add abstract

Author: lenis2000

_posts/IMS/VML/2025-01-09-Jordan-Ellenberg.md

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 # image-address:
 image-alt: Jordan Ellenberg
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-# more-text: Abstracts
+more-text: Abstracts
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 <p>Jordan Ellenberg will deliver the Virginia Mathematics Lectures on March 4-5, 2025.</p>
 
 - **Lecture 1**: Tuesday, March 4, 2025, *Time and room TBA*
-  **Title:** Three is harder than two
 
-  **Abstract:**
-  Ambitious linear algebra students sometimes ask:  "Now that we're so good at studying nxn matrices, are we going to study nxnxn boxes of numbers?" This is a good question, and at an undergraduate level the answer is usually, "Good lord, no."  The theory of tensors is substantially harder and messier than the theory of matrices (there are about a dozen competing definitions for what "rank" is, for instance.)  In the same way, the combinatorics of sets of three things is harder than the combinatorics of sets fo two things.  How big can a subset of the unit disc be such that no two points are at distance less than .001?  Very manageable problem, the beginning of the subject of sphere packing.  How big can a subset of the unit disc be such that no three points form a triangle of area less than .001?  Very hard, still the subject of active research.  I'll give an overview of some problems on sets of three (sets of integers with no 3-term arithmetic progression, the cap set problem, the "no-isosceles" problem, Smyth's conjecture on Galois conjugates, etc.) and if there's time, I'll talk about how the resolution of the cap set problem ended up involving notions of ranks for tensors.
+  ### Three is harder than two
+
+  (Abstract is below)
 
 - **Lecture 2**: Wednesday, March 5, 2025, *Time and room TBA*
-  **Title:** What does machine learning have to offer pure mathematics?
 
-  **Abstract:**
-  The interaction of machine learning with math has attracted a lot of attention, because mathematics is in some respects a closed world with well-defined rules (like chess, and unlike poetry-writing) but also a domain where success is ultimately judged by human assessments of ingenuity and importance, not rigid criteria (like poetry-writing, and unlike chess). Can machines prove theorems? Can they have mathematical ideas?  In what areas of mathematics and in what aspects of mathematical practice is machine learning likely to have the biggest impact?  I'm not going to answer any of these questions, obviously.  But I'm going to speculate about them, based on my own experience working on applications of ML to pure math with both industry and academic collaborators.   Many of the problems discussed in my first lecture will appear again here, but each lecture will make sense without the other.
+  ### What does machine learning have to offer pure mathematics?
 
+  (Abstract is below)
 
+<!--more-->
 
+<h2 class="mt-4 mb-4">Abstracts</h2>
 
-<!--more-->
+- **Lecture 1**: Tuesday, March 4, 2025, *Time and room TBA*
+
+  ### Three is harder than two
+
+   **Abstract:**
+  Ambitious linear algebra students sometimes ask:  "Now that we're so good at studying nxn matrices, are we going to study nxnxn boxes of numbers?" This is a good question, and at an undergraduate level the answer is usually, "Good lord, no."  The theory of tensors is substantially harder and messier than the theory of matrices (there are about a dozen competing definitions for what "rank" is, for instance.)  In the same way, the combinatorics of sets of three things is harder than the combinatorics of sets fo two things.  How big can a subset of the unit disc be such that no two points are at distance less than .001?  Very manageable problem, the beginning of the subject of sphere packing.  How big can a subset of the unit disc be such that no three points form a triangle of area less than .001?  Very hard, still the subject of active research.  I'll give an overview of some problems on sets of three (sets of integers with no 3-term arithmetic progression, the cap set problem, the "no-isosceles" problem, Smyth's conjecture on Galois conjugates, etc.) and if there's time, I'll talk about how the resolution of the cap set problem ended up involving notions of ranks for tensors.
+
+- **Lecture 2**: Wednesday, March 5, 2025, *Time and room TBA*
+
+  ### What does machine learning have to offer pure mathematics?
+
+
+  **Abstract:**
+  The interaction of machine learning with math has attracted a lot of attention, because mathematics is in some respects a closed world with well-defined rules (like chess, and unlike poetry-writing) but also a domain where success is ultimately judged by human assessments of ingenuity and importance, not rigid criteria (like poetry-writing, and unlike chess). Can machines prove theorems? Can they have mathematical ideas?  In what areas of mathematics and in what aspects of mathematical practice is machine learning likely to have the biggest impact?  I'm not going to answer any of these questions, obviously.  But I'm going to speculate about them, based on my own experience working on applications of ML to pure math with both industry and academic collaborators.   Many of the problems discussed in my first lecture will appear again here, but each lecture will make sense without the other.