher or Secondary Education Establishments
Non-EU contribution
€ 0,00
UNIVERSITAET LEIPZIG

Participation ended

Germany
Net EU contribution
€ 30 000,00
Address
Ritterstrasse 26
04109 Leipzig

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Region
Sachsen Leipzig Leipzig
Activity type
Higher or Secondary Education Establishments
Non-EU contribution
€ 0,00
THE HEBREW UNIVERSITY OF JERUSALEM

Participation ended

Israel
Net EU contribution
€ 389 705,60
Address
Edmond J Safra Campus Givat Ram
91904 Jerusalem

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Activity type
Higher or Secondary Education Establishments
Non-EU contribution
€ 0,00