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SUMMARY:Man Cheung Tsui (University of Pennsylvania)
DTSTART:20210129T154500Z
DTEND:20210129T164500Z
DTSTAMP:20260404T111448Z
UID:KolchinSeminar/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Kolch
 inSeminar/1/">Differential Essential Dimension</a>\nby Man Cheung Tsui (Un
 iversity of Pennsylvania) as part of Kolchin Seminar in Differential Algeb
 ra\n\n\nAbstract\nRoughly speaking\, the essential dimension of an algebra
 ic object counts the number of parameters needed to describe the object. I
 n this talk\, we define an analogue of essential dimension in differential
  Galois theory. As application\, we show that the number of coefficients i
 n a general homogeneous linear differential equation over a field cannot b
 e reduced via gauge transformations over the given field. We also give low
 er bounds on the number of parameters needed to write down certain generic
  Picard-Vessiot extensions.\n
LOCATION:https://stable.researchseminars.org/talk/KolchinSeminar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Khalil Ghorbal (INRIA)
DTSTART:20210212T154500Z
DTEND:20210212T164500Z
DTSTAMP:20260404T111448Z
UID:KolchinSeminar/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Kolch
 inSeminar/2/">Characterizing Positively Invariant Sets: Inductive and Topo
 logical Methods</a>\nby Khalil Ghorbal (INRIA) as part of Kolchin Seminar 
 in Differential Algebra\n\n\nAbstract\nSet positive invariance is an impor
 tant concept in the theory of dynamical systems and one which also has pra
 ctical applications in areas of computer science\, such as formal verifica
 tion\, as well as in control theory. Great progress has been made in under
 standing positively invariant sets in continuous dynamical systems and pow
 erful computational tools have been developed for reasoning about them\; h
 owever\, many of the insights from recent developments in this area have l
 argely remained folklore and are not elaborated in existing literature. Th
 is presentation contributes an explicit development of modern methods for 
 checking positively invariant sets of ordinary differential equations and 
 describes two possible characterizations of positive invariants: one based
  on the real induction principle\, and a novel alternative based on topolo
 gical notions. The two characterizations\, while in a certain sense equiva
 lent\, lead to two different decision procedures for checking whether a gi
 ven semi-algebraic set is positively invariant under the flow of a system 
 of polynomial ordinary differential equations.\n
LOCATION:https://stable.researchseminars.org/talk/KolchinSeminar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincenzo Mantova (University of Leeds)
DTSTART:20210219T154500Z
DTEND:20210219T164500Z
DTSTAMP:20260404T111448Z
UID:KolchinSeminar/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Kolch
 inSeminar/3/">Intersecting the graph of exp with complex algebraic varieti
 es</a>\nby Vincenzo Mantova (University of Leeds) as part of Kolchin Semin
 ar in Differential Algebra\n\n\nAbstract\nA conjecture of Zilber\, motivat
 ed by the model theory of complex exponentiation\, predicts the existence 
 of many intersections between the graph of the exponential function (exten
 ded pointwise to several variables) and algebraic varieties\, as long as t
 he varieties satisfy some geometric conditions related to Schanuel's conje
 cture. Some instances are known to be true\, most notably when the project
 ions of the varieties onto the domain side of the graph have maximal dimen
 sion\, i.e. equal to the number of variables (by work of Brownawell-Masser
  and D'Aquino-Fornasiero-Terzo).\n\nI will discuss the case of varieties w
 hose projection on the domain has dimension one\, that is\, it is a curve.
  Then such intersections always exist if the curve is not contained in a t
 ranslate of a rational hyperplane (and if it is\, a trivial geometric cond
 ition determines when there are no intersections). We can prove this by ap
 pealing to the classical theory of differentials on compact Riemann surfac
 es and a suitable instance of the Ax-Lindemann-Weierstrass theorem. This i
 s joint work with David Masser.\n
LOCATION:https://stable.researchseminars.org/talk/KolchinSeminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincent Bagayoko (École Polytechnique)
DTSTART:20210226T154500Z
DTEND:20210226T164500Z
DTSTAMP:20260404T111448Z
UID:KolchinSeminar/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Kolch
 inSeminar/4/">Three flavors of H-fields</a>\nby Vincent Bagayoko (École P
 olytechnique) as part of Kolchin Seminar in Differential Algebra\n\n\nAbst
 ract\nThe model theory of H-fields\, introduced by van den Dries and Asche
 nbrenner\, provides a general framework to study differential equations in
  ordered fields. This theory admits in particular geometric\, formal\, and
  number theoretic models: Hardy fields\, which are fields of differentiabl
 e real-valued germs at infinity\, transseries\, which are formal series in
 volving exponentials and logarithms\, and surreal numbers\, which are abst
 ract ordered quantities that can mimic both germs and transseries. I will 
 give an overview of the theory\, these different types of models and their
  connections.\n
LOCATION:https://stable.researchseminars.org/talk/KolchinSeminar/4/
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