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DTSTART;TZID=America/Los_Angeles:20251007T121500
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SUMMARY:The integer point transform as a complete invariant (Sinai Robins\, University of São Paulo\, Brazil)
DESCRIPTION:Given any finite set of integer points S\, there is an associated function f_S that encodes S\, which we call its integer point transform.   One can think of this integer point transform f_S algebraically or analytically.  Here we focus on its analytic properties\, showing that it is a complete invariant.   In fact\, we prove that it is only necessary to evaluate f_S at one algebraic point in order to uniquely determine the finite set S\, by employing the Lindemann-Weierstrass theorem.    Similarly\, we prove that it’s only necessary to evaluate the Fourier transform of a rational polytope P (as well as rational cones) at a single algebraic point\, in order to uniquely determine S.   Finally\, by relating the integer point transform to finite Fourier transforms\, we show that a finite number of integer point evaluations of f_S suffice in order to uniquely determine S.  
URL:https://colleges.claremont.edu/ccms/event/antc-talk-sinai-robins-university-of-sao-paulo-brazil/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20251021T121500
DTEND;TZID=America/Los_Angeles:20251021T131000
DTSTAMP:20260430T132406
CREATED:20250807T222137Z
LAST-MODIFIED:20251007T213217Z
UID:3777-1761048900-1761052200@colleges.claremont.edu
SUMMARY:Singularities in characteristic p and the Riemann–Hilbert correspondence (Robert Cass\, CMC)
DESCRIPTION:The Riemann–Hilbert correspondence relates algebra to differential equations on complex algebraic varieties. In characteristic p\, there is an analogous correspondence due to Emerton–Kisin and later generalized by Bhatt–Lurie\, where the derivative operator is replaced by the p-th power Frobenius operator. In this talk we will explain a relation between the mod p Riemann–Hilbert correspondence and the study of singularities of algebraic varieties in characteristic p. This talk is mostly about commutative algebra\, and we will introduce concepts such as local cohomology and perverse sheaves along the way. This is joint work with João Lourenço.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-robert-cass-cmc/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20251028T121500
DTEND;TZID=America/Los_Angeles:20251028T131000
DTSTAMP:20260430T132406
CREATED:20250813T050114Z
LAST-MODIFIED:20251023T042930Z
UID:3784-1761653700-1761657000@colleges.claremont.edu
SUMMARY:From sparsity of rational points on curves to the generic positivity of Beilinson-Bloch height (Ziyang Gao\, UCLA)
DESCRIPTION:It is a fundamental question to find rational solutions to a given system of polynomials\, and in modern language this translates into finding rational points in algebraic varieties.  It is already very deep for algebraic curves defined over Q.  An intrinsic natural number associated with the curve\, called its genus\, plays an important role in studying rational points on curves.  In 1983\, Faltings proved the famous Mordell Conjecture (proposed in 1922)\, which asserts that any curve of genus at least 2 has only finitely many rational points.  Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness\, bound\, uniform bound\, effectiveness.  An answer to each grade requires a better understanding of the distribution of the rational points.\n\nIn my talk\, I will explain the historical and recent developments of this problem according to the different grades.  I will also mention a recent work (joint with Shouwu Zhang) about a generic positivity property and a Northcott property of the Beilison-Bloch height of the Gross-Schoen cycles and the Ceresa cycles.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-ziyang-gao-ucla/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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