Department of

Mathematics


Seminar Calendar
for events the day of Thursday, February 12, 2015.

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Thursday, February 12, 2015

11:00 am in 241 Altgeld Hall,Thursday, February 12, 2015

New Pathways and Connections in Analysis and Analytic Number Theory Motivated by Two Incorrect Claims of Ramanujan

Arindam Roy   [email] (UIUC Math)

Abstract: In this talk, we concentrate on only one of the two incorrect “identities” in page 336 of Ramanujan's lost notebook. This identity may have been devised to attack the extended divisor problem. We provide a corrected version of Ramanujan’s identity, which contains the convergent series appearing in the incorrect claim. Our identity is admittedly quite complicated, and we do not claim that what we have found is what Ramanujan originally had in mind. But there are simple and interesting special cases as well as analogues of this identity, one of which very nearly resembles Ramanujan’s version. The aforementioned convergent series in Ramanujan’s incorrect identity is similar to one used by G. F. Voronoi, G. H. Hardy, and others in their study of the classical Dirichlet divisor problem, and so we are motivated to study further series of this sort. This now brings us to page 335 of Ramanujan's lost notebook, which comprises two formulas featuring doubly infinite series of Bessel functions. Although again not obvious at a first inspection, one is conjoined with the classical Circle Problem initiated by Gauss, while the other is associated with the Dirichlet Divisor Problem. Bruce Berndt and Alexandru Zaharescu, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this talk, we return to these two formulas and examine them in more general settings. This is a joint work with B. Berndt, A. Dixit, and A. Zaharescu.

1:00 pm in Altgeld Hall 243,Thursday, February 12, 2015

A Liouville theorem on Kaehler Ricci flat metric in $C^n$ with conical singularities along $\{0\}\times C^{n-1}$

Xiuxiong Chen (Stony Brook)

Abstract: We prove a new Liouville type theorem which goes back to E. Calabi and Pogorelov. The theorem of Calabi and Pogrelov can be formulated as follows: In C^n, any Kaehler Ricci flat metric which depends on R^n (i.e., admitting standard toric symmetry) must be trivial. We prove that, in C^n, any Kaehler Ricci flat metric which has conical singularity along {0}\times C^{n-1} and is quasi conformal to a standard flat conical metric must be trivial. This has important application in conical Kaehler geometry (such as regularity for conical Kaehler Einstein metrics). Joint with Yuanqi Wang. View talk at http://youtu.be/ejmhtgPY49c

2:00 pm in 243 Altgeld Hall,Thursday, February 12, 2015

Quasiconformal non-parametrization of almost Riemannian spheres

Pekka Pankka (University of Jyväskylä)

Abstract: By Semmes's examples there are metric 3-spheres which are Ahlfors 3-regular and linearly locally contractible but not quasisymmetric to the standard 3-sphere. The metric in Semmes's examples can be taken to be a completion of the length metric induced by a Riemannian metric on a punctured sphere. In this talk I will discuss similar examples in higher dimensions and their relation to examples of Heinonen and Wu. This is joint work with Vyron Vellis.

2:00 pm in 140 Henry Administration Bldg,Thursday, February 12, 2015

Open problems in Number Theory, III

Bruce Berndt (UIUC Math)

3:00 pm in 243 Altgeld Hall,Thursday, February 12, 2015

On Lech's Conjecture

Linquan Ma (Purdue University)

Abstract: I will talk on a long-standing conjecture of Lech on the multiplicities of a faithfully flat extension of local rings. I will discuss several attempts to attack this conjecture. I will show how this conjecture is related to some natural questions on modules of finite length and projective dimension and discuss some recent progress.

4:00 pm in 245 Altgeld Hall,Thursday, February 12, 2015

On the Kaehler Ricci flow

Xiuxiong Chen (Stony Brook)

Abstract: There is a long standing conjecture on Kaehler Ricci flow in Fano manifolds that the Ricci flow converges sub-sequentially to a Kaehler Ricci solution with at most codimension 4 singularities, with perhaps a different complex structure (so called "Hamilton Tian conjecture"). In this lecture, we will outline a proof for this conjecture. This is a joint work with Bing Wang.