Lecture 32, As-Salaam-Alaikum. Welcome to the virtual university's course on business and technical communication. In the last lecture, I had given you a general overview of visual aids. Today, we are going to look at creating 12 types of visual aids. In this lecture today, you will learn how to construct tables, bar graphs, pictographs, line graphs, pie charts, photographs, drawings, diagrams, flow charts, organizational charts, schedule charts and budget statements. We will look at them. We will talk about how they are constructed, where they are appropriate, in what kind of communication they are appropriate and what purpose they fulfill. Let us have a look at tables. The table is one of the most versatile and widely used visual aids, especially in business and technical communication. Tables are used so often because they can help writers achieve several common objectives. For example, they are an excellent tool for groups of detailed facts in a concise and readable form. The most commonly used visual aids in communication are tables and that is why we are first and foremost talking about them. Tables are widely used because they are so popular that if there is a lot of data and information, they can be expressed in a very concise way. It is also easy to read them. Tables are very readable. Tables also help readers to find particular facts quickly, such as the total amount of money spent in a particular column, etc. The superiority of the table would be even more evident if you try to find and compare two or more facts, for instance, in comparing amounts spent on healthcare in different years. Tables can also be just as effective in presenting information in words as in numbers and this is one other key element of tables which is very useful. In most other visual aids, in a lot of other visual aids, you can only represent numbers and in a lot of visual aids you can only represent words or information that would not be so easily accessible through numbers. However, in tables you can use both with the same ease. For example, tables are often used in reports and advertising to compare the features of competing products or processes. In that you could either use figures to show the superiority of one product over another, maybe by comparing sales figures or you could use words to show the key qualities of one product as opposed to or as compared to another product. Tables are also often used to display information in the troubleshooting sections of instructions manuals. So, tables are most frequently used in such troubleshooting sections where you have only reported problems and the corresponding in one column and the corresponding column talks about how to remedy those problems. Let us have a look at an example of a table where the information is displayed in words rather than figures. Now, how will you construct a table? To create a table, you systematically arrange information in rows and columns. You should adjust the basic structure as needed according to the information that you need to put in and also according to what your readers will find easy to understand. For example, if your table is crowded, you can separate columns or groups of columns with vertical lines and help your readers to read across rows. You can place horizontal lines or leave a blank row after every five lines or so. So, these are some of the strategies that you can use to make sure that your table looks more readable and more video friendly. Basically, if you have a lot of information in one column, you can divide that column into two or three columns or you can also group columns according to your distinctions. And if you feel that it is difficult to read rows in one row and you have a lot of information in one row, then you can also divide rows and you can also do this so that you can leave a blank row after a few rows so that the distinctions are more clear. Of course, if you are creating a table where the rows are grouped logically where all things dealing with a particular area are grouped together in some rows and then another area is grouped together in other rows, then you can insert the blank line between logical groupings rather than at arbitrary fixed intervals. That would make much more sense. If your table has too many columns to fit legibly and attractively onto one page, then you can turn the table sideways or you can turn the page sideways and have the table in such a way that the bottom faces to the right or if you are making the table on a computer, you can use the landscape option when you are setting your page rather than the portrait option. Note that the fancy variations of the basic design may be used where attractiveness is especially important as in advertising brochures and annual reports to stockholders etc. When deciding how to display your information within the framework of your table, you have several basic decisions to make. You need to decide how to order the rows and columns. You need to decide which labels to use for columns and which labels to use for rows, basically the headings that you will use for columns and for rows. You also need to decide how to align entries within the columns and where to place special notes if you have any special notes, if you have any special comments to make about the entries. Sometimes you can create tables that are much simpler than what we just talked about. These informal tables are useful where the sentences that precede the table provide the reader with all the information that they need to interpret the table or if the interpretation is otherwise obvious. Sometimes when you are writing the pros of the table, the pros tell you what is in the table. Then the tables are very simple. Basically, tables can be either formal or informal depending on the purpose where you are using them with your pros. Now, like a table, a bar graph can represent numerical quantities which is done using rectangles called bars. The greater the quantity, the longer the bar. Some uses of the bar graph are as follows. You can use a bar graph to compare quantities at a glance. You can use a bar graph to show trends or to indicate the composition of a whole. Now, how will you construct a bar