Events

Kostiantyn Drach, Universitat de Barcelona

Reverse inradius inequalities for ball-bodies

A ball-body, also called a $\lambda$-convex body, is an intersection of congruent Euclidean balls of radius $1/\lambda$ in $\mathbb{R}^n$, $n \geq 2$. Such bodies arise naturally in optimization problems in combinatorial and convex geometry, in particular when the number of generating balls is finite. In recent years, ball-bodies have also played a central role in an active research program on reverse isoperimetric-type problems under curvature constraints. The general objective of this program is to understand how prescribed curvature bounds restrict the extremal behavior of geometric functionals (e.g., volume, surface area, or mean width), and to identify sharp inequalities between them that reverse the existing classical isoperimetric-type inequalities. In this talk, we focus on the inradius minimization problem for $\lambda$-convex bodies with prescribed surface area or prescribed mean width. Here, the inradius of a convex body $K$ is the radius of the largest ball contained in $K$. In this setting, we establish sharp lower bounds for the inradius and show that equality is attained only by lenses, that is, intersections of two balls of radius $1/\lambda$. This solves a conjecture of Karoly Bezdek. We will outline the main ideas of the proof and pose several open problems. This is joint work with Kateryna Tatarko.

MC 5417

Duncan McCoy, Université du Québec à Montréal

The unknotting number of positive alternating knots

The unknotting number is simultaneously one of the simplest classical knot invariants to define and one of the most challenging to compute. This intractability stems from the fact that typically one has no idea which diagrams admit a collection of crossing changes realizing the unknotting number for a given knot. For positive alternating knots, one can show that if the unknotting number equals the lower bound coming from the classical knot signature, then the unknotting number can be calculated from an alternating diagram. I will explain this result along with some of the main tools in the proof, which are primarily from smooth 4-dimensional topology. This is joint work with Paolo Aceto and JungHwan Park.

MC 5417

Matt Young, Rutgers University

The shifted convolution problem for Siegel modular forms

The shifted convolution problem for Fourier coefficients of cusp forms has seen a lot of attention due to applications towards moments of L-functions and the subconvexity problem. However, the problem for higher rank automorphic forms (beyond GL_2) has been a notorious bottleneck towards progress on the sixth moment of the Riemann zeta function. In this talk, I will discuss recent progress on the problem for Siegel cusp forms on Sp_4. This is joint work with Wing Hong (Joseph) Leung.

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Caleb Suan, Chinese University of Hong Kong

Hull-Strominger Systems and Geometric Flows

The Hull-Strominger system is a system of partial differential equations stemming from heterotic string theory in physics. Mathematically, these equations lead us to consider special structures with torsion and have been proposed as a natural generalization of the Ricci-flat condition on non-Kahler Calabi-Yau threefolds. In this talk, we discuss a geometric flow approach to the system, known as the anomaly flow. We shall also look at 7-dimensional analogues of the system and flow.

MC 5417

Evan Sundbo, University of Waterloo

Broken Toric Varieties and Balloon Animal Maps

We will see the definition and some examples of broken toric varieties and balloon animal maps between them. After an overview of some of the many different areas in which they appear, we look at how their geometry can be studied via complexes of sheaves on an associated complex of polytopes. This yields results such as a version of the Decomposition Theorem and some explicit formulas for dimensions of rational cohomology groups of broken toric varieties.

MC 5417

Damaris Schindler, University of Göttingen

Density of rational points near manifolds

Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? How does this counting function behave if we let the size of the denominator go to infinity? The study of the density of rational points near manifolds has seen significant progress in the last couple of years. In this talk I will explain why we might be interested in this question, focusing on applications in Diophantine approximation and the (quantitative) arithmetic of projective varieties.

MC 5403

Damaris Schindler, University of Göttingen

Quantitative weak approximation and quantitative strong approximation for certain quadratic forms

In this talk we discuss recent results on optimal quantitative weak approximation for certain projective quadrics over the rational numbers as well as quantitative results on strong approximation for quaternary quadratic forms over the integers. We will also discuss results on the growth of integral points on the three-dimensional punctured affine cone and strong approximation with Brauer-Manin obstruction for this quasi-affine variety. This is joint work with Zhizhong Huang and Alec Shute.

MC 5479