Upcoming Events | Department of Mathematics

Upcoming Events

Friday, April 5, 2024 - 1:00pm
Friday, April 12, 2024 - 3:00pm

We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over the Gaussian quadratic field, and Suzuki's evaluation of the Fourier-Whittaker coefficients of quartic theta functions at squares. We also conjecture asymptotics for certain moments of quartic Gauss sums over primes. This is a joint work with C.David, A.Hamieh and H.Lin.

Friday, April 19, 2024 - 1:00pm

Based on joint with Adam Keilthy.

In isolated examples that appear in the literature, deformation coboundaries constructed geometrically for modular curves feature non-critical periods. These periods cancel in the associated deformation cocycle, but it was not clear whether non-critical periods are an artifact of a specific construction or genuinely contribute to the theory. In recent work, Bogo used an explicit hypergeometric uniformization to enable a calculation of deformation cocycles. He revealed a connection to quasi-modular forms, and confirmed the appearance of non-critical periods.

Using a formal deformation approach to the topic, we provide unique logarithmic extensions of any first order deformation, canonical and universal deformation families, and establish uniqueness of logarithmic deformation coboundaries. In particular, non-critical periods are not an artifact previous methods.

Friday, April 26, 2024 - 1:00pm
Friday, May 3, 2024 - 1:00pm
Friday, March 29, 2024 - 1:00pm

Perfect ideals of grade 3 can be classified based on algebra structures on their minimal free resolutions. The classification is incomplete in the sense that it remains open which of the possible algebra structures actually occur; this realizability question was formally posed by Avramov in 2012. Of five classes of algebra structures, the realizability question has been answered for one class. In this talk, we discuss and answer the realizability question for two classes and a partial answer for a third.

Friday, April 19, 2024 - 2:00pm

Nathan Dalaklis (Dynamics Seminar, TBD)

Wednesday, April 10, 2024 - 4:00pm

How many circles are tangent to 3 fixed circles? I'll explain how moduli spaces, a central topic in algebraic geometry, help shed light on the answer to this question and others like it. We'll start by discussing the moduli space of circles. This is an example of a moduli space of "embedded curves." Next, I'll explain the distinction between an "embedded curve" and its associated "abstract curve" (also called a "Riemann surface"). I'll finish by sharing some recent results about moduli spaces of abstract curves, which are joint work with Samir Canning.

Monday, April 15, 2024 - 5:00pm

The basic Equations of Number theory are the Diophantine Equations, which are polynomial equations with integer coefficients. One looks for integral, or rational, solutions, and makes use of the structure of real or complex solutions. Simplest non-trivial examples are the Pythagorean and Pell's Equations. Congruent numbers, which are positive integers which arise as areas of rational right triangles, have been understood quite well due to some striking recent results (due to others). At the end of the lecture, we will discuss a generalization to the setting of right rational tetrahedra, which leads to some intriguing geometry.

Friday, March 29, 2024 - 1:00pm

Perfect ideals of grade 3 can be classified based on algebra structures on their minimal free resolutions. The classification is incomplete in the sense that it remains open which of the possible algebra structures actually occur; this realizability question was formally posed by Avramov in 2012. Of five classes of algebra structures, the realizability question has been answered for one class. In this talk, we discuss and answer the realizability question for two classes and a partial answer for a third.

Friday, April 5, 2024 - 1:00pm
Wednesday, April 10, 2024 - 4:00pm

How many circles are tangent to 3 fixed circles? I'll explain how moduli spaces, a central topic in algebraic geometry, help shed light on the answer to this question and others like it. We'll start by discussing the moduli space of circles. This is an example of a moduli space of "embedded curves." Next, I'll explain the distinction between an "embedded curve" and its associated "abstract curve" (also called a "Riemann surface"). I'll finish by sharing some recent results about moduli spaces of abstract curves, which are joint work with Samir Canning.

Friday, April 12, 2024 - 3:00pm

We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over the Gaussian quadratic field, and Suzuki's evaluation of the Fourier-Whittaker coefficients of quartic theta functions at squares. We also conjecture asymptotics for certain moments of quartic Gauss sums over primes. This is a joint work with C.David, A.Hamieh and H.Lin.

Monday, April 15, 2024 - 5:00pm

The basic Equations of Number theory are the Diophantine Equations, which are polynomial equations with integer coefficients. One looks for integral, or rational, solutions, and makes use of the structure of real or complex solutions. Simplest non-trivial examples are the Pythagorean and Pell's Equations. Congruent numbers, which are positive integers which arise as areas of rational right triangles, have been understood quite well due to some striking recent results (due to others). At the end of the lecture, we will discuss a generalization to the setting of right rational tetrahedra, which leads to some intriguing geometry.

Friday, April 19, 2024 - 1:00pm

Based on joint with Adam Keilthy.

In isolated examples that appear in the literature, deformation coboundaries constructed geometrically for modular curves feature non-critical periods. These periods cancel in the associated deformation cocycle, but it was not clear whether non-critical periods are an artifact of a specific construction or genuinely contribute to the theory. In recent work, Bogo used an explicit hypergeometric uniformization to enable a calculation of deformation cocycles. He revealed a connection to quasi-modular forms, and confirmed the appearance of non-critical periods.

Using a formal deformation approach to the topic, we provide unique logarithmic extensions of any first order deformation, canonical and universal deformation families, and establish uniqueness of logarithmic deformation coboundaries. In particular, non-critical periods are not an artifact previous methods.

Friday, April 19, 2024 - 2:00pm

Nathan Dalaklis (Dynamics Seminar, TBD)

Friday, April 26, 2024 - 1:00pm
Friday, May 3, 2024 - 1:00pm