UIUC Dept. of Mathematics Seminar Calendar

      July 2009             August 2009           September 2009   
 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                      1          1  2  3  4  5
  5  6  7  8  9 10 11    2  3  4  5  6  7  8    6  7  8  9 10 11 12
 12 13 14 15 16 17 18    9 10 11 12 13 14 15   13 14 15 16 17 18 19
 19 20 21 22 23 24 25   16 17 18 19 20 21 22   20 21 22 23 24 25 26
 26 27 28 29 30 31      23 24 25 26 27 28 29   27 28 29 30         
                        30 31

Tuesday, August 25, 2009

Number Theory Seminar
1:00 pm Tuesday, August 25, 2009

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Submitted by ahlgren.

No talks this week.
Abstract: The Seminar will begin on Tuesday, September 1. Contact Scott Ahlgren or Jeremy Rouse to volunteer a talk.

Thursday, August 27, 2009

Friday, August 28, 2009

Model Theory and Descriptive Set Theory
4:00 pm in 345 Altgeld Hall, Friday, August 28, 2009

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Submitted by w-henson.

van den Dries / Henson / Solecki (UIUC Math)
Sums and Products -- organizational meeting
Abstract: This semester we will focus our Friday logic seminars on a new development of general mathematical interest, involving descriptive set theory, combinatorics, number theory, and model theory. At this organizational meeting, a plan for the semester will be worked out. The first paper we'll read is "Borel subrings of the reals" by G.A. Edgar and Chris Miller. A main result of this paper is that if E is a proper subring of the field of real numbers that is also a Borel set, then the Hausdorff dimension of E is 0. Indeed, they show that for all n, the set E^n has Hausdorff dimension 0 as a subset of R^n. Further work to which our seminar will aim involves papers by Jean Bourgain, Terry Tao, and others, and especially a preprint by Udi Hrushovski. All interested people are warmly invited to participate.

Tuesday, September 1, 2009

Wednesday, September 2, 2009

Special Topology, Group Theory, and Number Theory Seminar
3:00 pm in 345 Altgeld Hall, Wednesday, September 2, 2009

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Submitted by nmd.

Jordan Ellenberg (University of Wisconsin, Madison)
Congruence subgroups of mapping class groups (especially braid groups.)
Abstract: Abstract: There is a notion of "congruence subgroup" of a mapping class group which appears to have certain properties in common with congruence subgroups of arithmetic groups. We will discuss several open problems in the theory of congruence subgroups, some progress, and applications: for instance, the proof of a "strong congruence subgroup property" for the mapping class group of a once-punctured elliptic curve answers an open question from dynamics about Teichmuller curves in moduli spaces of curves (joint work with D.B. McReynolds). In number theory, theorems about stabilization of cohomology of congruence subgroups yield results towards the Cohen-Lenstra heuristics for ideal class groups over function fields over finite fields. (joint work with A. Venkatesh, C. Westerland.)

Thursday, September 3, 2009

Tuesday, September 8, 2009

Thursday, September 10, 2009

Friday, September 11, 2009

Tuesday, September 15, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, September 15, 2009

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Submitted by ahlgren.

Harold Diamond (UIUC Math)
A biased coin calculation with a biased number theoretic outcome
Abstract: The topic of the talk is the probability that one will be ahead after n tosses of a coin whose probability of heads is 3/5 and that of tails is 2/5. To compensate, each occurrence of heads wins $2 and each tails costs $3. While the expectation is 0, it turns out that there are some interesting patterns; there is a (small) probability of being ahead after n tosses if n is congruent to 1, 3, or 5 mod 5, and being behind in the other cases, and also there is an allover bias toward being ahead. The methods being used involve generating functions, a little Fourier series, and a saddle point type calculation. The details will be given in reasonable detail.

Thursday, September 17, 2009

Tuesday, September 22, 2009

Thursday, September 24, 2009

Tuesday, September 29, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, September 29, 2009

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Submitted by jarouse.

Kenneth Stolarsky (Department of Mathematics, University of Illinois)
Zeros of Polynomials in Analysis and Number Theory
Abstract: If an analytic (esp. meromorphic) function is "well-approximated" by a polynomial near the origin, we may expect its zeros to be "well-approximated" by the zeros of that polynomial near the origin. We also generally expect this to break down outside of the vicinity of the origin. But what are the transitional details? Robert Jentzsch lived just long enough to start a chain reaction of surprising discoveries starting from this question. Rational approximation theory, statistical functions, the Riemann zeta function and the number Pi*e have all shown up as actors on this stage. We shall mostly survey, but hope to sketch a few proofs. The only prerequisite is some basic complex analysis.

