Past Seminars- Algebra

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  • Bryden Cais
    Thu Dec 13, 2012
    3:00 pm
    What is the probability that a random abelian variety over F_q is ordinary? Using (semi)linear algebra, we will answer an analogue of this question, and explain how our method can be used to answer similar statistical questions about p-rank and a-number. The answers are perhaps surprising, and deviate from what one might expect via naive reasoning...
  • Andrei Jorza
    Thu Nov 15, 2012
    3:00 pm
    To a Hilbert modular form one may attach a p-adic analytic L-function interpolating certain special values of the usual L-function. Conjectures in the style of Mazur, Tate and Teitelbaum prescribe the order of vanishing and first Taylor coefficient of such p-adic L-functions, the first coefficient being controlled by an L-invariant which has...
  • Daniel Litt
    Thu Nov 8, 2012
    3:00 pm
    There are beautiful and unexpected connections between algebraic topology, number theory, and algebraic geometry, arising from the study of the configuration space of (not necessarily distinct) points on a variety. In particular, there is a relationship between the Dold-Thom theorem, the analytic class number formula, and the "motivic...
  • Shahed Sharif
    Thu May 31, 2012
    3:00 pm
    Let $C$ be a curve over a local field whose reduction is totally degenerate. We discuss the related problems of 1) determining the group structure of the torsion subgroup of the Jacobian of $C$, and 2) determining if a given line bundle on $C$ is divisible by a given integer $r$. Under certain hypotheses on the reduction of $C$, we exhibit...
  • Mitya Boyarchenko
    Tue May 8, 2012
    3:00 pm
    In the early 1970s Drinfeld introduced a family of rigid analytic spaces parameterizing deformations of certain formal groups with level structure. This family is called the Lubin-Tate tower. He found an open affinoid in the first level of this tower whose reduction is isomorphic to what is now known as a Deligne-Lusztig variety for GL_n over...
  • Maurice Rojas
    Thu Apr 12, 2012
    3:00 pm
    We show how to efficiently count exactly the number of solutions of a system of n polynomials in n variables over certain local fields L, for a new class of polynomials systems. The fields we handle include the reals and the p-adic rationals. The polynomial systems amenable to our methods are made up of certain A-discriminant chambers, and our...
  • Vladimir Baranovsky
    Thu Dec 1, 2011
    1:00 pm
    A very useful technique in homological algebra, is the Basic Peturbation Lemma which tells how cohomology of a complex changes when a differential is perturbed. It has numerous applications in algebra, topology, and computational methods in algebra; some of which will be reviewed in the talk.