Department of

# Mathematics

Seminar Calendar
for events the week of Monday, April 10, 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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Monday, April 10, 2006

3:00 pm in 343 Altgeld Hall,Monday, April 10, 2006

#### Vertex operators and correlation functions

###### Sheldon Katz   [email] (UIUC Math and Physics)

Abstract: I will describe vertex operators and correlation functions, in preparation for a geometric application of mirror symmetry on Wednesday. As time permits, I will round out last week's material by explaining why the type II string compactified on a Calabi-Yau threefold has N=2 supersymmetry, and how the massless spectrum is organized into hypermultiplets and vector multiplets.

3:00 pm in 3405 Siebel Center,Monday, April 10, 2006

#### Mimetic Discrete Models with Weak Material Laws, or Least Squares Principles Revisited

###### Pavel Bochev (Sandia National Lab, Computational Mathematics and Algorithms)

Abstract: To a casual observer, compatible (or mimetic) methods and least squares principles for PDEs are polar opposites. Mimetic methods inherit key conservation properties of the PDE, can be related to a naturally occurring optimization problem, and require specially selected, dispersed degrees of freedom. The conventional wisdom about least squares is that they rely on artificial energy principles, are only approximately conservative, but can work with standard C0 nodal (or collocated) degrees of freedom. The latter is considered to be among the chief reasons to use least squares methods. In this talk we demonstrate that exactly the opposite is true about least-squares methods. First, we will argue that nodal elements, while admissible in least squares, do not allow them to realize their full potential, should be avoided and are, perhaps, the least important reason to use least squares! Second, we will show that for an important class of problems least squares and compatible methods are close relatives that share a common ancestor. In fact, we will prove that in some circumstances, least squares and compatible methods compute identical answers. The price paid for gaining favorable conservation properties is that one has to give up what is arguably the least important advantage attributed to least squares methods: one can no longer use C0 nodal elements for all variables. To carry out this agenda we use algebraic topology to guide our analysis and develop a common framework for compatible discretizations. Using a reduction and a reconstruction maps between differential forms and cochains we define mutually consistent sets of natural and derived discrete operations that preserve the invariants of the DeRham homology groups and obey a discrete Stokes theorem. By choosing a specific reconstruction operator we obtain well-known mixed FE, mimetic FD and covolume methods and explain when they are equivalent. The key concept in our approach is the natural inner product on cochains. This inner product is sufficient to generate a combinatorial Hodge theory on cochains but avoids complications attendant in the construction of robust discrete Hodge-star operators. For problems that require approximations of material laws we employ equivalent constrained optimization problems that enforce the laws weakly, instead of using their explicit discretization. We then consider three possible mimetic discretizations of the optimization problem. Two of them give familiar Galerkin and/or mixed type methods. The third one reduces the optimization problem to a mimetic least squares principle whose minimizers are, under certain conditions, identical with the solutions of the other two mimetic discretizations. We conclude by a series of numerical examples that assert our findings. This talk is based on joint work with M. Gunzburger (CSIT, Florida State University) and M. Hyman (Theoretical Division, Los Alamos National Laboratory). Host : Anil Hirani (UIUC Computer Science)

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.

Wednesday, April 12, 2006

2:00 pm in 143 Altgeld Hall,Wednesday, April 12, 2006

#### Topological conformal field theories and Frobenius A-infinity algebras

###### Kevin Costello (U. Chicago, Mathematics)

Abstract: I'll show how a kind of "open string theory" is the same as a Frobenius A-infinity algebra, and also how to construct a "closed string theory" structure on Hochschild homology of such an algebra. This implies that this Hochschild homology carries natural operations coming from the homology of moduli spaces of Riemann surface.

3:00 pm in 343 Altgeld Hall,Wednesday, April 12, 2006

#### A Mathematical Example of Mirror Symmetry

###### Yong Fu (UIUC Math)

Abstract: Originating in physics, mirror symmetry in mathematics relates two types of spaces and different theories on these spaces. We study an example of this phenomenon and see how computation on one space leads to results on the other which is otherwise hard to get.

