Unless stated otherwise, colloquia are scheduled for Thursdays 4:15-5:15pm in WH-100E with refreshments served from 4:00-4:15 pm in WH-102.
A Special Event: Professor Ken Ono (University of Virginia), who is a Phi Beta Kappa Visiting Scholar this year, is virtually visiting our department on March 11-12. He will present three lectures via zoom, a Math Club Talk for undergraduate and graduate students with interest in mathematics, a Colloquium Talk for our faculty and graduate students, and a Public Lecture intended for a general audience. Details of these talk are given below. The talks are sponsored by our local Phi Beta Kappa Chapter. Many thanks to Professor Alex Feingold for organizing this.
Abstract: The Riemann hypothesis provides insights into the distribution of prime numbers, stating that the nontrivial zeros of the Riemann zeta function have a “real part” of one-half. A proof of the hypothesis would be world news and fetch a $1 million Millennium Prize. In this lecture, Ken Ono will discuss the mathematical meaning of the Riemann hypothesis and why it matters. Along the way, he will tell tales of mysteries about prime numbers and highlight new advances.
Abstract: In 1798 Gauss wrote Disquisitiones Arithmeticae, the first
rigorous text in number theory. This book laid the groundwork for modern
algebraic number theory and arithmetic geometry. Perhaps the most
important contribution in the work is Gauss's theory of integral
quadratic forms, which appears prominently in modern number theory (sums
of squares, Galois theory, rational points on elliptic
curves,L-functions, the Riemann Hypothesis, to name a few). Despite the
plethora of modern developments in the field, Gauss’s first problem
about quadratic forms has not been optimally resolved. Gauss's class
number problem asks for the complete list of quadratic form
discriminants with class number h. The difficulty is in effective
computation, which arises from the fact that the Riemann Hypothesis
remains open. To emphasize the subtlety of this problem, we note that
the first case, where h=1, remained open until the 1970s. Its solution
required deep work of Heegner and Stark, and the Fields medal theory of
Baker on linear forms in logarithms. Unfortunately, these methods do not
generalize to the cases where h>1. In the 1980s, Goldfeld, and Gross and
Zagier famously obtained the first effective class number bounds by
making use of deep results on the Birch and Swinnerton-Dyer Conjecture.
This lecture will tell the story of Gauss’s class number problem, and
will highlight new work by the speaker and Michael Griffin that offers
new effective results by different (and also more elementary) means.
Friday March 12, 4:00-5:00pm, THIS IS THE PUBLIC LECTURE
Speaker: Ken Ono (University of Virginia)
Topic: Why does Ramanujan, “The Man Who Knew Infinity”, matter?
Since this talk is open to the general public, we require registration in advance for this meeting: Use this link to preregister. After registering, you will receive a confirmation email containing information about joining the meeting.
Abstract: This lecture is about Srinivasa Ramanujan, “The Man Who Knew Infinity.” Ramanujan was a self-trained two-time college dropout who left behind 3 notebooks filled with equations that mathematicians are still trying to figure out today. He claimed that his ideas came to him as visions from an Indian goddess. This lecture gives many reasons why Ramanujan matters today. The answers extend far beyond his legacy in science and mathematics. The speaker was an Associate Producer of the film “The Man Who Knew Infinity” (starring Dev Patel and Jeremy Irons) about Ramanujan. He will share several clips from the film in the lecture, and will also tell stories about the production and promotion of the film.
Friday January 31, 4:30-5:30pm, WH-100E (NOTE SPECIAL DATE AND TIME)
Speaker: Daniel Studenmund (University of Notre Dame)
Topic: Hidden symmetries of groups
Many infinite discrete groups fail to have nice properties only
because of obstructions that disappear on passage to a finite-index
subgroup. Examples of such properties include superrigidity of
representations of linear groups and nilpotence of groups of polynomial
growth. The collection of all finite-index subgroups of a fixed group
Gamma has algebraic and geometric structures that can reflect properties
of Gamma. We will discuss some of these structures, including the abstract
commensurator of Gamma and commensurator growth of Gamma.
Thursday February 6, 4:15-5:15pm
Speaker: Selim Sukhtaiev (Rice University)
Topic: Anderson localization for disordered quantum graphs
Disorder is one of the central topics in modern science.
In this talk, we will discuss a mathematical treatment of a particular
disordered system modeling localization of quantum waves in random media.
The model in question was introduced by P. W. Anderson in his Nobel prize
winning work in physics which led to a rich mathematical theory of random
Schrodinger operators. We will show that the transport properties of several
natural Hamiltonians on metric trees with random branching numbers are
suppressed by disorder. This phenomenon is called Anderson localization.
Friday February 7, 4:30-5:30pm (NOTE SPECIAL DATE AND TIME)
Speaker: Li Chen (University of Connecticut)
Topic: On several functional inequalities for Markov semigroups and their applications
Abstract: Markov semigroups lie at the interface of analysis, PDEs, probability and geometry. Markov semigroup techniques, from both analytic and probabilistic viewpoints, have important applications in the study of functional inequalities coming from harmonic analysis, PDEs and geometry.
In this talk, we discuss regularization properties of heat semigroups and
their applications to the study of Sobolev type inequalities, isoperimetric
inequalities and $L^p$ boundedness of Riesz transforms in different
geometric settings. Fractal examples without differential structures are
emphasized. Besides, we also discuss sharp and dimension-free $L^p$ bounds
of singular integral operators via the martingale transform method.
Monday February 10
Speaker: Benjamin Schmidt (Michigan State University)
Topic: Preserve one, preserve all: Aleksandrov's problem in the context of Riemannian spaces.
Abstract: A classical theorem of Beckman and Quarles asserts that a function $F$ from a Euclidean space of dimension at least two to itself and having the property that $||F(x)-F(y)||=1$ whenever $||x-y||=1$ is necessarily an isometry. Aleksandrov has been credited with the problem of determining those metric spaces having this “preserve one distance, then preserve all distances” property.
Examples show that Riemannian manifolds need not have this property.
However, it is expected that self-functions of complete Riemannian
manifolds that preserve a sufficiently small distance are isometries.
I'll formulate a precise conjecture and will discuss supporting results
proved jointly with Meera Mainkar.
Thursday March 19, 4:15-5:15pm, WH-100E
Speaker: Kathryn Mann (Cornell University)
Thursday April 23, 4:15-5:15pm, WH-100E
Speaker: Thomas Hartman (Cornell University)
PETER HILTON MEMORIAL LECTURE
Thursday April 30, 3:00-4:00pm, LH009 (NOTE SPECIAL TIME AND LOCATION)
Speaker: Robert Gompf (University of Texas at Austin)
Topic: Exotic Smooth Structures on $\mathbb R^4$
Abstract: One of the most surprising discoveries in 4-manifold topology was the existence of smooth manifolds homeomorphic, but not diffeomorphic, to Euclidean 4-space. For fundamental reasons, this phenomenon can only occur in 4 dimensions. We will survey the subject, from its origin to recent developments regarding symmetries of such manifolds.