Analysis   Seminar

Unless specified, all seminars are Wednesday 4-5pm at 250 Math Building.

September 11th           Josh Isralowitz,    SUNY at Albany

Compactness criteria and essential norm estimates for operators in generally weighted Fock and Bergman spaces

Abstract: In a deep paper of D. Suárez published in 2007, the classical Axler-Zheng theorem was extended to show that an operator in the
Toeplitz algebra of the Bergman space on the unit ball is compact if and only if its Berezin transform vanishes on the boundary. Very
recently, a small flurry of results (such as extensions to the Fock space, new essential norm estimates, and vast simplifications of his
original proof) have appeared which extend this theorem. In this talk, we will discuss these recent results and the techniques used to prove
them. Furthermore, we will discuss some very interesting open problems related to these results. This is partly joint work with Brett Wick
and Mishko Mitkovski.

September 25th           Tim Austin,   Courant Institute
Partial difference equations over compact Abelian groups

Abstract: Given a compact Abelian group Z, an element z of that group, and a measurable function from it to another such group, one can form
a new function by taking the difference of the original function and its translate by z.  This is the obvious discrete analog of

differentiation, and defines an operator on functions called a differencing operator.

Recent work in additive combinatorics, related to Gowers' proof of Szemerédi's Theorem, leads naturally to the study of certain higher-
order' partial difference equations involving such operators.  Given several elements of Z, one asks for a description of those functions on
Z which vanish when one applies all of the resulting differencing operators.  It turns out that as the order of the difference equation
increases, one can sometimes find surprising extra structure among these solutions, which amounts to a first step towards understanding the
inverse problem for the directional Gowers norms'.

September 26th           John Friedlander,  University of Toronto
(Thursday, Colloquium)
The Spin of Prime Ideals

Abstract: We discuss joint work with H. Iwaniec, B. Mazur and K. Rubin. Fixing a nontrivial automorphism of a number field K, we associate to
ideals in K an invariant (with values in {0,±1}) which we call the spin. Using techniques from analytic number theory, we show the spin
values ±1 occur asymptotically equally often over primes, with a rather sharp error term. We apply the theorem to the arithmetic statistics
of the Selmer groups of elliptic curves.

October 2nd              Yunping Jiang,   CUNY at Queens College & Graduate Center
Symmetric Invariant Measures and Symmetric Rigidity

Abstract: In the study of modern dynamical systems, an invariant measure is an important topic and in the study of modern complex analysis,
the quasisymmetric condition on a map is an important topic. In this talk, I will combine these two topics together introducing a new
interesting topic, symmetric invariant measure. I will discuss the existence and the uniqueness of a symmetric invariant probability measure
for a uniformly symmetric circle endomorphism. I will also discuss symmetric rigidity related with the uniqueness. I will also discuss the
Teichmuller structure and complex manifold structure on the space of symmetric invariant probability measures.

October 16th             Djordje Milicevic,   Bryn Mawr College
(3-4pm, Math 150)        Sub-Weyl subconvexity and short p-adic exponential sums

Abstract: One of the principal questions about L-functions is the size of their critical values. In this talk, we will present our recent
subconvexity bound for the central value of a Dirichlet L-function of a character to a prime power modulus, which breaks a long-standing
barrier known as the Weyl exponent. The results are obtained by developing a new general method to estimate short exponential sums involving
p-adically analytic fluctuations, which can be naturally seen as a p-adic analogue of the method of exponent pairs. We will present the main
results of this method and the key points in its development, and discuss the structural relationship between the p-adic analysis and the
so-called depth aspect.

October 23rd             Bingbing Liang,  SUNY at Buffalo
Mean Dimension and von Neumann-Lück Rank

Abstract: Mean dimension is a numerical invariant in topological dynamics and related to entropy. The von Neumann-Lück rank is a
$L^2$-invariant and related to $L^2$ Betti number.
I will establish an equality between the von Neumann-Lück rank of a module of the
integral group ring of an amenable group and the mean dimension of the associated algebraic action. Also I will give some applications of
this equality. This is a joint work with Hanfeng Li.

October 30th             Yichao Zhang,  University of Connecticut
Vector-Valued and Scalar-Valued Modular Forms

Abstract: Vector-valued modular forms were employed by Borcherds to formulate his theory of automorphic products and other results, but
unlike scalar-valued modular forms, they do not possess nice structures. For example, they do not behave well under multiplication. In this
talk, we prove an isomorphism between spaces of both kinds of modular forms, generalizing a result of Bruinier and Bundschuh. If time
permits, we will describe canonical bases, derive a beautiful duality (Zagier duality), and mention the problem of integrality of Fourier
coefficients.

November 6th             Jaehyun Cho,  SUNY at Buffalo
Low-lying zeros of Artin L-functions

Abstract: With Malle's conjecture or a parametric polynomial, we construct families of Artin L-functions. For these families we prove
one-level density results unconditionally or under reasonable assumptions. Our results shows that the symmetry type of these families agrees
with the symplectic type. This is a joint work with Henry Kim.

November 13th            Yoonbok Lee,  Rochester University
Zero-density estimates of L-functions

Abstract : In the theory of zeta or L- functions it is useful to have good estimates for $N(\sigma, T, 2T)$, the number of zeros  $\rho = \beta+i\gamma$ for which $\beta> \sigma$ and $T \lt \gamma \lt 2T$.
In this talk we study zero-density estimates of various zeta or L- functions. In particular we introduce that zero-density estimates of a Hecke L-function are related to a
universality theorem and fractional moments of the L-function.

Past Analysis Seminars