A person needing a kidney transplant may have a friend or relative who volunteers to be a living donor, but whose kidney is incompatible, forcing the person to wait for a transplant from a deceased donor. In the U.S. alone, thousands of people die each year without ever finding a suitable kidney. A new technique applies graph theory to groups of incompatible patient-donor pairs to create the largest possible number of paired-donation exchanges. These exchanges, in which a donor paired with Patient A gives a kidney to Patient B while a donor paired with Patient B gives to Patient A, will dramatically increase transplants from living donors. Since transplantation is less expensive than dialysis, this mathematical algorithm, in addition to saving lives, will also save hundreds of millions of dollars annually. Naturally there can be more transplants if matches along longer patient-donor cycles are considered (e.g., A.s donor to B, B.s donor to C, and C.s donor to A). The problem is that the possible number of longer cycles grows so fast hundreds of millions of A >B>C>A matches in just 5000 donor-patient pairs that to search through all the possibilities is impossible. An ingenious use of random walks and integer programming now makes searching through all three-way matches feasible, even in a database large enough to include all incompatible patient-donor pairs. For More Information: Matchmaking for Kidneys, Dana Mackenzie, SIAM News, December 2008. Image of suboptimal two-way matching (in purple) and an optimal matching (in green), courtesy of Sommer Gentry.

What.s in store for our climate and us? It.s an extraordinarily complex question whose answer requires physics, chemistry, earth science, and mathematics (among other subjects) along with massive computing power. Mathematicians use partial differential equations to model the movement of the atmosphere; dynamical systems to describe the feedback between land, ocean, air, and ice; and statistics to quantify the uncertainty of current projections. Although there is some discrepancy among different climate forecasts, researchers all agree on the tremendous need for people to join this effort and create new approaches to help understand our climate. It.s impossible to predict the weather even two weeks in advance, because almost identical sets of temperature, pressure, etc. can in just a few days result in drastically different weather. So how can anyone make a prediction about long-term climate? The answer is that climate is an average of weather conditions. In the same way that good predictions about the average height of 100 people can be made without knowing the height of any one person, forecasts of climate years into the future are feasible without being able to predict the conditions on a particular day. The challenge now is to gather more data and use subjects such as fluid dynamics and numerical methods to extend today.s 20-year projections forward to the next 100 years. For More Information: Mathematics of Climate Change: A New Discipline for an Uncertain Century, Dana Mackenzie, 2007.

What.s in store for our climate and us? It.s an extraordinarily complex question whose answer requires physics, chemistry, earth science, and mathematics (among other subjects) along with massive computing power. Mathematicians use partial differential equations to model the movement of the atmosphere; dynamical systems to describe the feedback between land, ocean, air, and ice; and statistics to quantify the uncertainty of current projections. Although there is some discrepancy among different climate forecasts, researchers all agree on the tremendous need for people to join this effort and create new approaches to help understand our climate. It.s impossible to predict the weather even two weeks in advance, because almost identical sets of temperature, pressure, etc. can in just a few days result in drastically different weather. So how can anyone make a prediction about long-term climate? The answer is that climate is an average of weather conditions. In the same way that good predictions about the average height of 100 people can be made without knowing the height of any one person, forecasts of climate years into the future are feasible without being able to predict the conditions on a particular day. The challenge now is to gather more data and use subjects such as fluid dynamics and numerical methods to extend today.s 20-year projections forward to the next 100 years. For More Information: Mathematics of Climate Change: A New Discipline for an Uncertain Century, Dana Mackenzie, 2007.

Mathematics has helped investigators in several major cases of human rights abuses and election fraud. Among them: The 2009 election in Iran. A mathematical result known as Benford's Law states that the leading digits of truly random numbers aren't distributed uniformly, as might be expected. Instead, smaller digits, such as 1's, appear much more frequently than larger digits, such as 9's. Benford's Law and other statistical tests have been applied to the 2009 election and suggest strongly that the final totals are suspicious. Ethnic cleansing. When Slobodan Milosevic went on trial, it was his contention that the mass exodus of ethnic Albanians from Kosovo was due to NATO bombings and the activities of the Albanian Kosovo Liberation Army rather than anything he had ordered. A team collected data on the flow of refugees to test those hypotheses and was able to refute Milosevic's claim in its entirety. Guatemalan disappearances. Here, statistics is being used to extract information from over 80 million National Police archive pages related to about 200,000 deaths and disappearances. Sampling techniques give investigators an accurate representation of the records without them having to read every page. Families are getting long-sought after proof of what happened to their relatives, and investigators are uncovering patterns and motives behind the abductions and murders. Tragically, the people have disappeared. But because of this analysis, the facts won't. For More Information: Killings and Refugee Flow in Kosovo, March-June 1999, Ball et al., 2002.

