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| Kyle Binder | |
Singular Cohomology Rings of Uniform Matroids |
| Abstract:
The Chow ring of a matroid is a well-studied object in matroid theory, originally introduced to prove the
Heron–Rota–Welsh Conjecture.
It is defined to be the Chow ring of a certain quasi-projective toric variety associated to the Bergman fan of the matroid,
and although it is not the cohomology ring of a projective variety, it satisfies Poincaré duality, Hard Lefschetz,
and the Hodge–Riemann relations.
We introduce the singular cohomology ring of a matroid to be the singular cohomology ring of the same toric variety.
This is a richer invariant of the matroid which contains the Chow ring.
In the case of uniform matroids, we give a combinatorial basis for the singular cohomology ring and show that it
satisfies a Strong Lefschetz property.
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| Hasitha Geekiyanage | |
The Serre-Swan Correspondence: From Geometry to Algebra | |
| Abstract:
This poster explores a classical connection between geometry and algebra; focusing on the Serre–Swan correspondence,
which shows that geometric objects such as vector bundles can be studied through algebraic objects called projective
modules.
The goal is to highlight how this correspondence allows ideas from algebra to be used in understanding geometric objects.
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| Philip Speegle | |
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| Abstract:
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| David Yu | |
On a tamely ramified relative local geometric Langlands conjecture |
| Abstract:
The relative Langlands program predicts that harmonic analysis on a spherical variety should be controlled by
geometry on a dual space.
For spherical varieties, this picture has become increasingly concrete in the unramified setting.
Our work is a first step beyond that case, to the Iwahori, or tamely ramified, level.
Under mild hypotheses, the paper gives an explicit Langlands dual description of a natural category attached to the
loop space of a smooth affine spherical variety, using Springer-type geometry that records the extra ramification.
A key new ingredient in the proof is a categorical approach: ideas from categorical representation theory recover the
relevant Iwahori-level category, while integral transforms from derived algebraic geometry identify the spectral
side.
This is the first result confirming a variant of a recent conjecture by Devalapurkar and adds new evidence for
relative Langlands duality beyond the unramified setting.
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|
| Yinbang Lin | |
Expected behaviors of sheaves on algebraic surfaces |
| Abstract:
Motivated by the Brill--Noether problems and enumerative geometry over surfaces, we study the expected behaviors of coherent sheaves.
We estimate the dimension of global sections of stable sheaves.
We also prove some cases of an analogue of Lange's conjecture over curves, which states that general extensions of two vector bundles
are stable under some obvious conditions.
These are closely related to Segre invariants of sheaves, which studies maximal subsheaves of a fixed rank.
This can be understood as to determine when Grothendieck's Quot schemes are non-empty.
This is based on work in progress jointly with Thomas Goller and Zhixian Zhu.
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|
| Shilpi Mandal | |
Strong u-invariant |
| Abstract:
The u-invariant of a field Kis the maximal dimension of anisotropic quadratic forms over K.
For example, the u-invariant of complex numbers is 1, the u-invariant of a non-real global or local field is
1, 2, 4, or 8, etc.
In this poster, I will first present the ideas of field patching to obtain a local-to-global principle in the
setting of Berkovich curves. Then use it to obtain a bound for the u-invariant of complete non-Archimedean valued fields.
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|
| Paul Dessauer | |
Full Volume Chemical Reaction Networks |
| Abstract:
An important research problem for chemical reaction networks is determining the maximum number of steady states for
a network.
Many researchers used the mixed volume bound to give an upper bound for the maximum number of steady states; however,
the Newton-Okounkov bound (introduced by Obatake and Walker) is known to be able to give sharper bounds in some
cases.
In this poster, we investigate a class of networks for which the Newton-Okounkov bound will always be at least as
sharp as the mixed volume bound; we call such networks Full Volume.
We show that checking whether a network is Full Volume can be done easily using the Newton polytope, and introduce
reduced networks, a class of networks that requires significantly fewer checks to verify Full Volume.
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