4:00 pm in 245 Altgeld Hall,Wednesday, April 12, 2006
###### No 499 lecture today - visit the Coble Lecture Series, April 11-13

4:00 pm in 314 Altgeld Hall,Wednesday, April 12, 2006

#### On Golod-Shafarevich groups

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

Abstract: The talk will focus on groups presented by generators and relators. In 1964 Golod and Shafarevich found a sufficient condition for such groups to be infinite. We will discuss various properties of Golod - Shafarevich groups and their connections with Geometry and Number Theory.

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).

Thursday, April 13, 2006

1:00 pm in 347 Altgeld Hall,Thursday, April 13, 2006

#### No meeting this week

1:00 pm in 241 Altgeld Hall,Thursday, April 13, 2006

#### Covering systems and periodic arithmetical functions

###### Zhi-Wei Sun (Nanjing University)

Abstract: If the ring $\Z$ of integers is the union of finitely many residue classes $a_1(\mo\ n_1),\ldots,a_k(\mo\ n_k)$, then the system $A=\{a_s(\mo\ n_s)\}_{s=1}^k$ is called a cover of $\Z$ or a covering system. There are many problems and results on this topic initiated by Paul Erd\"os. In this talk we introduce progress on the main open problems in this field, as well as the algebraic theory of periodic arithmetical maps motivated by the speaker's research on covering systems. The talk is related to number theory, combinatorics, algebraic structures and special functions.

1:00 pm in Altgeld Hall,Thursday, April 13, 2006

#### To Be Announced

2:00 pm in 443 Altgeld Hall,Thursday, April 13, 2006

#### Folded symplectic structures

###### Christopher Lee   [email] (UIUC Math)

Abstract: I will introduce the notion of a folded symplectic form on an even dimensional smooth manifold. Essentially, a folded symplectic form is a closed two form whose top power is satisfies a certain transversality condition, making its zero locus a smooth hypersurface of the manifold. We then consider the two-form to be symplectic "away from the hypersurface". Following a formalization of these notions, we will consider examples.

2:00 pm in 57 Everitt Lab,Thursday, April 13, 2006

#### Lieb-Thirring Inequalities

###### Dirk Hundertmark (UIUC Math)

Abstract: We continue to study Lieb-Thirring inequalities, and their connection to the Stability of Matter.

2:00 pm in 241 Altgeld Hall,Thursday, April 13, 2006

#### Tools from Iwasawa Theory

###### Steve Ullom (UIUC Math)

Abstract: (RAP Elliptic Curves and Iwasawa Theory, Part 10) This week: Tools from Iwasawa Theory needed for the one-variable Main Conjecture.

3:00 pm in 243 Altgeld Hall,Thursday, April 13, 2006

#### Hilbert polynomials over regular local rings via intersection theory on the blow-up

###### Claudia Miller   [email] (Syracuse University)

Abstract: Hilbert polynomials have classically been shown to be dtermined by intersections with hyperplanes. We give another approach over regular local rings via Intersection Theory on the blow-up scheme.

4:00 pm in 314 Altgeld Hall,Thursday, April 13, 2006

#### Some Ring - Theoretic problems inspired by Combinatorial Group Theory

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

Abstract: I will list open problems concerning infinite dimensional algebras and explain how each problem arises in Combinatorial Group Theory or is inspired by it.

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).

Friday, April 14, 2006

12:00 am in 1005 Beckman Institute,Friday, April 14, 2006

#### "Estimation of Hybrid Models as Algebraic Sets"

###### Robert Fossum   [email] (UIUC Math, Beckman Institute)

Abstract: Lunch will be served. Subspace arrangements will be used to model hybrid data sets.

4:00 pm in 141 Altgeld Hall,Friday, April 14, 2006

#### Groups, measures, and the NIP - continuation

###### Ayhan Gunaydin (UIUC Math)

Abstract: Ayhan will continue his presentation of Section 6 (Groups and NIP) of the NIP paper.