Mathematics has helped investigators in several major cases of human rights abuses and election fraud. Among them: The 2009 election in Iran. A mathematical result known as Benford's Law states that the leading digits of truly random numbers aren't distributed uniformly, as might be expected. Instead, smaller digits, such as 1's, appear much more frequently than larger digits, such as 9's. Benford's Law and other statistical tests have been applied to the 2009 election and suggest strongly that the final totals are suspicious. Ethnic cleansing. When Slobodan Milosevic went on trial, it was his contention that the mass exodus of ethnic Albanians from Kosovo was due to NATO bombings and the activities of the Albanian Kosovo Liberation Army rather than anything he had ordered. A team collected data on the flow of refugees to test those hypotheses and was able to refute Milosevic's claim in its entirety. Guatemalan disappearances. Here, statistics is being used to extract information from over 80 million National Police archive pages related to about 200,000 deaths and disappearances. Sampling techniques give investigators an accurate representation of the records without them having to read every page. Families are getting long-sought after proof of what happened to their relatives, and investigators are uncovering patterns and motives behind the abductions and murders. Tragically, the people have disappeared. But because of this analysis, the facts won't. For More Information: Killings and Refugee Flow in Kosovo, March-June 1999, Ball et al., 2002.

Normally when creating a digital file, such as a picture, much more information is recorded than necessary-even before storing or sending. The image on the right was created with compressed (or compressive) sensing, a breakthrough technique based on probability and linear algebra. Rather than recording excess information and discarding what is not needed, sensors collect the most significant information at the time of creation, which saves power, time, and memory. The potential increase in efficiency has led researchers to investigate employing compressed sensing in applications ranging from missions in space, where minimizing power consumption is important, to MRIs, for which faster image creation would allow for better scans and happier patients. Just as a word has different representations in different languages, signals (such as images or audio) can be represented many different ways. Compressed sensing relies on using the representation for the given class of signals that requires the fewest bits. Linear programming applied to that representation finds the most likely candidate fitting the particular low-information signal. Mathematicians have proved that in all but the very rarest case that candidate-often constructed from less than a tiny fraction of the data traditionally collected-matches the original. The ability to locate and capture only the most important components without any loss of quality is so unexpected that even the mathematicians who discovered compressed sensing found it hard to believe. For More Information: "Compressed Sensing Makes Every Pixel Count," What's Happening in the Mathematical Sciences, Vol. 7, Dana Mackenzie.

As difficult as it is to do the census, the ensuing process of determining the number of congressional seats for each state can be even harder. The basic premise, that the proportion of each state's delegation in the House should match its proportion of the U.S. population, is simple enough. The difficulty arises when deciding what to do with the fractions that inevitably arise (e.g., New York can't have 28.7 seats). Over the past 200 years, several methods of apportioning seats have been used. Many sound good but can lead to paradoxes, such as an increase in the total number of House seats actually resulting in a reduction of a state's delegation. The method used since the 1940s, whose leading proponent was a mathematician, is one that avoids such paradoxes. A natural question is Why 435 seats? Nothing in the Constitution mandates this number, although there is a prohibition against having more than one seat per 30,000 people. One model, based on the need for legislators to communicate with their constituents and with each other, uses algebra and calculus to show that the ideal assembly size is the cube root of the population it represents. Remarkably, the size of the House mirrored this rule until the early 1900s. To obey the rule now would require an increase to 670, which would presumably both better represent the population and increase the chances that the audience in the seats for those late speeches would outnumber the speaker. For More Information: "E pluribus confusion", Barry Cipra, American Scientist, July-August 2010.

As difficult as it is to do the census, the ensuing process of determining the number of congressional seats for each state can be even harder. The basic premise, that the proportion of each state's delegation in the House should match its proportion of the U.S. population, is simple enough. The difficulty arises when deciding what to do with the fractions that inevitably arise (e.g., New York can't have 28.7 seats). Over the past 200 years, several methods of apportioning seats have been used. Many sound good but can lead to paradoxes, such as an increase in the total number of House seats actually resulting in a reduction of a state's delegation. The method used since the 1940s, whose leading proponent was a mathematician, is one that avoids such paradoxes. A natural question is Why 435 seats? Nothing in the Constitution mandates this number, although there is a prohibition against having more than one seat per 30,000 people. One model, based on the need for legislators to communicate with their constituents and with each other, uses algebra and calculus to show that the ideal assembly size is the cube root of the population it represents. Remarkably, the size of the House mirrored this rule until the early 1900s. To obey the rule now would require an increase to 670, which would presumably both better represent the population and increase the chances that the audience in the seats for those late speeches would outnumber the speaker. For More Information: "E pluribus confusion", Barry Cipra, American Scientist, July-August 2010.

