UIUC Dept. of Mathematics Seminar Calendar

      July 2012             August 2012           September 2012   
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
  1  2  3  4  5  6  7             1  2  3  4                      1
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    2  3  4  5  6  7  8
 15 16 17 18 19 20 21   12 13 14 15 16 17 18    9 10 11 12 13 14 15
 22 23 24 25 26 27 28   19 20 21 22 23 24 25   16 17 18 19 20 21 22
 29 30 31               26 27 28 29 30 31      23 24 25 26 27 28 29
                                               30

Wednesday, February 1, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Wednesday, February 1, 2012

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Joseph Rosenblatt (Department of Mathematics, University of Illinois)
Averaging over Rectangles in Euclidean Spaces

Wednesday, February 8, 2012

Wednesday, February 15, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Wednesday, February 15, 2012

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Peter Loeb (Department of Mathematics, University of Illinois)
Infinitesimals in Analysis and Probability Theory
Abstract: The notion of an infinitesimal quantity has been used in mathematics for over 2200 years. It eluded rigorous treatment until the work using model theory of Abraham Robinson in 1960 established a rigorous foundation for the use of infinitesimals in mathematics. Recent extensions and applications of his theory, now called nonstandard analysis, have produced new results in many areas including operator theory, stochastic processes, mathematical economics and mathematical physics. In all of these areas, infinitely small and infinitely large quantities can play an essential role in the creative process. At the level of calculus, the integral can now be correctly defined as the nearest ordinary number to an infinitely large sum of infinitesimal quantities. In Probability theory, Brownian motion can now be rigorously parameterized by a random walk with infinitesimal increments. In economics, an ideal economy can be formed from an infinite number of agents each having an infinitesimal influence on the economy. Spaces formed with nonstandard analysis give the simplest probability spaces for a continuum of independent random variables or traders in an economy. The talk gives an introduction to this fruitful area of mathematics.

Wednesday, February 29, 2012

Wednesday, March 14, 2012

Wednesday, April 11, 2012

Wednesday, April 18, 2012

Wednesday, April 25, 2012

Wednesday, May 2, 2012

Wednesday, August 29, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Wednesday, August 29, 2012

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Richard Laugesen [email] (UIUC Math)
Eigenvalues of the Laplacian
Abstract: The fundamental processes of classical physics are diffusion and wave motion. Quantum mechanics too is a kind of wave phenomenon. Mathematically, these processes can all be decomposed in terms of eigenfunctions (states) and eigenvalues (frequencies) of the Laplacian (the sum of pure second partial derivatives). What do we know about those eigenvalues? What would we like to know? I will concentrate on isoperimetric problems that stretch back to Queen Dido in antiquity and Lord Rayleigh in the 19th century, and continue into the future.

Wednesday, September 5, 2012

Wednesday, September 12, 2012

Wednesday, September 19, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Wednesday, September 19, 2012

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Alexandr Kostochka [email] (UIUC Math)
On k-color-critical n-vertex graphs with fewest edges
Abstract: A graph $G$ is $k$-{\em critical} if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. We give a lower bound, $f_k(n) \geq F(k,n)$, that is sharp for every $n=1\,({\rm mod }\,$ It is also sharp for $k=4$ and every $n\geq 6$. The result improves the classical bounds by Gallai and Dirac and some subsequent bounds. It establishes the asymptotics of $f_k(n)$ for every fixed $k$. It also proves that the conjecture by Ore from 1967 that for every $k\geq 4$ and $n\geq k+2$, $f_k(n+k-1)=f(n)+\frac{k-1}{2}(k - \frac{2}{k-1})$ holds for each $k\geq 4$ for all but at most $k^3/12$ values of $n$. The result has interesting applications. One of the corollaries of the theorem is a half-page proof of the theorem due to Gr\" otzsch that every triangle-free planar graph is $3$-colorable.

Wednesday, September 26, 2012

Wednesday, October 3, 2012

Wednesday, October 10, 2012

Wednesday, October 24, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Wednesday, October 24, 2012

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Ilya Kapovich [email] (UIUC Math)
Groups as geometric objects
Abstract: Geometric group theory is a vibrant and rapidly developing area of mathematics that lies at the juncture of group theory, low-dimensional topology, differential geometry and several other subjects. A crucial idea in the area is to view a finitely generated group as a geometric and not just as an algebraic object. One of the key tools for realizing this goal is the notion of the Cayley graph of a group, which is a metric space associated to a finitely generated group together with a finite generating set. Geometric group theory studies the connections between large-scale geometric properties of groups on one side and their algebraic and algorithmic properties on the other side. In this introductory talk we will explore the basic ideas and notions of the subject and demonstrate how the above connections manifest themselves in a number of representative results.

Wednesday, October 31, 2012

Wednesday, November 7, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Wednesday, November 7, 2012

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Kay Kirkpatrick (UIUC Math)
A random walk through statistical mechanics: scaling limits and supercool phenomena
Abstract: One of the main goals in statistical mechanics is to derive a macroscopic description of matter from the microscopic interacting particles via scaling limits. I will discuss recent advances on a supercooled phase of matter called Bose-Einstein condensation, in which a gas of quantum particles condenses and behaves like a giant quantum particle. I'll also discuss some challenges for understanding superconductors mathematically, and I'll mention related work in progress.

Wednesday, November 14, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Wednesday, November 14, 2012

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Renming Song [email] (UIUC Math)
An Introduction to Subordinate Brownian Motions
Abstract: A Levy process is a process with stationary and indpendent increments. Levy processes have been widely used in various fields. However, the class of Levy processes is very large and not very tractable. A subordinate Brownian motion is a Levy process that can be obtained by replacing the time parameter of a Brownian motion by an increasing Levy process (i.e.,subordinator). The subordinator can be thought of as the "operational time" or "intrinsic time". Subordinate Brownian motions form a large sublclass of Levy processes and yet they much more tractable than general Levy processes. In this talk, I will give a brief introduction to subordinate Brownian motions.

Wednesday, November 28, 2012

Wednesday, December 5, 2012

Wednesday, December 12, 2012