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BEGIN:VEVENT
SUMMARY:Antoine Ducros (Sorbonne University)
DTSTART;VALUE=DATE-TIME:20211001T120000Z
DTEND;VALUE=DATE-TIME:20211001T130000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/1
DESCRIPTION:Title: Non-standard analysis and non-archimedean geometry\nby Antoine Duc
ros (Sorbonne University) as part of Non-Archimedean and Tropical Geometry
\n\n\nAbstract\nIn a joint work with E. Hrushovski and F. Loeser\, we show
that certain one-parameter families of complex integrals have a limit tha
t can be expressed as an integral on a Berkovich space over the field $\\m
athbb C((t))$ (in the sense of the theory of integration of real different
ial forms on Berkovich spaces\, developed by Chambert-Loir and myself). In
this talk I will present this result\, but rather focus on the general me
thod we introduced to prove it\, which we hope will be useful for a lot of
other situations involving a "t-adic limit of one-parameter families of c
omplex objects". It consists in introducing a huge non-standard model of $
\\mathbb C$ also equipped with a non-archimedean absolute value\; working
on such a model enables by design to deal at the same time with limits of
usual complex objects (through non-standard analysis) as well as with non-
archimedean objects\, allowing for a direct comparison between these two w
orlds.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joe Rabinoff (Duke University)
DTSTART;VALUE=DATE-TIME:20211001T131500Z
DTEND;VALUE=DATE-TIME:20211001T141500Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/2
DESCRIPTION:Title: Weakly smooth forms and Dolbeault cohomology of curves\nby Joe Rab
inoff (Duke University) as part of Non-Archimedean and Tropical Geometry\n
\n\nAbstract\nGubler and I work out a theory of weakly smooth forms on non
-Archimedean analytic spaces closely following the construction of Chamber
t-Loir and Ducros\, but in which harmonic functions are forced to be smoot
h. We call such forms "weakly smooth". We compute the Dolbeault cohomology
groups of rig-smooth\, compact non-Archimedean curves with respect to thi
s theory\, and show that they have the expected dimensions and satisfy Poi
ncaré duality. We carry out this computation by giving an alternative cha
racterization of weakly smooth forms on curves as pullbacks of certain "sm
ooth forms" on a skeleton of the curve. This yields an isomorphism between
the Dolbeault cohomology of the skeleton\, which can be computed using st
andard combinatorial methods\, and the Dolbeault cohomology of the curve.\
nThis work is joint with Walter Gubler.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Yue Yu (California Institute of Technology)
DTSTART;VALUE=DATE-TIME:20211001T143000Z
DTEND;VALUE=DATE-TIME:20211001T153000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/3
DESCRIPTION:Title: Generalizing GKZ secondary fan using Berkovich geometry\nby Tony Y
ue Yu (California Institute of Technology) as part of Non-Archimedean and
Tropical Geometry\n\n\nAbstract\nGelfand-Kapranov-Zelevinski introduced th
e notion of secondary fan in the study of the Newton polytopes of discrimi
nants and resultants. It also controls the geometric invariant theory for
toric varieties. We propose a generalization of the GKZ secondary fan to g
eneral Fano varieties using ideas from Berkovich geometry and Mori theory.