The heart.s function of pumping blood may seem fairly simple but the underlying mechanisms and electrical impulses that maintain a healthy rhythm are extremely complex. Many areas of mathematics, including differential equations, dynamical systems, and topology help model the electrical behavior of cardiac cells, the connections between those cells and the heart.s overall geometry. Researchers aim to gain a better understanding of the normal operation of the heart, as well as learn how to diagnose the onset of abnormalities and correct them. Of the many things that can go wrong with a heart.s rhythm, some measure of unpredictability is (surprisingly) not one of them. A healthy heartbeat is actually quite chaotic not regular at all. Furthermore, beat patterns become less chaotic as people age and heart function diminishes. In fact, one researcher recommends that patients presented with a new medication should ask their doctors, "What is this drug going to do to my fractal dimensionality?" For More Information: Taking Mathematics to Heart: Mathematical Challenges in Cardiac Electrophysiology, John W. Cain, Notices of the AMS, April 2011, pp. 542-549.

There.s more mathematics involved in juggling than just trying to make sure that the number of balls (or chainsaws) that hits the ground stays at zero. Subjects such as combinatorics and abstract algebra help jugglers answer important questions, such as whether a particular juggling pattern can actually be juggled. For example, can balls be juggled so that the time period that each ball stays aloft alternates between five counts and one? The answer is Yes. Math also tells you that the number of balls needed for such a juggling pattern is the average of the counts, in this case three. Once a pattern is shown to be juggleable and the number of balls needed is known, equations of motion determine the speed with which each ball must be thrown and the maximum height it will attain. Obviously the harder a juggler throws, the faster and higher an object will go. Unfortunately hang time increases proportionally to the square root of the height, so the difficulty of keeping many objects in the air increases very quickly. Both math and juggling have been around for millennia yet questions still remain in both subjects. As two juggling mathematicians wrote, .A juggler, like a mathematician, is never finished: there is always another great unsolved problem.

There.s more mathematics involved in juggling than just trying to make sure that the number of balls (or chainsaws) that hits the ground stays at zero. Subjects such as combinatorics and abstract algebra help jugglers answer important questions, such as whether a particular juggling pattern can actually be juggled. For example, can balls be juggled so that the time period that each ball stays aloft alternates between five counts and one? The answer is Yes. Math also tells you that the number of balls needed for such a juggling pattern is the average of the counts, in this case three. Once a pattern is shown to be juggleable and the number of balls needed is known, equations of motion determine the speed with which each ball must be thrown and the maximum height it will attain. Obviously the harder a juggler throws, the faster and higher an object will go. Unfortunately hang time increases proportionally to the square root of the height, so the difficulty of keeping many objects in the air increases very quickly. Both math and juggling have been around for millennia yet questions still remain in both subjects. As two juggling mathematicians wrote, .A juggler, like a mathematician, is never finished: there is always another great unsolved problem.

James Sethian and Frank Morgan talk about their research investigating bubbles.

Muthu Alagappan explains how topology and analytics are bringing a new look to basketball.

Van Wedeen talks about the geometry of the brain's communication pathways.

Nazareth Bedrossian talks about using math to reposition the International Space Station.

Researcher: Christopher Butson, Scientific Computing and Imaging Institute, University of Utah. Christopher Butson talks about work he's done to help treat Parkinson's disease.

Researcher: Michel C. Molinkovitch, University of Geneva Description: Michel C. Milinkovitch used math, physics, and biology for an amazing discovery about the patterns on a lizard's skin.

Tom Patterson and Bojan Savric discuss the Equal Earth projection map that they created with Bernhard Jenny.

Rob Schneiderman talks about the metaphorical connections between math and music

euromicron AG, a medium-sized technology group and specialist for networking business and production processes, has taken on Funkwerk AG, Kölleda, as a strategic anchor investor. Funkwerk AG is a leading provider of innovative communication, information and security systems and intends to acquire a total of up to around 28% of euromicron AG’s increased share capital following the implementation of the resolved capital measures.

Chris Gayle had every reason to celebrate West Indies' seven-wicket victory over Australia on Saturday

From the moment he became a trainer, Frederick Watson says his dream was to win the Derby.

Water scarcity, a socio-environmental threat to anthropogenic activities and ecosystems alike, affects large regions of the globe. However, it is often the most vulnerable and disadvantaged populations that suffer the severest consequences, highlighting the role of economic and institutional factors in water scarcity. In this way, researchers generally consider not only the physical constraints but socio-economic determinants as well.

