MATH 301 EXAM

Notable Fields Medalist

- Caucher Birkar, Alessio Figalli, Peter Scholze, Ashley Venkatesh (most recent field medalists 2018) - Maryam Mirzakhani was the first woman to receive it (2014) died in 2017 (age 40) - Grigori Perelman 2006 - solving the Poincare Conjecture (one millenium problem)

Peer Review Process

- DOUBLE BLIND system: journal's editors transmit all back and forth communication between peer reviewers and authors without revealing names

Fields Medal

- awarded every 4 years up to 4 young mathematicians (under 40 yrs old) - awarded at the international congress of mathematicians

Abel Prize

- awarded to one mathematician every year by the King of Norway - it is equivalent of the Nobel Prize in other disciplines

Intro of an article of OG research in applied math

- background info about topic and field of research - scope, description, goal, relevance of research project - review of previous published works that are used as starting points for the current research project - explanation of the way in which the current research claims to be original

Mathematization Sciences

- before 20th century - applied math was mostly restricted to applications in physics - this refers to the gradual increase of math applications to all other fields of natural and social science during 2nd half of the 20th century

APPLIED MATH

- branch of math connected to science - empirical study of how physical world works - requires using experimental data collected by observing the physical world - primary activity: writing and working with math models, which describes the laws of nature and other equations that govern how the world works

Role of Peer Reviewers

- carefully read a submitted manuscript and check it for correctness and originality - make recommendations to authors and editors about whether it deserves to be published or nawh

ADV of incorporating math modeling into scientific method

- compared at the cost of experimental science, math models can be built and solved very FAST and very LITTLE COST - reveal patterns in experimental data that are invisible to the naked eye - would go undiscovered by experimental science alone - simulate experiments quickly and cheaply to help predict future experimental results and decide how to best invest money, time, effort towards future experiments

Euclid's Elements

- compiled all known math in his day (300 BC) into 13 books called the elements. - begins with primitive terms (23) - axioms (5) - 465 theorems in logical sequence - each proof is justified only by previously proved theorems and axioms

Besides peer reviewed journals, what are other sources of published mathematical research?

- conference proceedings - dissertations - books - working papers or manuscripts that have not yet completed the peer-review process

Properties of Axioms

- consistent - complete - independent

MATH STATEMENT

- declarative statement that must be either T/F - decidable - satisfy rule of noncontradiction (cannot be both true and false) no middle (cannot be neither true or false) - purpose is NOT to describe, clarify, or explain

Math Theory

- established by axiomatic method - consists sets of theorems which are proved on the basis of a set of axioms - logical deduction based on previously proved theorems - only true within the context of the axiomatic system - may fall apart if axioms are changed

WOMEN in the history of math

- heavily discriminated in Western academia until 20th century - not allowed to study math or even audit math in uni. - only a small number of wealthy women had access to math by hiring tutors

MATH MODEL

- math equation whose solution describes a natural phenomenon - math modeling = using a math model within the context of a scientific method to help the progress of science by discovering patterns and mechanisms that govern how the physical world works

Rhetorical Style

- math written in only words and sentences - SYMBOLS - 14th century (300 years for them to be standardized)

Origin of MATH

- mesopotamia (5000 BC) - egypt (4000 BC) - china (2000 BC) - mesoamerica (2000 BC) - greece (600 AD) - india (400 AD) - arab empire (800 AD)

PURE MATH

- non-empirical - does not use scientific method - uses axiomatic method - does not have to be practical or consistent with physical reality

Progress made in Applied Math

- partners with an experimental science (natural/social) to build a math model that describes a real-life system and use the model to study the system. - uses experimental data collected within the framework of the scientific method

Who can access articles in a peer reviewed journal?

- reader who has a subscription - pay-per-view price (PPV) - open access (OA) if the author chooses (free for all)

Natural PHILO

- refers to the study of how the physical world and universe work - 17th century, it started using the scientific method and experiments to study the physical world, and using math tools to describe patterns in the experimental data

Original Research

- refers to working on a problem that has not been solved before such as proving a theorem or developing a new method - expands the bounds of human knowledge, must be published by the persons who did the research

Scientific Theorey

- scientific method - confirms/rejects hypothesis about a system of behavior - constantly subject to revision and correction as new experimental results become available

Axiomatic Method

- template for organizing ideas and concepts in all fields of math - used in many other fields - philo, theology, law, poli sci

5 axioms in Euclidean geometry

1. each pair of points defines a line 2. a finite segment can be extended indefinitely in a straight line 3. given 2 distinct points O and A, there is a circle centered at O with radius OA. 4. all right angles are equal 5. if L1 is a line and P is a point not on L1, there is a unique line L2 through P parallel to L1.

