- What is the minimum number of distinct distances that are spanned by a set of points in general position?
- What is the minimum number of convex -gons determined by points in general position?
- For even , what is the maximum number of distinct ways in which one can cut a set of points in general position by a line into two halves, each of cardinality ?
- What is the maximum number of points that can be placed on an grid so that no three points are collinear. What happens when we also forbid four cocircular points?
An example of the first construction, created by Gábor Damásdi.
- Erdős and Purdy asked for the size of the largest subset of the integer lattice with no three points on a line and no four on a circle. The above construction provides the current best lower bound for this problem.
- Erdős, Hickerson, and Pach asked whether there exists a planar set of points in general position that spans no parallelograms and still determines distinct distances. Dumitrescu provided a positive answer to this question by using the above construction.
- The Heilbronn triangle problem asks for the smallest such that every set of points in the unit square determine a triangle of area at most . Erdős used a variant of the above construction to produce one of the first non-trivial lower bounds for the problem.
We are pleased to announce that we will provide pooled, subsidized
child care at STOC 2018. The cost will be $40 per day per child for
regular conference attendees, and $20 per day per child for students.
For more detailed information, including how to register for STOC 2018
childcare, see http://acm-stoc.org/stoc2018/childcare.html
“The way to get to the top of the heap in terms of developing original research is to be a fool, because only fools keep trying. You have idea number 1, you get excited, and it flops. Then you have idea number 2, you get excited, and it flops. Then you have idea number 99, you get excited, and it flops. Only a fool would be excited by the 100th idea, but it might take 100 ideas before one really pays off. Unless you’re foolish enough to be continually excited, you won’t have the motivation, you won’t have the energy to carry it through. God rewards fools.”
- Frank de Zeeuw – working mainly on polynomial methods in Discrete Geometry. Frank has very impressive results concerning expanding polynoamials, and the current best bound for the Szemerédi–Trotter theorem over finite fields.
- Pablo Soberon – working mainly on central ideas in convex geometry, such as Tverberg’s theorem and Helly’s theorem. Pablo also has an impressive background in the world of math competitions.
- Yumeng Ou – working in Harmonic Analysis and more specifically on restriction problems and related topics. Personally, I’m very interested in her works involving polynomial methods and Falconer’s distance problem.
- Andrew Obus – working in Algebraic Geometery and Number Theory. I can write more but I’m afraid of getting the details wrong. So just look at Andrew’s webpage to see his impressive works.
- Unlike learning material or doing an assignment, it is not clear whether what you are trying to do is possible. It might be that the math problem you are trying to solve is unsolvable. Or perhaps the problem is solvable but the tools for handling it would only be discovered in 200 years. Or perhaps it is solvable now but after several months of making slow progress, some renowned mathematician will publish a stronger result that makes your work obsolete. These scenarios are not that rare in mathematics and related theoretical fields. Are you still going to spend months of hard work on a problem with these possibilities in mind?
- Unlike exams and most jobs, there are no clear deadlines. It is likely that nothing horrible will happen if you will not work on research today, or this week, or this month. There might not be any short term consequences when spending a whole month watching the 769 episodes of “Antique Roadshow”.
- It’s hard! Working on an unsolved problem tends to require more focus and deeper thinking than learning a new topic. Also, part of the work involves trying to prove some claim for weeks/months/years and not giving up. It is surprising to discover that reading a textbook or doing homework becomes a way of procrastinating – it is easier than thinking hard on your research.
- Brainstorming is not a solution. For most people it is much easier to discuss a problem with others than to focus on it on their own. Sessions of working with someone obviously have many advantages, but they are not a solution for the willpower problem. One needs to spend time and frustration thinking hard on the problem on their own. Otherwise, they are unlikely to get a good understanding of the topic and get to the deeper issues. Brainstorming sessions become much more effective after first spending time alone and obtaining some deeper understanding and intuition.
- Collaborations do help. Unlike a brainstorming session, long-term collaborations do seem to help with the willpower problem. Not wanting to disappoint a collaborator that I respect, I will have extra motivation to work hard. Having someone else that is interested in the problems also helps keep the motivation high.
- Reserve long stretches of time for research work. Like most people, I constantly have a large amount of non-research tasks, from preparing lectures to babyproofing the house. It is tempting to focus on the non-research tasks since these require less focus and are easier to scratch of the to-do list. When this happens I try to place in my schedule long stretches of time dedicated to research. I try to find times when I am unlikely to be tired or distracted. Sometimes I turn off the wireless and phone during these times. To quote Terence Tao:
“Working with high-intensity requires a rather different “mode” of thought than with low-intensity tasks. (For instance, I find it can take a good half-hour or so of uninterrupted thinking before I am fully focused on a maths problem, with all the relevant background at my fingertips.) To reduce the mental fatigue of transitioning from one “mode” to another, I find it useful to batch similar low-intensity tasks together, and to separate them in time (or space) from the high-intensity ones.”
- Procrastination with writing tasks is a separate issue. While beginners often have a hard time sitting to write and revise their work, this seems to be a simpler problem. The magic solution seems to be writing a lot (not necessarily research work). After a lot of practice, writing becomes a task that does not require a lot of mental energy or deep concentration, is easy to do, and is mostly fun.
- Find the research environment that works best for you. This is an obvious observation, but I would still like to state it. Different people have different environments that work better for them: Some need a quiet environment while others focus better in a crowded coffee shop, some focus better in the morning while others prefer the middle of the night, and so on.
- Find ways to keep yourself highly motivated. Everyone seems to be at least somewhat motivated by being successful and by their ego. Everyone seem to be at least somewhat motivated by an urge to discover the mathematical truth. However, most people seem to need additional motivation when things are not going well. Some people get extra motivation by being surrounded with hard working people. Others become more motivated by reading biographies of successful mathematician and scientists. And so on.