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REVISED WITH ADDED SEMINARS |
APRIL 2011 |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
Parabolic equations and the Ricci flow on manifolds with boundary |
| Presenter: |
Artem Pulemotov, University of Chicago |
| Date: |
Friday, April 29, 2011, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
In the first part of the talk, we will focus on a second-order quasilinear parabolic equation in a vector bundle over a compact manifold~$M$ with boundary. Our goal is to explore the short-time existence of solutions to this equation. In the second part, we will discuss the Ricci flow on~$M$. The objective is to propose a new boundary condition for the flow and state two short-time existence results. |
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| MAY 2011 |
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| Ergodic Theory and Statistical Mechanics Seminar **Please note special day and time** |
| Topic: |
Relative Entropy Indicators and applications to Authorship Attribution |
| Presenter: |
Prof. Mirko Degli Esposti, Universita' di Bologna |
| Date: |
Monday, May 2, 2011, Time: 4:00 p.m., Location: Fine Hall 801 |
| Abstract: |
Estimating the entropy of an unknown (ergodic, stationary) stochastic source with unknown memory, only relying on finite samples, poses interesting and challenging mathematical questions. In this talk we aim first to review some recent and less recent relative entropy indicators, such as the zipper's based methods. Then we will recall the main definitions and results on non-sequential recursive pair substitutions and their use as relative entropy indicators. Finally, we will discuss some recent applications to authorship attribution of literary texts. |
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| Algebraic Geometry Seminar |
| Topic: |
On the three compactifications of Siegel space |
| Presenter: |
Valery Alexeev, University of Georgia |
| Date: |
Tuesday, May 3, 2011, Time: 4:30 p.m., Location: Fine Hall 322 |
| Abstract: |
The moduli space of $A_g$ of abelian varieties has three classical toroidal compactifications: (1) perfect, (2) 2nd Voronoi, and (3) Igusa blowup, each with its own distinct geometric meaning. It is an interesting problem to understand exactly how these compactifications are related. I will show that (1) and (2) are isomorphic in a neighborhood of the image of a regular map from the Deligne-Mumford's moduli space $\bar M_g$, and that the rational map $\bar M_g \to \bar A_g$ for (3) is not regular for $g>8$. This is a joint work with Adrian Brunyate. |
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| Topology Seminar |
| Topic: |
L-spaces and left-orderability |
| Presenter: |
Liam Watson, UCLA |
| Date: |
Thursday, May 5, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Various families of examples suggest an interesting correspondence between L-spaces and 3-manifolds with non-left-orderable fundamental group. This motivates the study of left-orderability in the context of Dehn surgery. In particular, since every knot group is left-orderable, we study the phenomenon of when a left-order descends to the quotient group associated with the surgery. This leads to the notion of a decayed knot; such knots have the property that all sufficiently large surgeries have non-left-orderable fundamental group. It can also be shown that sufficiently positive cables of decayed knots are decayed knots. Both of these properties mirror the behaviour of L-spaces under Dehn surgery. This is joint work with Adam Clay. |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
Geometry of level sets of elliptic equations and a conjecture of De Giorgi |
| Presenter: |
Yannick Sire, Universite de Aix-Marseille III |
| Date: |
Friday, May 6, 2011, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
I will describe recent results on the rigidity of level sets of solutions of local and non local elliptic equations on the euclidean space and on riemannian manifolds in connection with a conjecture by De Giorgi. |
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| Department Colloquium |
| Topic: |
Higher order Fourier analysis |
| Presenter: |
Balazs Szegedy, University of Toronto |
| Date: |
Wednesday, May 11, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
In a famous paper Timothy Gowers introduced a sequence of norms U(k) defined for functions on abelian groups. He used these norms to give quantitative bounds for Szemeredi's theorem on arithmetic progressions. The behavior of the U(2) norm is closely tied to Fourier analysis. In this talk we present a generalization of Fourier analysis (called k-th order Fourier analysis) that is related in a similar way to the U(k+1) norm. Ordinary Fourier analysis deals with homomorphisms of abelian groups into the circle group. We view k-th order Fourier analysis as a theory which deals with morphisms of abelian groups into algebraic structures that we call "k-step nilspaces". These structeres are variants of structures introduced by Host and Kra (called parallelepiped structures) and they are close relatives of nil-manifolds. Our approach has two components. One is an uderlying algebraic theory of nilspaces and the other is a variant of ergodic theory on ultra product groups. Using this theory, we obtain inverse theorems for the U(k) norms on arbitrary abelian groups that generalize results by Green, Tao and Ziegler. As a byproduct we also obtain an interesting limit theory for functions on abelian groups in the spirit of the recently developed graph limit theory. |
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| Topology Seminar |
| Topic: |
A Heegaard Floer characterization of Borromean knots |
| Presenter: |
Yi Ni, Caltech |
| Date: |
Thursday, May 12, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
If the total rank of the knot Floer homology of a knot is equal to the total rank of the Heegaard Floer (hat) homology of the ambient 3-manifold, we say that the knot has simple knot Floer homology (or the knot is Floer simple). It is known that the unknot is the only Floer simple knot in S3. However, the question of determining all the Floer simple knots in general 3-manifolds is far from being solved. In this talk we will answer this question for the connected sums of S1\times S2: Such knots are essentially the Borromean knots. |
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| Algebraic Geometry Seminar |
| Topic: |
Theta Functions |
| Presenter: |
Hershel Farkas, Hebrew University of Jerusalem |
| Date: |
Tuesday, May 17, 2011, Time: 4:30 p.m., Location: Fine Hall 322 |
| Abstract: |
In this talk we consider Weierstrass points for the linear space of meromorphic functions on a compact Riemann surface whose divisors are multiples of $\frac{1}{P_0^{\alpha}P_1..P_{g-1}} where the points $P_i$ are points on the surface and $\alpha$ is a positive integer for which there is no holomorphic differential whose divisor is a multiple of $P_0^{\alpha)P_1..P_{g-1}$. Thus by the Riemann Roch theorem the dimension of the space is precisely $\alpha$. It develops that here are two different ways to define the Weierstrass points for this space. One way is to consider the Wronskian determinant of a basis for the space and to define the Weierstrass points as the zeros of the Wronskian and the weight of the Weierstrass point as the order of the zero. Another way is to consider, for a suitably chosen $ e \in C^g$, the Riemann theta function $ \theta(\alpha \Phi_{P_0}(P)-e)$ as a multivalued function on the surface, to define the Weierstrass points as its zeros and the weight of the Weierstrass point as the order of the zero. In this talk we deal with the question of whether to two definitions agree. We show that the sets of zeros are indeed the same and the problem is to detrmine whether the weights are the same. We give a partial solution to this problem. |
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