Thursday, October 1, 2009

Tuesday, October 6, 2009

Thursday, October 8, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Thursday, October 8, 2009

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Submitted by jarouse.

Bruce Berndt (UIUC Math)
Ramanujan's Lost Notebook
Abstract: In the spring of 1976, while searching through papers of the late G. N. Watson at Trinity College, Cambridge, George Andrews found a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan, generally regarded as India's greatest mathematician. In view of the fame of Ramanujan's earlier notebooks, Andrews naturally called these papers Ramanujan's "lost notebook." This work, comprising about 650 results with no proofs, arises from the last year of Ramanujan's life and represents some of his deepest work. First, we provide a history of the lost notebook. Second, a general description of the topics found in the lost notebook will be provided. For some of the topics, such as ranks and cranks of partitions, we offer some details. The third portion of the lecture will be devoted to a more detailed discussion of one of the topics prominently addressed in the lost notebook, namely continued fractions.

Tuesday, October 13, 2009

Thursday, October 15, 2009

Monday, October 19, 2009

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Monday, October 19, 2009

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Submitted by seminar.

Kenneth Stolarsky (Department of Mathematics, University of Illinois)
Polynomials in Number Theory and Analysis
Abstract: In algebra and analysis one needs eigenvalues of matrices, and in number theory one needs the set of all algebraic numbers, so roots of polynomials, with or without integer coefficients, are of central importance. There is much more to the study of polynomials than the fundamental theorem of algebra. Various easily stated problems (even for polynomials with coefficients not constrained to be integers) remain open. In particular, there are many ways to measure the size of a polynomial, and these lead to interesting results and open problems about the largest or smallest polynomial satisfying a given condition. Polynomials can enter into problems whose formulation does not mention polynomials. One example is finding universally optimal distributions of points on spheres. Another is the open problem of determining the mod 1 distribution of x^n for a given x > 1 (e.g. x = 3/2). This has led to the concept of a PV number.

Tuesday, October 20, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, October 20, 2009

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Submitted by jarouse.

Bruce Reznick (UIUC Math)
Sums of two cubes
Abstract: This will include a discussion of some very old formulas of Viete (1591) and Euler-Binet on equal sums of two cubes of rational numbers, polynomials and rational functions over C. A simple necessary and sufficient condition will be given for a polynomial to be a sum of two cubes of polynomials; for example, xy(x^4-y^4) has six essentially different representations as a sum of two cubes of quadratic forms and x^6 + y^6 has four such representations. There are some analogues to the addition of points on elliptic curves. The speaker will present a lot of questions which he cannot answer.

Thursday, October 22, 2009

Tuesday, October 27, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, October 27, 2009

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Submitted by ahlgren.

Harold Diamond (UIUC Math)
Weak logarithmic density estimates for Beurling generalized numbers
Abstract: An infinite collection of real numbers $p_1, p_2, ... $ satisfying $1 < p_1 \le p_2 \le ...$ is called a sequence $P$ of Beurling generalized (g-) primes, and the semigroup $N_P$ it generates under multiplication is called the associated Beurling g-integers. We say that $N_P$ satisfies an upper (resp. lower) logarithmic density condition if $(1/\log x) \sum_{n_i \le x} 1/n_i$ is bounded above (resp. below). We show that these conditions are connected with the behavior of the g-zeta function near its pole. The case of logarithmic density is the famous tauberian theorem of Hardy, Littlewood, and Karamata.

Thursday, October 29, 2009

Tuesday, November 3, 2009

Thursday, November 5, 2009

Tuesday, November 10, 2009

Thursday, November 12, 2009

Tuesday, November 17, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, November 17, 2009

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Submitted by jarouse.