Furthermore\, inspired by mirror symmetry\, we propose a synthetic constr
uction of a universal family of Kollár-Shepherd-Barron-Alexeev stable pai
rs over the toric variety associated to the generalized secondary fan. Thi
s generalizes the families of Kapranov-Sturmfels-Zelevinski and Alexeev in
the toric case. We gave a detailed construction and proved the stability
in the case of del Pezzo surfaces. This is joint work with Hacking and Kee
l.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kris Shaw (University of Oslo)
DTSTART;VALUE=DATE-TIME:20211015T120000Z
DTEND;VALUE=DATE-TIME:20211015T130000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/4
DESCRIPTION:Title: A tropical approach to the enriched count of bitangents to quartic cur
ves\nby Kris Shaw (University of Oslo) as part of Non-Archimedean and
Tropical Geometry\n\n\nAbstract\nUsing A1 enumerative geometry Larson and
Vogt have provided an enriched count of the 28 bitangents to a quartic cur
ve. In this talk\, I will explain how these enriched counts can be compute
d combinatorially using tropical geometry. I will also introduce an arithm
etic analogue of Viro’s combinatorial patchworking for real algebraic cu
rves which\, in some cases\, retains enough data to recover the enriched c
ounts. Finally\, I will comment on a possible tropical approach to the enr
iched count of the 27 lines on a cubic surface of Kass and Wickelgren. Thi
s talk is based on joint work with Hannah Markwig and Sam Payne.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melody Chan (Brown University)
DTSTART;VALUE=DATE-TIME:20211015T131500Z
DTEND;VALUE=DATE-TIME:20211015T141500Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/5
DESCRIPTION:Title: Homology representations of compactified configurations on graphs appl
ied to $M_{2\,n}$\nby Melody Chan (Brown University) as part of Non-Ar
chimedean and Tropical Geometry\n\n\nAbstract\nThe homology of a compactif
ied configuration space of a graph is\nequipped with commuting actions of
a symmetric group and the outer\nautomorphism group of a free group. We co
nstruct an efficient free\nresolution for these homology representations.
Using the Peter-Weyl\nTheorem for symmetric groups\, we consider irreducib
le representations\nindividually\, vastly simplifying the calculation of t
hese homology\nrepresentations from the free resolution.\nAs our main appl
ication\, we obtain computer calculations of the top\nweight rational coho
mology of the moduli spaces M_{2\,n}\, equivalently\nthe rational homology
of the tropical moduli spaces Δ_{2\,n}\, as a\nrepresentation of S_n act
ing by permuting point labels for all n≤10.\nWe further give new multipl
icity calculations for specific irreducible\nrepresentations of S_n appear
ing in cohomology for n≤17. Our approach\nproduces information about the
se homology groups in a range well\nbeyond what was feasible with previous
techniques. Joint work with\nChristin Bibby\, Nir Gadish\, and Claudia He
Yun.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniele Turchetti (University of Warwick)
DTSTART;VALUE=DATE-TIME:20211015T143000Z
DTEND;VALUE=DATE-TIME:20211015T153000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/6
DESCRIPTION:Title: Schottky spaces and moduli of curves over $\\mathbb{Z}$\nby Daniel
e Turchetti (University of Warwick) as part of Non-Archimedean and Tropica
l Geometry\n\n\nAbstract\nSchottky uniformization is the description of an
analytic curve as the quotient of an open dense subset of the projective
line by the action of a Schottky group.\nAll Riemann surfaces can be unifo
rmized in this way\, as well as some non-archimedean curves\, called Mu
mford curves.\nIn this talk\, I will present a construction of univ
ersal Mumford curves: analytic spaces that parametrize both archimedea
n and non-archimedean uniformizable curves of a fixed genus.\nThis result
relies on the existence of suitable moduli spaces for marked Schottky grou
ps\, that can be built using the theory of Berkovich spaces over rings of
integers of number fields developed by Poineau.\n\nAfter introducing Berko
vich analytic geometry from the beginning\, I will describe universal Mumf
ord curves and explain how these can be related to combinatorial structure
s arising from the theories of tropical moduli and geometric group theory.
