Departmental Colloquium
Fall 2013
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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September 26 |
Jonathan Rubin
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Baseline healthy activity in many brain regions is believed to be irregular, with little correlation in firing times of different neurons. On the other hand, excessive regularity has been associated with disorders such as Parkinson's disease. Mathematically, neuronal irregularity has been represented as an asynchronous state that can be stable under so-called balance conditions that yield cancellation of correlations. I will discuss a mechanism by which irregular, apparently chaotic activity emerges naturally in small networks, such as certain neuronal networks involved in parkinsonism, that lack such balance. Moreover, I will describe a one-dimensional map that captures this activity as well as analysis of how expansion, folding, and contraction emerge in a corresponding phase plane, as expected in chaotic dynamics. Once this irregularity is lost and parkinsonism arises, symptoms can be alleviated with implantation of an electrode to deliver periodic high frequency current pulses, known as deep brain stimulation (DBS). I will also present the mathematics of two possible mechanisms by which DBS may work, including a transformation from periodic stimulation to restored irregularity.
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October 10 |
Martin Deraux
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The theory of discrete groups has connections to many areas of mathematics (geometry, differential equations, number theory, dynamical systems, topology, representation theory). After an introduction to the subject based on simple examples, I will discuss recent developments on finding discrete groups with special arithmetic or geometric properties. Their construction makes use of sophisticated computational techniques and heavy computer power, thereby questioning us on the meaning of proof.
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November 7 |
Malgorzata Peszynska
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In various applications, the mathematical models based on traditional partial differential equations (PDEs), with their empirically determined coefficients, and associated traditional numerical approximations, are not sufficient to describe all the relevant dynamics and complexity. In the talk, we describe a few alternative, hybrid computational models, which combine traditional numerical PDEs with other computational methods from, e.g., statistical mechanics, and more broadly computational physics and geosciences. The new models account for complicated physics occuring at material interfaces in semiconductors, or rock-fluid boundaries in evolving porous media. Sometimes the interfaces are dynamically evolving, e.g., as in phase interfaces aarising due to phase transitions in methane hydrates, and sometimes they are artificial, such as in heterogeneous domain decomposition solvers for fluids. These new modeling approaches need to be analysed and validated. The former is the charge for computational mathematicians; we overview our recent analysis of these models and indicate the challenges, not the least of which is a substantial interdisciplinary nature of these projects.
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November 14 |
Dave Morris
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In combinatorial geometry (and engineering), it is important to know that certain scaffold-like geometric structures are rigid. (They will not collapse, and, in fact, have enough bracing that they cannot be deformed at all.) Replacing the geometric structure with an algebraic structure (namely, a group) leads to the following question: given a homomorphism that is defined on the elements of a subgroup, is it possible to extrapolate the homomorphism to the rest of the elements of the group? It is fairly obvious that every additive homomorphism from the group Z of integers to the real line R can be extended to a homomorphism that is defined on all of R, and we will see some other examples.
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December 5 |
Wei Ho
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The origins of "arithmetic invariant theory" come from the work of Gauss, who used integer binary quadratic forms to study ideal class groups of quadratic fields. The underlying philosophy---parametrizing arithmetic and geometric objects by orbits of group representations---has now been used to study higher degree number fields, curves, and higher-dimensional varieties. We will discuss some of these constructions and highlight the applications to topics such as bounding ranks of elliptic curves and dynamics on K3 surfaces. This talk is intended for a general mathematical audience.
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December 6 |
Bhargav Bhatt
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The cohomology groups of a complex algebraic manifold enjoy an extremely rich structure, traditionally the subject of Hodge theory. A key piece of this structure stems from Grothendieck's purely algebraic description of the de Rham cohomology of smooth varieties. But what about singular varieties? I will begin by reviewing the story for smooth varieties, and then explain how a modern perspective based on derived algebraic geometry provides an answer to the above question that was conjectured by Grothendieck.
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December 12 |
Tonghai Yang
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It was observed 90 years ago that the famous modular j -function has the following cool property j ( 1 + - 1 6 3 2 ) = - e 1 6 3 π + 7 4 4 + t i n y = - 2 1 8 3 3 5 3 2 3 3 2 9 3 --> is a huge integer with very small prime factor, and so is j ((1+√-163)/2)-1728. In the early 80s, Gross and Zagier proved that this is a general phenomenon-the difference of values of j at two quadratic integers, of discriminants a and b, has an explicit factorization formula with prime factors less than or equal to 4ab. It turns out that this phenomenon is much more general and works for a whole family of modular functions on a Shimura variety of orthogonal type (n, 2) and unitary type (n, 1). In this talk, we will give an informal account of this story.
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