Youness Lamzouri (The Institute for Advanced Study)
On the Distribution of large values of L-functions at the edge of the critical strip
Abstract: In this talk we will construct a class of probabilistic random Euler products to study large values of various families of $L$-functions at the edge of the critical strip. In particular this class includes the random models constructed recently by A. Granville and K. Soundararajan to study large values of the Riemann zeta function and Dirichlet $L$-functions on the $1$-line. Among new applications, we study families of symmetric power $L$-functions of holomorphic cusp forms in the level aspect (assuming the automorphy of these $L$-functions) at $s=1$, functions in the Selberg class (in the height aspect), and the family of $L$-functions of quadratic twists of a fixed $GL(m)/{\Bbb Q}$-automorphic cusp form at $s=1$.

Thursday, November 19, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Thursday, November 19, 2009

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Submitted by jarouse.

Andrew Shallue (Illinois Wesleyan)
Enumerating class group structures of real quadratic number fields
Abstract: There are many classic conjectures surrounding the structure of the ideal class group of real quadratic number fields. Among them are the Cohen-Lenstra heuristics, which give precise information about the expected structure of such class groups. Supporting such conjectures provides motivation to tabulate class numbers of quadratic number fields. I will discuss the present progress of a project to enumerate the structure of all class groups for discriminants of real quadratic number fields up to 10^11, and to do so without relying on the Extended Riemann Hypothesis. In essence, this has two main components. First, there are algorithms for computing the structure of an abelian group given group operations as a black box. Second, these group operations must be instantiated. This is nontrivial in the case of the ideal class group of a real quadratic number field. In addition to first computing the regulator, elements of the group do not have a unique representative, making identity testing a difficult proposition. We will see how these difficulties are overcome, and discuss exciting new developments in algorithms for finding the structure of a generic group.

Tuesday, December 1, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, December 1, 2009

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Submitted by jarouse.

Melissa Dennison (UIUC Math)
A Sequence Related to the Stern Sequence
Abstract: The general bow sequence is defined by b(0)=0, b(1)=alpha, b(2)=beta, b(2n)=b(n)+b(n+1) for n>1, b(2n+1)=b(n) for n>0. This recursion is similar to, but the reverse of the Stern sequence. A brief introduction to the bow sequences and their relation to the Stern sequence will be given, along with a discussion of several main properties of the bow sequences. The generating functions will be derived and a complete description of the bow sequences modulo 2 will be presented, along with several open questions. This seminar is part of my dissertation research.

Thursday, December 3, 2009

Friday, December 4, 2009

Graduate Analysis Seminar
4:00 pm in 341 Altgeld Hall, Friday, December 4, 2009

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Submitted by wgreen4.

Johann Thiel (UIUC Math)
The Gaussian Moat Problem
Abstract: A Gaussian integer is any complex number a+ıb with a and b integers. A Gaussian prime is any irreducible Gaussian integer. If we think of the Gaussian integers as a lattice in the complex plane, the Gaussian moat problem asks whether one can start at the origin and walk out to inﬁnity on Gaussian primes taking steps of bounded length. This question is still unresolved, and it is conjectured that no bounded step length will work. We will survey known results, including the largest known bounded step size that does not work, as well as a probabilistic model that suggests the smallest (unbounded) step size needed to be able to walk to inﬁnity along Gaussian primes. In this talk we will use tools from analytic number theory, algebraic number theory, combinatorics and probability.

Tuesday, December 8, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, December 8, 2009

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Submitted by ahlgren.

Jaebum Sohn (Yonsei University)
1 mod k lecture hall partitions
Abstract: A Lecture Hall Partition of length $n$ is a sequence $(b_1,b_2,\dots,b_n)$ of nonnegative integers satisfying $\frac{{b_1 }}{n}\ge \frac{{b_2 }}{{n - 1}} \ge\cdots\ge \frac{{b_n }}{1}\ge0$. M. Bousquet-M\'{e}lou and K. Eriksson showed that there is an one to one correspondence between the set of all lecture hall partitions of length $n$ and the set of all partitions of $N$ into $n$ odd parts less than $2n.$ G. E. Andrews also proved this result by using MacMahon's $\Omega$ operator. In this talk, we compute a generating function of the $1$ mod $k$ Lecture Hall Partition of length $n$ satisfying the condition $\frac{{b_1 }}{1+(n-1)k}\ge \frac{{b_2 }}{{1+(n-2)k}} \ge\cdots\ge \frac{{b_n }}{1}\ge0$. We utilize MacMahon's $\Omega$ operator to prove our result and then give some partition interpretation.