\nThis is based on joint work with Jérôme Poineau.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elise Goujard (Bordeaux University)
DTSTART;VALUE=DATE-TIME:20211029T120000Z
DTEND;VALUE=DATE-TIME:20211029T130000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/7
DESCRIPTION:Title: Counting (tropical) covers: quasimodularity of generating functions\nby Elise Goujard (Bordeaux University) as part of Non-Archimedean and T
ropical Geometry\n\n\nAbstract\nCounting (weighted) ramified coverings of
the sphere or the torus lead \nto several applications\, one of them is th
e (weighted) count of \ninteger points in some moduli spaces of flat surfa
ces\, leading to the \nevaluation of the Masur-Veech volumes or the Siegel
-Veech constants of \nthese moduli spaces. Both these quantities are relev
ant to the study \nof dynamics in polygonal billiards for instance\, but a
lso to other \ndynamical problems on flat surfaces (such as the count of c
losed \ngeodesics).\nWith that motivation in mind\, I will explain a joint
work with Martin \nMöller on the generating series of these counts : the
y are quasimodular and this property holds "graph by graph". Some of our r
esults can be stated in terms of tropical covers and I will detail the rel
ation of our results with a previous work of Böhm-Bringmann-Buchholz-Mark
wig.\nI will also give several questions and conjectures about the \n"comp
leted cycles" that appear naturally in this setting.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Walter Gubler (University of Regensburg)
DTSTART;VALUE=DATE-TIME:20211029T131500Z
DTEND;VALUE=DATE-TIME:20211029T141500Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/8
DESCRIPTION:Title: Forms on Berkovich spaces based on harmonic tropicalizations\nby W
alter Gubler (University of Regensburg) as part of Non-Archimedean and Tro
pical Geometry\n\n\nAbstract\nChambert-Loir and Ducros introduced smooth f
orms and currents on Berkovich spaces using tropicalization maps induced b
y morphisms to tori. In joint work with Philipp Jell und Joe Rabinoff\, we
allow more generally harmonic tropicalization maps to define a larger cla
ss of weakly smooth forms which has essentially the same properties as the
smooth forms\, but have a better cohomological behavior.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Navid Nabijou (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20211029T143000Z
DTEND;VALUE=DATE-TIME:20211029T153000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/9
DESCRIPTION:Title: Tropical expansions and rubber torus actions\nby Navid Nabijou (Un
iversity of Cambridge) as part of Non-Archimedean and Tropical Geometry\n\
n\nAbstract\nGiven a normal crossings pair (X\,D)\, its tropicalization ca
n be defined as the cone over the dual intersection complex of D. A polyhe
dral subdivision of the tropicalization induces a degeneration of X\, call
ed a tropical expansion. To first approximation\, this is obtained by atta
ching additional "bubble" irreducible components to X\, along strata in D.
Tropical expansions form a natural class of degenerations which admit nic
e combinatorial descriptions. They have been studied and exploited by many
authors\, in many contexts.\n\nWe investigate automorphisms of tropical e
xpansions covering the identity on X. We discover a purely tropical descri
ption: the so-called rubber torus is the torus associated to the moduli sp
ace of tropical edge lengths in the polyhedral subdivision\, and its actio
n on each component of the expansion is encoded in a linear "tropical posi
tion map." Our main application is to logarithmic enumerative geometry\, w
hich I will motivate\, but the talk will not focus on this. This is joint
work with Francesca Carocci.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Steffen Müller (University of Groningen)
DTSTART;VALUE=DATE-TIME:20211112T130000Z
DTEND;VALUE=DATE-TIME:20211112T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/10
DESCRIPTION:Title: $p$-adic Arakelov theory on abelian varieties and quadratic Chabauty<
/a>\nby Jan Steffen Müller (University of Groningen) as part of Non-Archi
medean and Tropical Geometry\n\n\nAbstract\nI will discuss a new construct
ion of p-adic height functions on abelian varieties over number fields usi
ng Besser's p-adic Arakelov theory. In analogy with Zhang's construction o
f Néron-Tate heights via adelic metric\, these heights are given in terms