graph? You can use a bar graph to show trends or to indicate the composition of a whole. Now, how will you construct a bar graph? You will begin by drawing your axis so that your graph will be roughly square along one axis. You will place tick marks at regular intervals to indicate quantities. For example, 5 million, 10 million, etc. You will place the tick mark so that the longest bar will extend nearly to the end of its parallel axis. You will draw the bar as if the longest bar is parallel to the axis. Because the quantities you are writing are at the axis, then the bar will go along the axis and correlate with the quantities. Now this axis may be the vertical axis or the horizontal axis. You may extend the bars vertically or horizontally. Vertical bars are often used for height and depth whereas horizontal bars are often used for distance, length and time. When vertical bars are used for height and depth, horizontal bars are used for time, space and length. Generally, readers will find it easier to use labels placed next to the bars than to use labels provided in a separate key. You may have seen that there is a separate key box on the side with charts. Shades are given and it is written that this is the design. But in bar charts, it is generally more useful to place the labels alongside the bars rather than to place them in a separate key. One exception to this rule however is when you use the same groups of bars repeatedly. If the same bars are repeated again and again in a bar graph, then you will place the labels in a bar graph. When you are repeating groups of bars, you will use distinctive hatching, shading or colouring different designs. To distinguish each repeating category within the groups of bars and then provide a key to those categories. Let us have a look at a sample bar graph where the particular groups are repeated. As you can see, there are distinct patterns or colours that are used for each bar and then those bars are repeated to show difference over time. And the key is given in the top right hand corner of the graph. Very frequently it happens that when people are using Vajranads specifically bar graphs, they tend to put information down in such a way that it is not very clear to the audience. In fact, they run the risk of misleading the readers. One current mistake that results in accidentally misleading the readers is to omit the zero point on the axis that shows the quantity. There are many times that we do not write zero quantity on the axis and due to that the values that are changed are mislead. So, the axis on which you are showing the quantity is zero and the next quantity is zero. Also, sometimes it happens that people cannot put all the entire quantity scale onto one axis and therefore they skip out some chunks. This however misleads the reader. You need to indicate this fact to your readers. If you are missing out any quantity then you need to indicate to your reader perhaps by using hash marks to signal a break in the quantity axis or in the bars themselves. So, basically if you have any quantities in your quantity that you are skipping because you think they are irrelevant or because of including them your axis is very long or your information is not fit on that axis. So, then you tell your readers that you have skipped some information or missed out. You can just tell them that you can use hash marks or dashes so that the reader can get the quantity or some figures that you have taken out. We are going to have a look at some misleading biographs. In the first biograph A, that is absolutely correct. That is an accurate version. The visual relationships reflect the numerical relationships. However, graphs B and C are overstated and understated respectively. Bar B is overstated because Z shows dramatic growth in relation to X and bar C is understated because X, Y and Z hardly show any change. So, you need to make sure that whatever biograph you are using, whatever values you are including actually show accurate relationships and the relationships are actually visible to the reader as well.  Another way to avoid misleading your readers is to make all the bars the same width. This is also a mistake that people make that the width of the bars is not equal and because of that the readers are misled. If you vary the width, readers may compare either the widths or the areas of the bars rather than just their lengths and thereby they can draw incorrect conclusions about the quantities that they are comparing. Remember that in a bar graph you have to compare the length of the bars, the width of the bars or the area that is covering the bars. If you don't keep the width of the bars equal then the readers may start comparing the widths rather than the height of the bars. Another form of visual aids is pictographs. Pictographs are a special kind of bar graph in which the bars are replaced by drawings that represent the things that are being described. The chief advantage of the pictograph is that it uses drawings to symbolize concretely the quantities that you are talking about in your graph. You will find pictographs especially useful where you want to do one or both of the following. They are useful when you want to either emphasize the practical consequences of the data represented and or they are also useful when you want to make your data visually interesting and memorable. The procedure for creating a pictograph is nearly identical to that of creating a bar chart. The difference is that you draw pictures instead of rectangles to represent quantities. In a bar graph you are also representing quantities and in a pictograph you are also representing quantities. But in a bar graph you have to make rectangles that are a little different. In a pictograph you make interesting pictures that represent the things that you are comparing. Like bar graphs, pictographs can also mislead the reader if they are not drawn properly. When using pictographs you can avoid misleading your readers if you keep all of your pictures the same size and use all of them to represent greater quantities. As you can see in this graph on your screen, if you make different sizes of pictures then you will get distorted information. But