Friday, January 22, 2010

Mathematics Colloquium - Special Lecture 2009-10
4:00 pm in 245 Altgeld Hall, Friday, January 22, 2010

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Submitted by seminar.

Kay Kirkpatrick (NYU, Courant Institute)
Bose-Einstein condensation: from many quantum particles to a quantum "super-particle."
Abstract: Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation, that behaves like a giant quantum particle. I will discuss results that provide the rigorous connection between the physics of the microscopic dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). In joint work with Benjamin Schlein and Gigliola Staffilani, we have handled two-dimensional Bose-Einstein condensation--and the periodic case is especially interesting, because of techniques from analytic number theory and applications to quantum computing. Time permitting I'll also mention work in progress on large deviations for quantum many-body systems, and phase transitions for the invariant measures of the NLS.

Please join us for refreshments at 3:30 p.m. in the Common Room, 321 Altgeld Hall.

Monday, January 25, 2010

Mathematics Colloquium - Special Lecture 2009-10
4:00 pm in 245 Altgeld Hall, Monday, January 25, 2010

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Florian Herzig (Northwestern University)
Modular representations of p-adic groups
Abstract: The Langlands program relates complex representations of GL_n(Q_p) to Galois representations. For n = 1 this is explained by class field theory and for n = 2 this is closely related to the theory of modular forms. For general n, this is now understood by the work of Harris-Taylor and Henniart. In the last decade, a mod-p (as well as a p-adic) version of the Langlands program have been emerging, and they have already played an important role in some recent progress in number theory. But so far understanding has been limited to n = 1 and 2. We survey some of the known story in the classical and in the mod p case, and then discuss some recent progress on the classification of mod p representations of GL_n(Q_p), as time permits.

Please join us for refreshments at 3:30 p.m. in the Common Room, 321 Altgeld Hall.

Tuesday, January 26, 2010

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, January 26, 2010

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Submitted by jarouse.

Dominic Klyve (Carthage College)
Twin Primes and an Upper Bound on Brun's Constant
Abstract: Determining the number of twin primes is one of the most easily-understood open questions in mathematics, and one of the hardest to answer. We shall approach this question via the related question: What is the sum of the reciprocals of the twin primes? This sum, which has been known to converge for almost a century, has been bounded explicitly only recently. In this talk, we will see how these explicit bounds are found, and demonstrate how they can be improved under the Extended Riemann Hypothesis. A good time will be had by all.

Mathematics Colloquium - Special Lecture 2009-10
4:00 pm in 245 Altgeld Hall, Tuesday, January 26, 2010

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Submitted by seminar.

Sam Payne (Stanford University and Clay Mathematics Institute)
Nonarchimedean algebraic geometry
Abstract: The usual norm on the complex numbers and its associated analytic geometry (holomorphic functions and differential forms) have been fundamental tools for understanding the geometry and topology of complex algebraic varieties since the beginnings of the subject. Nonarchimedean norms, such as the p-adic norm on the rational numbers, also have an associated analytic geometry which has been used in number theory, but is just beginning to be applied in other areas of mathematics, such as algebraic geometry and dynamical systems. This talk will be an introduction to nonarchimedean geometry with an explanation of its combinatorial manifestation in tropical geometry and relations to intersection theory.

Please join us for refreshments at 3:30 p.m. in the Common Room, 321 Altgeld Hall.

Thursday, January 28, 2010

Tuesday, February 2, 2010

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, February 2, 2010

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Submitted by jarouse.

Johann Thiel (UIUC Math)
The Gaussian moat problem
Abstract: A Gaussian integer is any complex number a+ib with a and b integers. A Gaussian prime is a Gaussian integer that cannot be written, in a nontrivial way, as a product of Gaussian integers. If we think of Gaussian integers as points in the complex plane, the Gaussian moat problem asks whether one can start at the origin and walk out to infinity on Gaussian primes taking steps of bounded length. This question is still unresolved, and it is conjectured that no bounded step length will work. We will survey known results, including the largest known bounded step size that does not work, as well as a probabilistic model that suggests the smallest (unbounded) step size needed to be able to walk to infinity along Gaussian primes. In this talk we will use tools from analytic number theory, algebraic number theory, combinatorics and probability.

Thursday, February 4, 2010

Thursday, February 11, 2010

Thursday, March 4, 2010