of canonical p-adic adelic metrics on line bundles. As an application\, I
will describe a new and simplified approach to the quadratic Chabauty met
hod for the computation of rational points on certain curves. This is join
t work in progress with Amnon Besser and Padmavathi Srinivasan.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Klaus Künnemann (University of Regensburg)
DTSTART;VALUE=DATE-TIME:20211112T141500Z
DTEND;VALUE=DATE-TIME:20211112T151500Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/11
DESCRIPTION:Title: Pluripotential theory for tropical toric varieties and non-archimedea
n Monge-Ampère equations\nby Klaus Künnemann (University of Regensbu
rg) as part of Non-Archimedean and Tropical Geometry\n\n\nAbstract\nTropic
al toric varieties are partial compactifications of finite dimensional rea
l vector spaces associated with rational polyhedral fans. We introduce plu
ripotential theory on tropical toric varieties. This theory provides a can
onical correspondence between complex and non-archimedean pluripotential t
heories of invariant plurisubharmonic functions on toric varieties. We app
ly this correspondence to solve invariant non-archimedean Monge-Ampère eq
uations on toric and abelian varieties over arbitrary non-trivially valued
non-archimedean fields. This is joint work with José Ignacio Burgos Gil\
, Walter Gubler and Philipp Jell.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Wewers (Ulm University)
DTSTART;VALUE=DATE-TIME:20211112T153000Z
DTEND;VALUE=DATE-TIME:20211112T163000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/12
DESCRIPTION:Title: Explicit models of curves via nonarchimedian geometry\nby Stefan
Wewers (Ulm University) as part of Non-Archimedean and Tropical Geometry\n
\n\nAbstract\nI will report on my long term efford to make the computation
of the semistable reduction of curves over p-adic fields effective and pr
actical. I will focus on some particular cases\, and on the use of methods
from nonarchimedian analytic geometry to achieve this goal.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vlerë Mehmeti (University Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20211126T130000Z
DTEND;VALUE=DATE-TIME:20211126T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/13
DESCRIPTION:Title: A Hasse Principle on Berkovich Analytic Curves\nby Vlerë Mehmeti
(University Paris-Saclay) as part of Non-Archimedean and Tropical Geometr
y\n\nLecture held in Seminar Room of the CMLS\, École Polytechnique (Pala
iseau\, France).\n\nAbstract\nPatching techniques\, under various forms an
d inspired from results in complex analysis\, have in the past been used a
s an approach to the inverse Galois problem. \nRecently\, these techniques
have become a very important tool in the study of local-global principles
. I will explain how patching can be adapted to Berkovich \nanalytic curve
s. Working in this setting\, one can then obtain several local-global prin
ciples\, all of which are applicable to quadratic forms.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Erwan Brugallé (Nantes University)
DTSTART;VALUE=DATE-TIME:20211126T140000Z
DTEND;VALUE=DATE-TIME:20211126T150000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/14
DESCRIPTION:Title: Euler characteristic and signature of real semi-stable degenerations<
/a>\nby Erwan Brugallé (Nantes University) as part of Non-Archimedean and
Tropical Geometry\n\nLecture held in Seminar Room of the CMLS\, École Po
lytechnique (Palaiseau\, France).\n\nAbstract\nIt is interesting to compar
e the Euler characteristic of the real part of a real algebraic variety to
the signature of its complex part. For example\, a theorem by Itenberg an
d Bertrand states that both quantities are equal for "primitive T-hypersur
faces". After defining these latter\, I will give a motivic proof of this
theorem via the motivic nearby fiber of a real semi-stable degeneration. T
his proof extends in particular the original statement by Itenberg and Ber
trand to non-singular tropical varieties.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jérôme Poineau (University of Caen Normandy)
DTSTART;VALUE=DATE-TIME:20211126T153000Z
DTEND;VALUE=DATE-TIME:20211126T163000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/15
DESCRIPTION:Title: Analytic dynamics over $\\mathbf{Z}$ and torsion points of elliptic c