if you keep all the pictures the same size then the information is correct and if you want to show quantities then increase the number of pictures instead of increasing their size. Just like we said earlier that when you draw a bar chart you will not increase the size of the bars, you will increase the height. Similarly pictures that are not of the same size, you will increase the size of the bars. You will stack them up in height using more than one picture. Now coming to another type of graph which is a line graph. A line graph shows how one quantity changes as a function changes in another quantity. You can use line graphs in many ways including to show trends and cycles. When you want to show a pattern of change over time, line graphs can be very helpful especially when compared to a table. In a table you cannot show a trend with time but you can easily show a changing trend over time. You can also use line graphs very effectively to compare trends. If you want to show readers how to or no trends compare with one another then line graphs are very effective. If you are comparing more than one thing over the same amount of time or over the same kind of time period then you use line graphs. You must have seen frequently in cricket matches that when two different teams are showing progress, then they show it in line graphs and when they are comparing then they also compare in line graphs that the speed of the wickets or the speed of the runs etc. There are two lines and both teams are comparing the performance. You can also use line graphs to show how two or more variables interact. Line graphs are very well suited to display interaction amongst variables. Let's have a look at a line graph that shows trends. As you can see the x-axis has the names of the months of a year and the y-axis has different quantities and the trend is shown over time.  Here you have more than one line that is going across the x-axis. Again the x-axis has months, the y-axis has the quantities but they are for more than one county. This third line graph shows the interaction of different variables. As you can see on the x-axis you have years, on the y-axis you have quantities in dollars and the three different lines are looking at three different variables. Balance on goods and services and income, balance on current account and merchandise trade balance and all these three are then compared. How will you create a line graph? In a line graph you generally show how variation in one thing called the dependent variable is affected by variation in another thing called the independent variable. So you have a dependent variable and an independent variable and you show the relationship of the two. The dependent variable is affected by the independent variable. Line graphs almost always show the dependent variable on the vertical axis and the independent variable on the horizontal axis. Time is usually treated as an independent variable. Time comes on the horizontal axis because it has an independent variable. And any other quantity that is being looked at goes on the vertical axis. Now when you have your different variables then and you have decided what goes on which axis. As I said on the horizontal axis there will be generally time, the second quantity, the second variable will be on the vertical axis. Now after this you will mark on your axis, on both the axis, in regular intervals, you will mark to show the changes. In some situations your readers may find it easier to read your line graph if you extend the tick marks all the way across the graph to form a grid. If you do this you will make the grid with a thinner line than you use to represent the quantities you are describing so that your data stands out. Just like in the last graph I had seen on the screen. All the quantities written on x axis and y axis are on regular intervals and then a graph is plotted on them. Many times it is said that you put the ticks on the entire graph till the end and then make a grid so that it is easier to understand the graph in reading it. But for that grid you will use a thin line because the line you have used for the variables will be as much as the line for the grid and the line for the variables. Generally the vertical axis should indicate a zero point. Otherwise readers can be misled about the proportions of change shown in the graph. Just like we talked about graphs that it is very important to include a zero point. Similarly on the line graphs the vertical axis is very important to show a zero point zero quantity. Generally there are a few situations where you do not need a zero quantity if you have the same amount of quantities but generally you will use zero quantity. If you do not begin the vertical axis at zero, alert your users to that by using hash marks or similar symbols to indicate that the scale is not continuous from zero. You will have to tell your readers by using hash marks or any other scale that you are not starting from zero and your quantities are starting from a bigger number. Where you need to indicate some negative quantities such as losses use both positive and negative values on whichever of the axis needs them. If you want to show negative values on x axis or y axis such as losses then you will show both positive and negative values. Remember to label the plotted lines on your graph preferably immediately next to the lines. Whatever you gave the labels was given along with the lines. Now coming to pie charts. Pie charts are unsurpassed in their ability to depict the composition of a whole. There is no other type of visual aid that has a complete composition or different parts of it that is so effective. For example, you can use a pie chart to show how much each of several food sources contributes to the total amount of dietary fat consumed by the average American. Whatever you are showing as an average American diet you can show it in this video. Different food sources contribute to the amount of fat consumed. There are many different parts of a pie chart to show how much each of the food sources contributes to the amount of dietary fat consumed by the average American. You can also use a pie chart to show how much each of the food sources contributes to the amount of fat consumed by the average American. a pie chart, you draw a circle and then draw lines that slice that circle into wedges. Each wedge occupies a portion of the circle's circumference proportional to the amount of the total pie that the wedge represents, and then we will arrange the wedges in a way that helps your readers determine the rank order of the wedges and compare the relative sizes of particular wedges as well.  