In set theory one encounters statements which cannot be decided within a background theory one works in. Some of these statemetns can be shown to be consistent within standard background theories widely accepted in mathematics, but some of them require the use of very large sets, known under the term ``large cardinals". In this talk I will discuss how large cardinas naturally arise when studying relative consistencies and also give examples from mainstream mathematics which lead to relative consistencies and large cardinals.

We can think of a line as the shortest curve joining its endpoints and of a circle as the shortest closed curve enclosing a fixed area (isoperimetric problem). In this talk we will discuss what happens when curves are replaced by surfaces in such problems, and how the solutions of these problems can shed light on the differential geometry of curved spaces. 

Location: Zoom Address https://zoom.us/j/8473088589

In this talk, I will first give a brief history of the non-Euclidean geometry. After that, I will present the Riemann's point of view of geometry which led the modern differential geometry.

The talk will be given by zoom, the link is

https://uci.zoom.us/j/97641313435

Zoom Address: 

https://zoom.us/j/8473088589

The goal of this talk is to explain how enumerative geometry can be used to simplify the solution of polynomials in one variable. Given a polynomial in one variable, what is the simplest formula for the roots in terms of the coefficients? Hilbert conjectured that for polynomials of degree 6,7 and 8, any formula must involve functions of at least 2, 3 and 4 variables respectively (such formulas were first constructed by Hamilton). In a little-known paper, Hilbert sketched how the 27 lines on a cubic surface should give a 4-variable solution of the general degree 9 polynomial. In this talk I’ll recall Klein and Hilbert's geometric reformulation of solving polynomials, explain the gaps in Hilbert's sketch and how we can fill these using modern methods. As a result, we obtain best-to-date upper bounds on the number of variables needed to solve a general degree n polynomial for all n, improving results of Segre and Brauer.