urves\nby Jérôme Poineau (University of Caen Normandy) as part of No
n-Archimedean and Tropical Geometry\n\nLecture held in Seminar Room of the
CMLS\, École Polytechnique (Palaiseau\, France).\n\nAbstract\nLet $Y$ be
a Berkovich space over $\\mathbf{Z}$. Recall that such a space naturally
contains non-Archimedean parts (such as usual $p$-adic Berkovich spaces) a
nd Archimedean parts (such as complex analytic spaces). Denote by $X$ the
relative projective line over $Y$. For each point $y$ in $Y$\, let $\\mu_
y$ be a measure defined on the fiber $X_y$ (which is an analytic projectiv
e line over the complete residue field associated to $y$). Inspired by th
e work of Favre on endomorphisms on hybrid Berkovich spaces\, we prove gen
eral continuity results for families of measures of the form $(\\mu_y)_{y\
\in Y}$ coming from dynamical systems on $X$. Following a strategy by DeMa
rco-Krieger-Ye\, we then deduce new cases of a conjecture of Bogomolov-Fu-
Tschinkel on uniform bounds on the number of common images on $P^1$ of tor
sion points of two elliptic curves.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudia Fevola (Max Planck Institute Leipzig)
DTSTART;VALUE=DATE-TIME:20211210T130000Z
DTEND;VALUE=DATE-TIME:20211210T140000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/16
DESCRIPTION:Title: Kp Solitons from Tropical Limits\nby Claudia Fevola (Max Planck I
nstitute Leipzig) as part of Non-Archimedean and Tropical Geometry\n\n\nAb
stract\nIn this talk\, we study solutions to the Kadomtsev-Petviashvili eq
uation whose underlying algebraic curves undergo tropical degenerations. R
iemann’s theta function becomes a finite exponential sum that is support
ed on a Delaunay polytope. We introduce the Hirota variety which parametri
zes all tau functions arising from such a sum. After introducing solitons
solutions\, we compute tau functions from points on the Sato Grassmannian
that represent Riemann-Roch spaces.\nThis is joint work with Daniele Agost
ini\, Yelena Mandelshtam and Bernd Sturmfels.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Holmes (Leiden University)
DTSTART;VALUE=DATE-TIME:20211210T141500Z
DTEND;VALUE=DATE-TIME:20211210T151500Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/17
DESCRIPTION:Title: Piecewise polynomials and intersection theory\nby David Holmes (L
eiden University) as part of Non-Archimedean and Tropical Geometry\n\n\nAb
stract\nRather than studying the Chow ring of a variety X\, it is becoming
\nincreasing popular to study some kind of limit of Chow rings over\nblowu
ps of X\; for example this arises in enumerative geometry and in the\nstud
y of singular hermitian metrics. Philosophically this limit might be\nthou
ght of as the Chow ring of the Riemann-Zariski space of X\, or\nperhaps of
the valuativisation of X\; but we will not dwell on such\nquestions. Rath
er\, we will explain how piecewise-polynomial functions on\nthe tropicalis
ation of X give an efficient way to write down certain\n'tautological' ele
ments of this 'limit Chow ring'\, and describe\napplications to enumerativ
e problems\, and a Sage implementation for the\nmoduli space of stable cur
ves.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitri Wyss (École Polytechnique Fédérale de Lausanne)
DTSTART;VALUE=DATE-TIME:20211210T153000Z
DTEND;VALUE=DATE-TIME:20211210T163000Z
DTSTAMP;VALUE=DATE-TIME:20240329T005814Z
UID:NonArchTrop/18
DESCRIPTION:Title: DT-invariants from non-archimedean integrals\nby Dimitri Wyss (É
cole Polytechnique Fédérale de Lausanne) as part of Non-Archimedean and
Tropical Geometry\n\n\nAbstract\nLet $M(\\beta\,\\chi)$ be the moduli spac
e of one-dimensional semi-stable sheaves on a del Pezzo surface $S$\, supp
orted on an ample curve class $\\beta$ and with Euler-characteristic $\\ch
i$.\n\nWorking over a non-archimedean local field $F$\, we define a natura
l measure on the $F$-points of $M(\\beta\,\\chi)$. We prove that the integ
ral of a certain gerbe on $M(\\beta\,\\chi)$ with respect to this measure
is independent of $\\chi$ if $S$ is toric. A recent result of Maulik-Shen
then implies that these integrals compute the Donaldson-Thomas invariants
of $M(\\beta\,\\chi)$. A similar result holds for suitably twisted Higgs b
undles. This is joint work with Francesca Carocci and Giulio Orecchia.\n
LOCATION:https://researchseminars.org/talk/NonArchTrop/18/
END:VEVENT
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