Department of

Mathematics


Seminar Calendar
for events the day of Thursday, February 11, 2016.

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Thursday, February 11, 2016

11:00 am in 241 Altgeld Hall,Thursday, February 11, 2016

PNT Equivalences and Nonequivalences for Beurling primes

Harold Diamond (UIUC Math)

Abstract: In classical prime number theory there are several asymptotic formulas that are said to be ``equivalent'' to the Prime Number Theorem. (This notion is colloquial, not mathematical: it means that the formulas can be deduced from each other by relatively simple arguments.) We show conditions under which analogues of these formulas do or do not hold for Beurling generalized numbers.

12:00 pm in Altgeld Hall 243,Thursday, February 11, 2016

Characterization of Aleksandrov Spaces of Curvature Bounded Above by Means of the Metric Cauchy-Schwarz Inequality.

David Berg (Illinois Math)

Abstract: Joint work of I.D.Berg and I.G.Nikolaev. We employ the previously introduced notion of the K-quadrilateral cosine,the cosine under parallel transport in model K-space,denoted by cosqK. In K-space, modulus cosqK bounded by 1 is equivalent to the Cauchy-Schwarz inequality for tangent vectors under parallel transport. Our principal result states that a geodesic space(of diameter bounded by half the hemisphere diameter for positive K) is a catK space if and only if cosqK is bounded by 1. If, in addition, 1 is actually achieved for two directed non-collinear segments, the geodesic span of the two segments is isometric to a section of K-plane. The diameter restriction is significant. This talk will be devoted to illustrating and explaining these results.If there is time, I will give the ideas of the proofs, veiling the difficult computations that arise,especially in the case of nonzero K, in a decent obscurity.

4:00 pm in 245 Altgeld Hall,Thursday, February 11, 2016

Global singularity theory

Richard Rimányi (University of North Carolina at Chapel Hill)

Abstract: The topology of the spaces A and B may force every map from A to B to have certain singularities. For example, a map from the Klein bottle to 3-space must have double points. A map from the projective plane to the plane must have an odd number of cusp points. To a singularity one may associate a polynomial (its Thom polynomial) which measures how topology forces this particular singularity. In this lecture, we will explore the theory of Thom polynomials and their applications to enumerative geometry. Along the way, we will meet a wide spectrum of mathematical concepts from geometric theorems of the ancient Greeks to the theory of diagrams of linear maps (quivers).