Time & location: All talks are on Thursday in Gibson 414 at 3:30pm unless otherwise noted. Refreshments in Gibson 426 after the talk.
Comments indicating vacations, special lectures, or change in location or time are made in green.
Organizer: Gustavo Didier
Joel OuaknineOxford University
Given a linear recurrence sequence over the reals, the Positivity Problem asks whether the terms of the sequence are all positive. Positivity (and closely related variants) have applications in many different areas, such as theoretical biology (analysis of L-systems, population dynamics), software verification (termination of linear programs), probabilistic model checking (reachability in Markov chains, stochastic logics), quantum computing (threshold problems for quantum automata), as well as combinatorics, term rewriting, and the study of generating functions.
The decidability of Positivity (for integer, rational, or algebraic sequences) is a long-standing open problem (mentioned, for example, in [Salomaa 1976], but probably going back much earlier). Until now, the only known result was that for integer linear recurrence sequences of order 2, Positivity is decidable [Halava 2007]. Our main results are as follows:
(i) Positivity is decidable for algebraic linear recurrence sequences of order 5 or less;
(ii) When the companion matrix of the linear recurrence sequence is diagonalisable, it becomespossible to decide Positivity for sequences of order 8 or less;
(iii) Extending (i) even to integer linear recurrence sequences of order 6 would entail major breakthroughs in the field of Diophantine approximation of transcendental numbers. The present results therefore appear unlikely to be improved in the absence of significant advances in number theory.
Finally, in terms of complexity, we show that our decision procedures lie within the counting hierarchy.
In this talk, I will introduce the main results, discuss their significance, and present some of the key ideas appearing in the proofs.
(This is joint work with James Worrell and Matt Daws.)
LOCATION: Gibson Hall 325
TIME: 2:00 PM
Alexander Ihleruniversity of california-irvine
Abstract: Message Passing for Estimation and Decisions in Graphical Models
Graphical models have become a key tool for modeling and reasoning about complex, large-scale systems, including communications, computer vision, and computational biology. Significant progress has been made in developing approximate methods, such as the family of belief propagation algorithms, for basic reasoning tasks such as summation (computing marginal probabilities) and MAP estimation (finding the most likely configuration). However, many useful reasoning problems are much more complex, including optimal estimation of a part of the model ("marginal MAP"), and single or multi-agent decision-making problems. We describe a new variational framework capturing these more general inference problems, in which analogues of the Bethe, tree-reweighted, and mean field approximations can be applied, resulting in new message-passing approximations with strong theoretical properties on these harder tasks.
About the Speaker:
Alexander Ihler is an Assistant Professor in the Department of Computer Science at the University of California, Irvine. He received his Ph.D. in Electrical Engineering and Computer Science from MIT in 2005 and a B.S. with honors from Caltech in 1998. His research focuses on machine learning, graphical models, and algorithms for exact and approximate inference, with applications to areas including sensor networks, computer vision, data mining, and computational biology. His papers have received a number of awards, at conferences including NIPS, IPSN, and AIStats.
Location: Gibson 414
Time: 3:30 PM
David VoganMassachusetts institute of technology
Gelfand's program of abstract harmonic analysis seeks to understand a manifold X with an action of a Lie group G in three steps:
1) understand all irreducible unitary representations of G;
2) construct the Hilbert space L^2(X);
3) decompose L^2(X) as a "direct integral" of irreducible unitary representations.
4) relate (hard) problems about X to L^2(X), and so (via (3)) to (easier) problems about irreducible unitary representations.
We are interested in two (very old) technical problems arising in Gelfand's program. Suppose for instance that we wish to understand a G-invariant differential operator \Delta_X. The first problem is that the eigenfunctions of \Delta_X may not be square-integrable, so relating them to L^2(X) (step (4)) is subtle.
The second problem appears already in step (2), the construction of irreducible unitary representations. Natural candidates for such representations are simultaneous eigenspaces for G-invariant differential operators on X; but if (as very often happens) the eigenfunctions are not square-integrable, then it is not easy in this way to find natural Hilbert space representations.
The general solution we propose is dualization: to replace kernels of differential operators acting on distributions by cokernels of differential operators acting on compactly supported densities.
I'll describe these problems and their resolution in some very familiar examples related to SL(2,R) and one complex variable.
(Joint with David Jerison)
Marty Golubitskyohio state university
A coupled cell system is a network of interacting dynamical systems. Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can emerge. This talk will discuss a result (motivated by quadruped locomotion) that classifies rigid patterns of phase-shift synchrony in time-periodic solutions of coupled cell systems and a recent application to binocular rivalry.
Mathew RogersUniversity of montreal
Some of Ramanujan's most important accomplishments are his formulas for $\pi$ and the evaluation of the Roger-Ramanujan continued fraction. I will give an overview of how modular functions provide the basis for understanding Ramanujan's work. Then I will discuss the relevance of modularity in solving modern problems such as the Bloch-Beilinson conjectures, and also for understanding various sums and integrals from statistical mechanics.
Chris Bailey-KelloggDartmouth college
The explosive growth of biotherapeutic agents is revolutionizing treatment of numerous diseases, but innovations in biotherapies have also created new challenges for drug design and development. One distinguishing risk factor of therapeutic proteins is the prospect of eliciting an immune response in humans. To meet this challenge, we have developed optimization algorithms that minimize a protein’s T cell epitope content while simultaneously ensuring that the engineered variant maintains a high level of stability and activity. Immunogenicity is assessed via T cell epitope predictors that score peptide binding potential against class II MHC molecules. The structural and functional consequences of deimmunizing mutations are evaluated with statistical sequence potentials and molecular mechanics force fields. Our algorithms then map the Pareto frontier of designs balancing both criteria. The application of these algorithms will be highlighted through comparative analysis with previously published deimmunization efforts as well as our collaborative experimental validation using beta-lactamase, a model therapeutic candidate with utility in ADEPT cancer therapies.
About the Speaker:
Chris Bailey-Kellogg is an associate professor of computer science at Dartmouth College. His research focuses on developing and applying integrated computational-experimental methods to study proteins and to engineer them, gaining insights into protein sequence-structure-function relationships and using those insights to design improved variants.
John FricksPenn State university
Molecular motors, such as kinesin and dynein, carry cargo through a cell along a microtubule network. The heads of these motors step along a microtubule and are on the order of nanometers, while the cargo size and the distance traveled can be on the order of hundreds of nanometers. Stochastic models of motors that describe and bridge these spatial scales will be discussed along with how data can be incorporated into these models at the various scales. A stochastic model for variable-length stepping of kinesins engineered with extended neck linkers was developed. This requires consideration of the separation in microtubule binding sites between the heads of the motor at the beginning of a step. The separation is a stationary process and can be included in the calculation of standard experimental quantities though a semi-Markov framework by using weak convergence results from stochastic processes theory. In order to better understand how both single and multiple motors can influence the motion of a cargo, a stochastic model for microtubule-motor-cargo dynamics is presented that emphasizes the spatial configuration and the resultant distribution of forces generated on and by the cargo. Using stochastic averaging, a simplified model parameterized with in vitro data can be studied to gain insight into the in vivo behavior of the motor-cargo system.Location: Norman Mayer 106
Time: 3:30 PM
Mark M. MeerschaertMichigan State University
Fractional derivatives were invented by Leibnitz soon after their integer-ordercousins. They are nonlocal operators. Fractional differential equations have recently become popular in applications, especially for anomalous diffusion. We will review the basic notions of fractional calculus, fractional diffusion equations, and connections to random walk models in probability.
Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 firstname.lastname@example.org