If you draw a circle that is larger than the circle, then a bigger slice will come, and then the biggest slice will come, and then it will be easier for your readers to see and compare all those sections. You can start the largest wedge at the 12 o'clock position on a clock and then measure clockwise. Generally, this works very well and this helps in making the comparisons clear. In some situations, of course, you may need to arrange your slices in some other way to make your point effectively or to achieve a visually pleasing design. But those will be exceptions. Generally, this design where you have the largest slice first and then you go into a descending order is very attractive and also very legible. Now, it is said that you will arrange the slices in a way that you also need to label each wedge or each slice, and you will label it with its type. We are looking at percentages in the pie chart. We are looking at how much or how much is more proportional to the other slices. So, we will tell that in the percentages. Depending on the size of the wedge, you might place the labels inside or outside the circle. Obviously, if it is a very small wedge, then the label cannot fit inside the wedge. So, then you will place it outside. Each wedge typically has a different color, shade or pattern to help you readers clearly distinguish each portion. If you have the same pattern or color of each slice, then obviously, they will not see each other. So, you will have to give each wedge a different design or pattern and then label them. You will label them inside or outside the circle so that you can find out which design or pattern is signifying which thing. Let's have a look at a sample pie chart which shows college expenses of a student. These pie charts also show college expenses and they are just shown differently. They are just different models of pie charts and how the same thing can be expressed differently. In pie chart A, you see college expenses referred to in terms of total quantities in numbers and in pie chart B, the college expenses are expressed as percentages of a whole. But in either case, both the pie charts talk of different quantities, different sections of a whole. Another very effective visual aid is photographs. We use photographs in our normal life as well. We know that they are very pleasing to the eye and that they are very effective. But how do we use them in business communication? How do we use them in technical communications? With photographs, you can show your readers exactly what they would see if they were personally looking at the object that you are showing them. If they were physically there or they were personally looking at the object, that is what you depict in a photograph. Photographs can help you achieve a variety of communication purposes. One such purpose is to show the readers something that they have never seen before. This could include a new product or the inside of the human heart. Another purpose could be to show the condition of something. For example, the result of a skin treatment. You can see a lot of places in Ishtahara that it was like this before the hair and after the hair, etc. Or another purpose could be to help the readers recognize something. For example, in a lab manual, you might include photographs that would enable your readers to identify the imperfections they might encounter in a sheet steel, for example. Or to help readers find something. You can use photographs very effectively to help your readers find something. For example, you could show parts of a machinery in an instruction manual. How will you create a photograph? When you are taking a photograph, obviously you are taking a photograph from a camera, but you need to choose an appropriate angle of view, preferably the same angle that the readers would take when they would be looking at that object. The angle that the readers would look at the object, if it is physically looking at it, you can also try to eliminate unnecessary or distracting detail in the photograph. You can even crop or trim away the extra material that would take your readers' attention away from the object that you want them to see. If you feel that you have a photograph in which this amount of material is included, which is distracting your audience's attention, then you can cut the photograph from the sides and only include the things that you want your audience to focus on. Also, you need to ensure that all the relevant parts show clearly. Don't let important parts be hidden or half hidden behind other parts. Try to make sure that all the things that you are looking at are clearly visible so that the important parts are hidden behind something else. When you are giving a photograph in your communication, provide whatever labels your audience will need. Similarly, you can label other visual aids. Similarly, you can label photographs as well. And make your labels stand out sufficiently from the background of the photograph. A photograph's background will be very dark. Your charts, bar charts, live graphs, etc. don't have to have a background that is dark. But in photographs, there will be something coming in the background as well. So, try to give whatever labels that you are looking at to make sure that they are clearly visible in the background. Another visual aid that we can use in business communication is diagrams. A diagram is very much like a drawing, except that drawings accurately convey the actual appearance of things. Diagrams depict subjects more abstractly. Drawings are shown in the same way. Diagrams are not shown so accurately. They are shown in some abstract ways. But diagrams can convey complex information, such as the system used for industrial processing and fruits without actually showing the actual machinery. It is not necessary that the machine that has the fruit processing is shown, but a diagram can show that process. Now, how will you create a diagram?  