Mathematics

Networks describe how things are connected, and are ubiquitous in science and society. Networks can be very concrete, like road networks connecting cities or networks of wires connecting computers. They can represent more abstract connections such as friendship on Facebook. Networks are widely used to model connections between things that have no real connections. For example, Biologists try to understand how cells work by studying networks connecting proteins that interact with each other, and Economists try to understand markets by studying networks connecting institutions that trade with each other.

Questions we ask about a network include "which components of the network are the most important?", "how well do things like information, cars, or disease spread through the network?", and "does the network have a governing structure?".

Professor Spielman will explain how mathematicians address these questions by modeling networks as physical objects, imagining that the connections are springs, electrical resistors, or pipes that carry fluid, and analyzing the resulting systems.

The classical theory of (phi, Gamma)-modules relates continuous p-adic representations of the Galois group of a p-adic field with modules over a certain mildly noncommutative ring. That ring admits a description in terms of a group algebra over Z_p which is crucial for Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We describe a method for applying a key property of perfectoid spaces, the analytic analogue of Drinfeld's lemma, to the construction of "multivariate (phi, Gamma)-modules" corresponding to p-adic Galois representations in more exotic ways. Based on joint work with Annie Carter and Gergely Zabradi.

Let E be an elliptic curve defined over a finite field with q elements. Hasse’s theorem says that #E(F_q) = q + 1 - t_E where |t_E| is at most twice the square root of q. Deuring uses the theory of complex multiplication to express the number of isomorphism classes of curves with a fixed value of t_E in terms of sums of ideal class numbers of orders in quadratic imaginary fields. Birch shows that as q goes to infinity the normalized values of these point counts converge to the Sato-Tate distribution by applying the Selberg Trace Formula.

In this talk we discuss finer counting questions for elliptic curves over a fixed finite field. We study the distribution of rational point counts for elliptic curves containing a specified subgroup, giving exact formulas for moments in terms of traces of Hecke operators. This leads to formulas for the expected value of the exponent of the group of rational points of an elliptic curve over F_q and for the probability that this group is cyclic. This is joint with work Ian Petrow (ETH Zurich).

Please see the event webpage for more information.

The talk will first present some classical results on the automorphisms of complex projective curves (or alternatively, of compact Riemann surfaces). We will then discuss the automorphism groups of projective algebraic varieties of higher dimensions; in particular, their "connected part" (which can be arbitrary) and their "discrete part" (of which little is known).

he Pennsylvania Department of Correction operates 29 correctional facilities (prisons) and about 50,000 prisoners (inmates) each year. The assignments of inmates to appropriate correctional facilities is a complex task. Well over 80 rules need to be considered. Many of them, such as the security of prison units, yield hard constraints; while others, such assigning the inmates to prisons close to their home, are arranged in a preference hierarchy because it is impossible to satisfy all for all inmates.

We are giving an overview of the complexity of the problem; discuss the data/rule collection phase of the project by using decision trees; discuss a how the MILO model is developed by using a weighted penalty objective function. Finally we discuss further extensions of the model, including waiting lists for mental/educational/job training programs, and transfers between facilities.
Based on joint work with L. Plebani, M. Shahabsafa, G. Wilson and K. Bucklen.