Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, March 9, 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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Thursday, March 9, 2006

11:00 am in 345 Altgeld Hall,Thursday, March 9, 2006

#### Orbifolds as localized groupoids II

###### Eugene Lerman (UIUC Math)

1:00 pm in 347 Altgeld Hall,Thursday, March 9, 2006

#### The Maier-Schmid Problem II

###### Derek Robinson (UIUC)

Abstract: I will survey various classes of infinite groups for which the Maier-Schmid Theorem is valid. I will also describe a solvable p-group for which the theorem fails.

1:00 pm in 57 Everitt Lab,Thursday, March 9, 2006

#### Weyl Asymptotics for the Laplacian

###### Richard S. Laugesen   [email] (UIUC Math)

Abstract: We prove Weyl's asymptotic formula for the eigenvalue counting function of the Dirichlet Laplacian, as the energy approaches infinity.

1:00 pm in 241 Altgeld Hall,Thursday, March 9, 2006

#### Perfect powers, almost squares and sums of two squares

###### Tsz Ho Chan (Central Michigan University)

Abstract: In this talk, I will discuss questions concerning perfect powers (e.g. 52, 65, ...), almost squares (e.g. 99 * 100, 201 * 203, ...), and sums of two squares (e.g. 32 + 52, ...). They are related to some other problems in number theory such as the abc-conjecture and the distribution of n2 \theta (mod 1).

2:00 pm in 241 Altgeld Hall,Thursday, March 9, 2006

#### Elliptic Units

###### Bill Hart (UIUC Math)

Abstract: (RAP Elliptic Curves and Iwasawa Theory, Part 6).

2:00 pm in 443 Altgeld Hall,Thursday, March 9, 2006

#### An Introduction to the method of moving frames via curves and surfaces

###### Joe Montgomery (UIUC Math)

Abstract: I will give a very brief introduction to E. Cartan's method of moving frames in differential geometry. Topics will include the structure equations for Euclidian space, Gauss' Theorema Egregium, the Gauss map, and the Gauss-Bonnet theorem. If time permits, I will examine some nice relations between complex geometry and mininmal surfaces.

2:00 pm in 243 Altgeld Hall,Thursday, March 9, 2006

#### An Elementary Proof of Lyapunov's Theorem

###### David A. Ross (University of Hawaii (visiting UIUC))

Abstract: By a theorem of Lyapunov, the range of an atomless finite-dimensional vector measure is convex. The proofs of this theorem have traditionally either been long and difficult, or have required very powerful tools from convexity theory. I'll give a short proof that uses nothing stronger than the Intermediate Value Theorem.

3:00 pm in 243 Altgeld Hall,Thursday, March 9, 2006

#### On The Upper Semi-Continuity of Hilbert-Kunz Multiplicity, cont.

###### Jinjia Li   [email] (UIUC Math)

Abstract: I will discuss a recent paper by Enescu and Shimomoto on the upper semi-continuity of the Hilbert-Kunz multiplicity. I will begin with a introduction/overview of the Hilbert-Kunz multiplicity. This talk only assumes minimum background in commutative algebra (e.g. MATH 502).

3:00 pm in 345 Altgeld Hall,Thursday, March 9, 2006

#### Hausdorff Dimension of the Contours of Symmetric Additive L\'evy Processes

###### D. Khoshnevisan   [email] (University of Utah)

Abstract: Let $X_1,\ldots,X_N$ denote $N$ independent, symmetric L\'evy processes on $\mathbf{R}^d$. The corresponding \emph{additive L\'evy process} is defined as the following $N$-parameter random field on $\mathbf{R}^d$: $$X(\mathbf{t}) :=X_1(t_1) + \cdots + X_N(t_N)\qquad(\mathbf{t}\in \mathbf{R}^N_+).$$ Khoshnevisan and Xiao (2002) have found a necessary and sufficient condition for the zero-set $X^{-1}(\{0\})$ of $X$ to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of $X^{-1}(\{0\})$ which hold with positive probability in the case that $X^{-1}(\{0\})$ can be non-void. Here, we prove that the Hausdorff dimension of $X^{-1}(\{0\})$ is a constant almost surely on the event $\{ X^{-1}(\{0\})\neq\emptyset \}$. Moreover, we derive a formula for the said constant. This portion of our work extend the one-parameter formulas of Horowitz (1968) and Hawkes (1974). More generally, we prove that for every non-random Borel set $F$ in $(0\,,\infty)^N$, the Hausdorff dimension of $X^{-1}(\{0\})\cap F$ is a constant almost surely on the event $\{ X^{-1}(\{0\})\cap F\neq\emptyset\}$. This constant is computed explicitly in many cases. (Joint work with Narn-Rueih Shieh and Yimin Xiao)

4:00 pm in 245 Altgeld Hall,Thursday, March 9, 2006

#### A Coupling and the Darling - Erdos Conjectures

###### Davar Khoshnevisan   [email] (University of Utah)

Abstract: We present a coupling of the 1-dimensional Ornstein-Uhlenbeck process with an i.i.d. sequence. We then apply this coupling to resolve two conjectures of Darling and Erdos (1956). Interestingly enough, we prove one and disprove the other conjecture. [This is joint work with David Levin.] Time-permitting, we may use the ideas of this talk to describe precisely the rate of convergence in the classical law of the iterated logarithm of Khintchine for Brownian motion (1933). [This portion is joint work with David Levin and Zhan Shi, and has recently appeared in the Electr. Comm. of Probab. (2005).