Department of Mathematics

Newton-Okounkov bodies are convex bodies associated to a line bundle \(L\) on a projective variety \(X\). We consider a variant \(\Delta_x(L)\) constructed from the degree lexicographic valuation determined by a general linear coordinate system around a point \(x\in X\). We prove that  \(\Delta_x(L)\) has a special shape that we call Borel-fixed, resembling a similar property of generic initial ideals. This shape imposes strong bounds on the body in terms of its widths. We also provide two algebro-geometric interpretations of these widths, first in terms of jumping behavior of base loci, and second as asymptotic measures of partial jet separation.

Our work recovers two known results: The size of the largest (resp. smallest) simplex contained in (resp. that contains) \(\Delta_x(L)\) is the Seshadri constant of \(L\) at \(x\) (resp. the width of \(L\) at \(x\)).

This is joint work with Victor Lozovanu.

The two main techniques of integration taught in calculus courses are integration by substitution and integration by parts. This talk will describe and illustrate a third technique of integration, almost never taught in math courses, called differentiation under the integral sign. It can handle integrals that appear inaccessible to simpler methods. The physicist Richard Feynman had great affection for differentiation under the integral sign, writing once “I caught on how to use that method, and I used that one damn tool again and again.”

Some familiarity with calculating partial derivatives from multivariable calculus will be assumed.

Note: Free refreshments. The talk starts at 5:40.

Title: Three approaches to building the real numbers.

Abstract: Historically, there have been two standard ways to make a rigorous definition of the real numbers: Dedekind’s method using Dedekind cuts and Cantor’s method using Cauchy sequences of rational numbers. We will review these approaches, indicate their advantages and disadvantages as well as generalizations that each admits, and then present a third way to define the real numbers that was worked out only in the past 50 years.

Postponed to September 16.

Abstract: TBA

Webex Link: https://uconn-cmr.webex.com/uconn-cmr/j.php?MTID=mf895c9b09fa6c4a5824628db5e65843d

Webex meeting number:2633 560 6690; Password:srBx3dzbq79

Speaker’s Biography: Marco Mirabella is the CEO of Ensuro, a decentralized reinsurance company licensed by the Bermuda Monetary Authority. Prior to Ensuro, Marco was the Chief Strategy Officer at BigGo, the biggest price comparison website in South East Asia. Marco is also the founder of Cartesi, a layer2 blockchain solution. He started his career in the venture capital space, working for SOSV a US$ 1b AUM venture capital fund.