Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, April 11, 2006.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, April 11, 2006

1:00 pm in 345 Altgeld Hall,Tuesday, April 11, 2006

#### Completing Hardy Fields and a Generalization of a Theorem of Borel

###### Isaac Goldbring (UIUC Math)

Abstract: Hardy fields (at 0+) are ordered differential fields of germs at 0+ of real-valued C^1 functions defined on intervals of the form (0,a) for some real number a. Every Hardy field comes equipped with a valuation and thus we can also consider Hardy fields as valued fields. It is well know that valued fields can be equipped with a valuation topology and that every valued field (K,v) possesses a completion with respect to this valuation topology. I will consider the question of whether this completion, which is a priori just a valued field, can be realized as a Hardy field. I will answer this question affirmatively for the simple case of the Hardy field R(t) and outline the proof. The proof requires two theorems from analysis and in order to answer the question affirmatively for more complicated Hardy fields, one would need generalizations of them to broader contexts. I will present a generalization of one of the theorems, Borel’s theorem on Taylor series, as half of the work necessary to complete the Hardy field R(t^1/d : d=1,2,…). I will define all of the notions involved and the talk should be accessible to anyone with knowledge of undergraduate real analysis.

1:00 pm in 347 Altgeld Hall,Tuesday, April 11, 2006

#### KAM Method and Limit Periodic Potential

###### Prof. Yulia Karpeshina   [email] (Univ. of Alabama at Birmingham)

Abstract: We consider an application of KAM (Kolmogorov-Arnold-Moser) method for spectral investigation of a polyharmonic operator $H=(-\Delta)^l+V(x)$ with a limit-periodic potential $V(x)$ in dimension two. We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i\langle \vec k,\vec x\rangle }$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec k$ corresponding to these eigenfunctions have a form of slightly distortedcircles with holes(Cantor type structure). Joint work with Young-Ran Lee

2:00 pm in 347 Altgeld Hall,Tuesday, April 11, 2006

#### Variations of the Solution to a Stochastic Heat Equation

###### Jason Swanson   [email] (University of Wisconsin-Madison, Department of Mathematics)

Abstract: We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, F(t), has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical It\^o calculus does not apply. Motivated by heuristic ideas about a possible new calculus for this process, we are led to study modifications of the quadratic variation. Namely, we modify each term in the sum of the squares of the increments so that it has mean zero. We then show that these sums, as functions of $t$, converge weakly to Brownian motion.

2:00 pm in 243 Altgeld Hall,Tuesday, April 11, 2006

#### Covers for angleworms

###### Prof. John E. Wetzel (UIUC Department of Mathemtics)

Abstract: We show that each triangle ABC with AB = 1, 45� ≤ MBAC 60, and altitude G3/4 from C to AB is a convex cover of least area G3/8 for the family of all angleworms (two-segment unit arcs), and we show that for each P > 0 there is a (non-convex) Jordan domain bounded by a rectifiable Jordan curve that is a cover for this family and has area less than P. The analogous questions for three-segment unit arcs in the plane remain open.

This reports joint work with Wacharin Wichiramala and Banyat Sroysang at Chulalongkorn University in Bangkok.

3:00 pm in 241 Altgeld Hall,Tuesday, April 11, 2006

#### The linear discrepancy of products of chains

###### Jeong-Ok Choi (UIUC Math)

Abstract: Given a linear extension of a poset P, let h be the function giving the height of each element on the extension. The linear discrepancy ld(P) is the least integer m for which there exists a linear extension such that |h(x)-h(y)| <= m if x and y are incomparable. The exact value of the linear discrepancy of a product of two chains is known; In 2003 Hong, Hyun, and Kim proved that the linear discrepancy of the product of n-element and m-element chains is the ceiling of (1/2)mn-2.

In this talk we present asymptotic bounds for the linear discrepancy of the product of three k-chains and the linear discrepancy of the product of four k-chains. In three dimensions, the value is (3/4+o(1))k3. In four dimensions, the value is (7/8+o(1))k4. The upper bound construction generalizes easily to d dimensions. This is joint work with Douglas B. West.

3:00 pm in 443 Altgeld Hall,Tuesday, April 11, 2006

#### Nonstandard epsilon-theories and related results.

###### Dr. Petr Andreev (Russia, visiting City College of CUNY)

Abstract: We consider the problem of constructing an axiomatic set theoretical framework for nonstandard analysis in the pure epsilon language. The main result is a construction of a "natural" epsilon-theory of external sets over a hyperfinite domain. Some related results will also be discussed.

4:00 pm in 314 Altgeld Hall,Tuesday, April 11, 2006

#### Infinite dimensional Lie (super)algebras

###### Efim Zelmanov (University of California at San Diego )

Abstract: We will discuss (i) infinite dimensional Lie (super)algebras of polynomial growth, and (ii) infinite dimensional Lie (super)algebras graded by root systems. Surprisingly both theories lead to very similar objects.

Efim Zelmanov will present the Coble Memorial Lectures April 11-13 at 4 p.m. each day in 314 Altgeld Hall. Tea, coffee and cookies will be served at 3:15 p.m. before the lecture. A reception will be held Tuesday, April 11, at 5 p.m. in 321 Altgeld (Common Room).

5:00 pm in 343 Altgeld Hall,Tuesday, April 11, 2006

#### Complexity results in Biological systems

###### Zoi Rapti (UIUC Math)

Abstract: In this talk I'm going to review some complexity results with applications to DNA and protein folding. I also will try to explain how hyperbolic geometry is useful when trying to obtain fast algorithms that solve the protein-folding and DNA-reconfiguration problems. These ideas are based on work by V. Peterson and R. Ghrist.