Department of

Mathematics


Seminar Calendar
for events the day of Thursday, April 23, 2015.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, April 23, 2015

11:00 am in 241 Altgeld Hall,Thursday, April 23, 2015

Random Quadratic Forms

Jayadev Athreya   [email] (UIUC Math)

Abstract: We give effective versions of the Quantitative Oppenheim results of Eskin-Margulis-Mozes on existence and counting of solutions of the form a < Q(x) < b, x \in Z^n, for random indefinite quadratic forms Q. This is joint work with G. Margulis.

11:00 am in 241 Altgeld Hall,Thursday, April 23, 2015

Quadratic Weyl sums, Automorphic Functions, and Invariance Principles

Francesco Cellarosi (UIUC Math)

Abstract: In 1914, Hardy and Littlewood published their celebrated approximate functional equation for quadratic Weyl sums (theta sums). Their result provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. We construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Joint work with Jens Marklof.

1:00 pm in Altgeld Hall,Thursday, April 23, 2015

Dynamical systems on dense graphs and graph limits

Georgi Medvedev (Drexel University, Math)

Abstract: The continuum limit is an approximate procedure, by which coupled dynamical systems on large graphs are replaced by an evolution integral equation on a continuous spatial domain. This approach has been useful for studying dynamics of diverse networks in physics and biology. We use the combination of ideas from the theories of graph limits and nonlinear evolution equations to develop a rigorous justification for using the continuum limit for dynamical models on deterministic and random graphs. As an application, we discuss stability of spatial patterns in the Kuramoto model on certain Cayley and random graphs.

3:00 pm in 243 Altgeld Hall,Thursday, April 23, 2015

Rees-Algebras and Implicitization

Eliana Duarte (UIUC Math)

Abstract: In the first part of this talk I will explain how to use approximation complexes to obtain the implicit equation of a parameterized hypersurface. The use of approximation complexes in this context is closely related to the problem of finding the defining equations of the Rees Algebra associated to the ideal generated by the defining polynomials of the parameterization. For the last part of the talk I will talk about ideals for which the defining equation of the Rees Algebras are known.

4:00 pm in 245 Altgeld Hall,Thursday, April 23, 2015

Discrete groups of affine transformations

Gregory Margulis (Yale University)

Abstract: About 40 years ago L. Auslander conjectured that if a discrete group H acts properly by affine transformations on R^n and R^n/H is compact then H is virtually polycyclic. I will talk about the current status of the Auslander conjecture. Another topic to be covered is affine proper actions without the assumption that the quotient space is compact. Video of lecture is at https://youtu.be/dByFQGMnwRQ