Western (European) civilization began pursuing math

1200 AD - importing math concepts to middle east and india - algebra - started using hindu-arabic basic 10 number system - more practical than roman numerals - christian church established the first unis, required all students to study math (logic, geom, arithmetic)

Millenium Problems

7 math problems proposed by the Clay Institute in the year 2000 as the most important unsolved problems in math. Promises 1 million dollars to the first person who solves each of these problems.

ADV and DISADV of peer reviewers

ADV - high quality, free of errors, guaranteed to be truly original DISADV - slow and expensive

Axiomatic Method VS Scientific Method

AM: does not have to be consistent with physical reality SM: true if it is consistent with physical reality and can be verified by repeated experiments

Hypothesis & Theory in AXIOMATIC & SCIENTIFIC method

Axiomatic: - hypothesis is the first part of a conditional statement - theory: set of theorems based on the same set of axioms Scientific - hypothesis is an educated guess about the behavior of the system, waiting to be supported/rejected by an experiment - explanation of a physical system that has been supported by repeated experiments

Name a 17th century NATURAL PHILOSOPHER who used experiments and math to study the world.

GALILEO - gravity KEPLER - planetary motion NEWTON - light diffraction and motion

Contrapositive

If not Q, then Not P - backs up conditional statement - they are either both true or both false

Math definitions

a theory must also provide precise Math definitions for the terms it uses - primitive terms (assumed to be clear enough - no need explanation)

John Nash

awarded the Abel Prize in 2015 for inventing Game Theory and for his research in partial differentiation equations (died shortly after)

Andrew Wiles

awarded the Abel Prize in 2016 for solving Fermat's Last Theorem (1995) - a problem that had been open for more than 350 years

Non Euclidean Geometry

axiomatic system that replaces the 5th axiom of Euclidean geometry - if L1 is a line and P is a point not on L1, there ARE NO LINES through P parallel to L1 - if L1 is a line and P is a point not on L1, there ARE MORE THAN ONE LINE through P parallel to L1.

Experimental Scientist

collects experimental data - involves designing experiments, making observations, taking measurements

Review Research

compiling, organizing, comparing the orginal research of other people

GREEK math

developed geometry into an abstract system used to organize true statements (theorems) into a coherent structure based on axioms - which holds together by chains of deductive reasoning (Axiomatic Method) - basis of math

Peer Reviewed Journal

enlists a team of outside experts in the publication process. they are not employed by the journal directly and they do not make the final determination on whether or not a manuscript gets published. but their comments are taken very seriously by the journal editors who do make the final decision

Progress made in Pure Math

expands a mathematical theory by proving a new theorem that generalizes and or extends an existing theorem. it begins with a conjecture, which becomes a theorem when it is proved deductively using the axiomatic method.

Expository Research

explaining in research of other people to a broader audience

Karen Uhlenbeck

first woman to ever receive the Abel Prize (2019)

Conditional Statement

if P, then Q

Secondary Literature

includes review literature and expository literature

Algebra

math objects (numbers, vectors, matrices) and properties of operations between them

Conjecture

math statement believed to be true, but for which no one has been able to provide a valid proof yet - once someone has a proof it becomes a theorem

Fermat's Last Theorem

no positive integers a, b, and c that satisfy the equation a^n+b^n=c^n for any integer n>2 - it was proposed in 1637 and solved in 1995 (andrew wiles)

Impact Factor of a Journal

number that represents the average number of times that an article published in that journal is cited in other works - seen as an indicator for the overall quality, prestige, and its importance of the journal

Primary Literature

original research written by the persons who did the research

Proof of a theorem

provides a sequence of equivalent statements that are true because they follow a valid rule of inference that connects them to a previously proved theorem or an axiom

Current fields of open research in pure and applied mathematics:

pure: number theory, algebra, topology, analysis applied: differential equations, stats, operations research, numerical analysis

Axiomatic System

set of axioms used as a foundation to prove theorems

Math Theory

set of theorems build on a common set of axioms

2 main types of MATH MODELS

similarities: require experimental data collected by experiments within the context of the scientific method mechanistic model - uses differential equations to discover the laws of nature or first principles that describe the mechanisms that drive the patterns observed in experimental data - these models are more common in the natural sciences empirical model - uses stats to discover and describe patterns in data and infer cause-effect relas. these models are more common in the social sciences (econ, psych, socio)

Computational Scientist

solves the model - most models are solved by designing an algorithm, implemented by a computer to solve a problem

Axiom

statement that is accepted as being true by definiton

Analysis

studies infinity, infinitesimals, and the related concepts encountered in calculus (limits, series, derivatives, integrals)

Topology

studies properties of geometrical objects that are preserved under a continuous deformation, that is, how their points are connected to each other regardless of their actual shape

Number Theory

study of integers

Descriptive Defintiion

vague or subjective

Math Modeler

writes math equations of the model - requires using the data to make informed decisions and making simplifying modeling assumptions to translate the behavior of a real-life system into a set of equations