A new study from University of Michigan climate researchers concludes that some of the latest-generation climate models may be overly sensitive to carbon dioxide increases and therefore project future warming that is unrealistically high.

Fishery collapses can be difficult to forecast and prevent due to hyperstability, a phenomenon where catch rates remain high even as fish abundance declines. In a recent Canadian Journal of Fisheries and Aquatic Sciences study, researchers conducted a whole-lake experiment to reveal the causes of hyperstability in recreational fisheries. Fish habitat preferences were found to leave them vulnerable to overexploitation.

In a ground breaking study, an international team of 21 scientists led by Sean Hoban, Ph.D., Conservation Biologist at The Morton Arboretum in Lisle, Illinois, evaluated five genera spanning the plant tree of life (Hibiscus, Magnolia, Pseudophoenix, Quercus and Zamia) to understand how much genetic diversity currently exists in collections in botanical gardens and arboreta worldwide.

Rampant deforestation, uncontrolled expansion of agriculture, infrastructure development and exploitation of wild species have created a 'perfect storm' for the spillover of diseases from wildlife to people.

"The coronavirus pandemic has made abundantly clear that if life is to thrive on this Earth, human and nonhuman, we need cooperation at all scales-global, regional, binational, within a nation, interstate, and in our local communities. And we need to learn how to coexist with and have compassion for our nonhuman relatives-and acknowledge in the midst of this pandemic that bats are not our enemies."

Sovereign nations typically measure economic success in terms of GDP (income) but this approach is risky as it fails to track and measure the impact of this on nature. Inclusive wealth, on the other hand captures financial and produced capital, but also the skills in our workforce (human capital), the cohesion in our society (social capital) and the value of our environment (natural capital).

The good thing about BirdLife is that, as a truly global organisation, we're already great at staying connected, even when we're thousands of miles apart. Here are just a few of the ways our work will carry on over the coming months, even if it's from our living rooms.

The earth we abuse and the living things we kill will, in the end, take their revenge; for in exploiting their presence we are diminishing our future. ~ Marya Mannes Each species represents a thread in the closely woven fabric of Nature. For centuries, we humans have prided ourselves on being the most 'evolved' species. Superior intelligence and technological capability have bred this arrogance.

In these unusual and tragic times, I am indeed grateful that we are able to connect virtually to continue the business of environmental governance. While the efforts of all your capitals are correctly focused on preventing human suffering, as parts of the world move slowly towards recovery, the environmental agenda remains one of our most powerful insurance policies in preventing future global pandemics like COVID-19.

When did you last see a glow worm? Most likely, quite some time ago. Depending on how young you are, you may have never seen one at all. Those light-emitting insects, Wordsworth's "earthborn stars", have been declining in the UK for decades. That means that scientists now see them in fewer places, and even in those pockets where conditions are right for them, there are fewer of them to be found.

To imagine earth without greenhouse gases in its atmosphere is to turn the familiar blue marble into a barren lump of rock and ice on which the average surface temperature hovers around -18ºC. Such a planet would not receive less of the sunlight which is the ultimate source of all Earth's warmth. But when the energy it absorbed from the sunlight was re-emitted as infrared radiation, as the laws of physics require, it would head unimpeded back out into space.

And against the backdrop of threatened lives, crippled businesses and damaged economies, the UN chief warned the Petersberg Climate Dialogue in Berlin that the Sustainable Development Goals (SDGs) are also under threat.

As the coronavirus pandemic continues to wreak havoc, the world's energies are rightly focused on efforts to contain the virus and manage the economic fallout. Yet, in the background, the climate emergency remains as urgent as ever.

"Pollinators were the key," says Edgar Mora, reflecting on the decision to recognise every bee, bat, hummingbird and butterfly as a citizen of Curridabat during his 12-year spell as mayor.

We asked Guardian readers living in cities and towns across the world to share their images of the wildlife they can see from their homes. You answered in your droves, from Canada to Cardiff, and here are some of the best.

In the wake of the COVID-19 crisis, any economic stimulus measures must safeguard nature or governments risk exposing humanity to further pandemics.

Growth at any cost is, well, costing us — and we can’t rely on the marketplace to solve the earth’s problems.

The surprise player in affecting workplace safety overseas? Multinational buyers.

Global action is also a matter of self-interest for developed economies.

The flu vaccine was developed through open science, without any intellectual-property considerations. Imagine if we could do the same for COVID-19.

The world's two largest economies are both partners and rivals, deeply intertwined but also with divergent interests. How will these contradictions resolve themselves following COVID-19?