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Title: Analytic aspects in the evaluation of multiple zeta and Hurwitz zeta values Speaker: Cezar Lupu, University of Pittsburgh Time: 4:00pm–5:00pm Place: CMC 130

The multiple zeta values (Euler-Zagier sums) were introduced independently by Hoffman and Zagier in 1992 and they play a crucial role at the interface between analysis, number theory, combinatorics, algebra and physics.

The central part of the talk is given by Zagier's formula for the multiple zeta values, \(\zeta(2,2,\dotsc,2,3,2,2,\dotsc,2)\). Zagier's formula is a remarkable example of both the strength and the limits of the motivic formalism used by Brown in proving Hoffman's conjecture where the motivic argument does not give us a precise value for the special multiple zeta values \(\zeta(2,2,\dotsc,2,3,2,2,\dotsc,2)\) as rational linear combinations of products \(\zeta(m)\pi^{2n}\) with $m$ odd. The formula is proven indirectly by computing the generating functions of both sides in closed form and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality.

By using the Taylor series of integer powers of arcsin function and a related result about expressing rational zeta series involving \(\zeta(2n)\) as a finite sum of \(\mathbb{Q}\)-linear combinations of odd zeta values and powers of \(\pi\), we derive a new and direct proof of Zagier's formula in the special case \(\zeta(2,2,\dotsc,2,3)\). If time allows, we present a more general formula for \(\zeta(2,2,\dotsc,m)\). We discuss similar results for multiple Hurwitz zeta values.

No seminar this week — Thanksgiving Holiday.

Title: On polynomials with vanishing Hessians Speaker: Tom McKinley Time: 4:00pm–5:00pm Place: CMC 130

I will talk about a recent result regarding a conjecture we made earlier about polynomials with vanishing Hessian determinant.

The conjecture states that for two homogeneous polynomials \(p\) and \(f\) in \(n\) variables the equality \(p(\mathrm{grad} f)=0\) implies that \(p(\mathrm{grad})f=0\). The conjecture is known in \(5\) variables and in case when the degree of \(f\) is \(2\).

In this talk I will present a proof of the conjecture when the degree of \(p\) is two. So, in particular, if the sum of the squares of first partial derivatives of a polynomial \(f\) is zero then the polynomial \(f\) is harmonic.

No seminar this week — Veteran's Day Holiday.

Title: On Fatou's theorem Speaker: Arthur Danielyan Time: 4:00pm–5:00pm Place: CMC 130

By Fatou's fundamental theorem, a bounded analytic function \(f\) in the open unit disc \(D\) has radial (even non-tangential) limits at the points of the unit circle \(T\) except a subset \(E\) of measure zero. It is a well-known elementary (topological) fact that the exceptional set \(E\) is a \(G_{\delta\sigma}\). But if \(f\) has unrestricted limits at each point of \(T\setminus E\), then obviously \(E\) becomes just an \(F_\sigma\) set. In this talk we show that the converse statement is true as well. Namely, we prove the following:

Let \(E\) be a subset on \(T\). There exists a bounded analytic function in \(D\) which has no radial limits on \(E\) but has unrestricted limits at each point of \(T\setminus E\) if and only if \(E\) is an \(F_\sigma\) set of measure zero.

An obvious corollary of Theorem 1 is the Lohwater-Piranian theorem: If \(E\) is an \(F_\sigma\) set of measure zero on \(T\) then there exists a bounded analytic function in \(D\) which has no radial limits exactly on \(E\). In 1994 S. V. Kolesnikov has proved: There exists a bounded analytic function in \(D\) which has no radial limits exactly on the given set \(E\) (on \(T\)) if and only if \(E\) is a \(G_{\delta\sigma}\) set of measure zero. Both Theorem 1 and Kolesnikov's theorem extend the Lohwater-Piranian theorem up to necessary and sufficient results, but in different directions. The question leading to Kolesnikov's theorem has been proposed by Lohwater and Piranian already in 1957. But the question leading to Theorem 1 has remained unnoticed by now. The method of proof of Theorem 1 is completely elementary but it offers some simplification even for the proof of Kolesnikov's theorem.

Title: Planar orthogonal polynomials Speaker: Seung-Yeop Lee Time: 4:00pm–5:00pm Place: CMC 130

I will show that planar orthogonal polynomials with finite number of logarithmic singularities in its orthogonality measure can be written in terms of Type II Hermite-Padé approximation (that appeared in the last week’s seminar). This will be an improved version of the affair that Meng Yang gave a talk last time.

Title: On zero distribution of Hermite-Padé polynomials, Part III Speaker: Evguenii Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

Title: On zero distribution of Hermite-Padé polynomials, Part II Speaker: Evguenii Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

Title: Inverse moment problems, Padé approximations, and linear operators Time: 4:00pm–5:00pm Place: CMC 130

This is an open problems discussion, and the speakers will consist of all those who wish to contribute.

Title: On zero distribution of Hermite-Padé polynomials Speaker: Evguenii Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

Last semester I discussed the Angelesco case when singularities of functions are well enough separated. Now I will try to discuss general case where everything is much less understood.

More exactly, the whole situation is a mystery. I will share my confusion with the public.

No seminar this week due to AMS Southeastern Sectional Meeting.

Title: On the boundary behaviour of Dirichlet series, Part III Speaker: Myrto Manolaki Time: 4:00pm–5:00pm Place: CMC 130

This week's seminar is cancelled due to a university closure.

Title: On the boundary behaviour of Dirichlet series, Part II Speaker: Myrto Manolaki Time: 4:00pm–5:00pm Place: CMC 130

In this talk, we will discuss the strong connections between the boundary behaviour of a holomorphic function representable as absolutely convergent Dirichlet series in a half-plane, and the limiting behaviour of subsequences of its partial sums. In particular, we will focus on a complementary result of the High Indices Theorem of Hardy and Littlewood. (Joint work with Stephen Gardiner.)