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

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

Partial difference equations over compact Abelian groups

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

(Thursday, Colloquium) The Spin of Prime Ideals

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

Symmetric Invariant Measures and Symmetric Rigidity

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

(3-4pm, Math 150) Sub-Weyl subconvexity and short p-adic exponential sums

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

Mean Dimension and von Neumann-Lück Rank

$L^2$-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.

Vector-Valued and Scalar-Valued Modular Forms

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

Low-lying zeros of Artin L-functions

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

Zero-density estimates of L-functions

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