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MAY 2011 |
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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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Joint Princeton University and IAS Number Theory Seminar |
Topic: |
Relative Homotopy type and obstructions to the existence of rational points |
Presenter: |
T. Schlank, Hebrew University |
Date: |
Thursday, May 12, 2011, Time: 4:30 p.m., Location: Fine Hall 214 |
Abstract: |
In 1969 Artin and Mazur defined the etale homotopy type Et(X) of scheme X, as a way to homotopically realize the etale topos of a X. In the talk I shall present for a map of schemes X-> S a relative version of this notion. We denoted this construction by Et(X/S) and call it the homotopy type of X over S. It turns out that the relative Homotopy type, can be especially useful in studying the sections of the map X-> S. In the special case where S = Spec K is the spectrum of a field, the set of sections are just the set of rational points X(K) and then the relative homotopy type Et(X/Spec K) can be used to define obstructions to the existence of a rational point on X. When K in a number fields it turns out that most known obstructions for the existence of rational points (such as Grothendieck's section obstruction , the regular and etale Brauer-Manin obstructions, etc.. ) can be obtained in this way and this point a view can be used to show new properties of these obstructions. In the case where K is a general field or ring this method allows one to get new obstructions that generalized the obstructions above. This is a joint work in progress with Y. Harpaz many of the results appear in our joint paper http://arxiv.org/abs/1002.1423 |
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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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