Seminars and Colloquia by Series

Where to place a hole to achieve fastest escape (What are the best sink and source in a network)

Series
School of Mathematics Colloquium
Time
Thursday, February 24, 2011 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Leonid Bunimovich – Georgia Institute of Technology
Consider any dynamical system with the phase space (set of all states) M. One gets an open dynamical system if M has a subset H (hole) such that any orbit escapes ("disappears") after hitting H. The question in the title naturally appears in dealing with some experiments in physics, in some problems in bioinformatics, in coding theory, etc. However this question was essentially ignored in the dynamical systems theory. It occurred that it has a simple and counter intuitive answer. It also brings about a new characterization of periodic orbits in chaotic dynamical systems. Besides, a duality with Dynamical Networks allows to introduce dynamical characterization of the nodes (or edges) of Networks, which complements such static characterizations as centrality, betweenness, etc. Surprisingly this approach allows to obtain new results about such classical objects as Markov chains and introduce a hierarchy in the set of their states in regard of their ability to absorb or transmit an "information". Most of the results come from a finding that one can make finite (rather than traditional large) time predictions on behavior of dynamical systems even if they do not contain any small parameter. It looks plausible that a variety of problems in dynamical systems, probability, coding, imaging ... could be attacked now. No preliminary knowledge is required. The talk will be accessible to students.

Evolution problem in General Relativity

Series
School of Mathematics Colloquium
Time
Thursday, January 27, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Igor Rodnianski – Princeton University
The talk will introduce basic mathematical concepts of General Relativity and review the progress, main challenges and open problems, viewed through the prism of the evolution problem. I will illustrate interaction of Geometry and PDE methods in the context of General Relativity on examples ranging from incompleteness theorems and formation of trapped surfaces to geometric properties of black holes and their stability.

An inverse problem arising in decoding of bar codes

Series
School of Mathematics Colloquium
Time
Thursday, January 20, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Fadil Santosa – University of Minnesota (Minneapolis)
Information encoded in a bar code can be read using a laser scanner or a camera-based scanner. For one-dimensional bar codes, which are in most prevalent use, the information that needs to be extracted are the widths of the black and white bars. The collection of black and white bars may be viewed as a binary one-dimensional image. The signal measured at the scanner amounts to the convolution of the binary image with a smoothing kernel. The challenge is that the smoothing kernel, in addition to the binary image, is also unknown. This presentation will review the technology behind bar code scanning and present several approaches to the decoding problem.

Convex Algebraic Geometry

Series
School of Mathematics Colloquium
Time
Thursday, November 11, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Bernd Sturmfels – UC Berkeley –
Convex algebraic geometry is an emerging field at the interface of convex optimizationand algebraic geometry. A primary focus lies on the mathematical underpinnings ofsemidefinite programming. This lecture offers a self-contained introduction. Startingwith elementary questions concerning multifocal ellipses in the plane, we move on todiscuss the geometry of spectrahedra and orbitopes, and we end with recent resultson the convex hull of a real algebraic variety.

Plank problems - the discrete geometric side

Series
School of Mathematics Colloquium
Time
Thursday, November 4, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Karoly Bezdek – University of Calgary –
In the 1930's, Tarski introduced his plank problem at a time when the field Discrete Geometry was about to born. It is quite remarkable that Tarski's question and its variants continue to generate interest in the geometric and analytic aspects of coverings by planks in the present time as well. The talk is of a survey type with some new results and with a list of open problems on the discrete geometric side of the plank problem.

Euler's pentagonal numbers theorem - refinements, variations and companions

Series
School of Mathematics Colloquium
Time
Thursday, October 28, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Krishnaswami Alladi – University of Florida –
Euler's celebrated pentagonal numbers theorem is one themost fundamental in the theory of partitions and q-hypergeometric series.The recurrence formula that it yields is what MacMahon used to compute atable of values of the partition function to verify the deep Hardy-Ramanujanformula. On seeing this table, Ramanujan wrote down his spectacular partition congruences. The author recently proved two new companions to Euler'stheorem in which the role of the pentagonal numbers is replaced by the squares.These companions are deeper in the sense that lacunarity can be achievedeven with the introduction of a parameter. One of these companions isdeduced from a partial theta identity in Ramanujan's Lost Notebook and theother from a q-hypergeometric identity of George Andrews. We will explainconnections between our companions and various classical results such asthe Jacobi triple product identity for theta functions and the partitiontheorems of Sylvester and Fine. The talk will be accessible to non-experts.

Group Dynamics in Phototaxis

Series
School of Mathematics Colloquium
Time
Tuesday, October 26, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Doron Levy – CSCAMM University of Maryland (College Park)
Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients, after which they swim or crawl into optimal conditions. This phenomenon is known as "chemotaxis" or "phototaxis." Using time-lapse video microscopy we have monitored the movement of phototactic bacteria, i.e., bacteria that move towards light. These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. Following these observations, in this talk we will present a hierarchy of mathematical models for phototaxis: a stochastic model, an interacting particle system, and a system of PDEs. We will discuss the models, their simulations, and our theorems that show how the system of PDEs can be considered as the limit dynamics of the particle system. Time-permitting, we will overview our recent results on particle, kinetic, and fluid models for phototaxis. This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute), Tiago Requeijo (Math, Stanford), and Seung-Yeal Ha (Seoul, Korea).

Generalized Borcherds Products and Two number theoretic applications

Series
School of Mathematics Colloquium
Time
Thursday, October 7, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Ken Ono – University of Wisconsin at Madison and Emory University –
n his 1994 ICM lecture, Borcherds famously introduced an entirely new conceptin the theory of modular forms. He established that modular forms with very specialdivisors can be explicitly constructed as infinite products. Motivated by problemsin geometry, number theorists recognized a need for an extension of this theory toinclude a richer class of automorphic form. In joint work with Bruinier, the speakerhas generalized Borcherds's construction to include modular forms whose divisors arethe twisted Heegner divisors introduced in the 1980s by Gross and Zagier in theircelebrated work on the Birch and Swinnerton-Dyer Conjecture. This generalization,which depends on the new theory of harmonic Maass forms, has many applications.The speaker will illustrate the utility of these products by resolving open problemson the following topics: 1) Parity of the partition function 2) Birch and Swinnerton-Dyer Conjecture and ranks of elliptic curves.

The Aleksandrov problem and optimal transport on $S^n$

Series
School of Mathematics Colloquium
Time
Thursday, September 2, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
249 Skiles
Speaker
Vladimir Oliker – Emory University –
The purpose of this talk is to describe a variational approach to the problemof A.D. Aleksandrov concerning existence and uniqueness of a closed convexhypersurface in Euclidean space $R^{n+1}, ~n \geq 2$ with prescribed integral Gauss curvature. It is shown that this problem in variational formulation is closely connected with theproblem of optimal transport on $S^n$ with a geometrically motivated cost function.

Noncommutative geometry and the field with one element

Series
School of Mathematics Colloquium
Time
Tuesday, April 27, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Matilde Marcolli – Caltech –
There are presently different approaches to definealgebraic geometry over the mysterious "field with one element".I will focus on two versions, one by Soule' and one by Borger,that appear to have a direct connection to NoncommutativeGeometry via the quantum statistical mechanics of Q-latticesand the theory of endomotives. I will also relate to endomotivesand Noncommutative Geometry the analytic geometry over F1,as defined by Manin in terms of the Habiro ring.

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