That is the first thing that you need to consider. You need to decide exactly what you want to show. Then you will create an appropriate means to represent your subject with geometric shapes or perhaps sketches that suggest their appearance. You will provide the explanations that people need in order to understand your diagram as a separate key in the title or as part of the diagram itself. It is not necessary that the diagrams are shown in the form. Let us look at a sample diagram. As you can see, it is very clearly labeled, so that the different elements, different paths and the relationships become clear to the reader. Another very commonly used visual aid is flow charts. Flow charts are excellent means of representing the succession of events in a process or procedure. The simplest flow charts use rectangles, circles, diamonds or other geometric shapes to represent events and arrows to show the progress from one event to another. Now, let us look at a sample diagram. Let us look at a sample diagram. It shows the progress from one event to another. Now, how will you create a flow chart? A few conventions govern the creation of flow charts and we need to keep these in mind when we are creating flow charts. The labels that identify the activities are placed inside the boxes that represent those activities. Now, let us look at some other things. For example, in the pie chart, you can write the labels outside the pie chart. But in the flow chart, it is important that the labels that are in the boxes, in the geometric shape, are written inside. Also, the boxes are arranged so that the activity flows from left to right or from top to bottom or both. Flow chart means that whatever things are flowing, whether they are left to right or top to bottom or both are flowing. In this flow chart, you can see that things are flowing from top to bottom and some things are also flowing sideways. Another important visual aid is organizational charts. An organizational chart uses rectangles and lines to represent the arrangement of people and departments in an organization. It reveals the organization's structure, hierarchy, indicating how the smaller units are combined to create larger units. It also indicates who reports to home and who gives direction to home. An organizational chart tells the structure of an organization. Let us have a look at an organizational chart. Here, as you see, the board of directors is at the top and then the president and then we have a cluster of people who report to the president and then different people who are reporting to the respective heads of their departments. Now, how will you create an organizational chart? Because of the hierarchical nature of most business organizations, organizational charts are usually pyramids. This means that organizational charts are usually in a pyramid shape where more boxes are at the bottom and more people are at the top. In an organization where more people are working, seniority-wise, more people are at the top. The president is at the top and vice-president is at the top and vice-managers are at the top and so on. You do not need to show every part of the organization only those relevant to your readers. Sometimes you may need to represent more than one kind of relationship by using different lines for the different relationships as well. Now, we are coming to schedule charts. A schedule chart identifies the major steps in a project and tells when they will be performed. We talked a few words earlier about the progress reports. In the progress reports, schedule charts are used a lot because you can tell what things will be used in it. You can use schedule charts in proposals as well because in the proposal, you can tell the readers what schedule will be useful. Now, how will you create a schedule chart? One of the principal considerations in creating a schedule chart is deciding how much detail to include which depends on the audience's needs and expectations. Whatever you are including, you will keep in mind what the audience is expecting from that schedule chart. Tasks are always listed along the vertical axis with indentations which are used to distinguish sub-tasks from major tasks. On the vertical axis, you will write tasks along the height of the task. You will divide the sub-tasks with indentations. That is, you will write them a little inside as we have seen in the last schedule chart. Also, the intervals, weeks and months are usually marked off with vertical lines to help the reader. You will make columns and show the intervals. Now, coming to budget statements. Budget statements are also used frequently in business writing. Obviously, in every business, you have to make budget statements. A budget statement is a table that shows how money will be gained or spent. A budget statement may be very simple or very elaborate depending on the reader's needs. Many times, you will see what kind of budget you have to make for your readers. You have to give a detailed budget as per its simplicity or complexity. On the job, you can use budget statements in situations like where you have to explain the expenses involved with the project purchase to summarize the savings to be realized by following the recommendations that you are making. Or to report the costs that have been incurred by a project for which you have responsibility or to explain the source of revenue associated with some project or activity. If you are working on it, where is the revenue coming from to give a detailed budget statement. Briefly about some steps which will enable you to prepare simple, informative budget statements. First of all, you will divide your page into two vertical columns. In the left-hand column, you will list the major categories of expenses. In the right-hand column, you will show the income and align these along decimal points. Then, you will clearly indicate the total. If you want to show income, you will also follow the same steps. In this lecture, we have learnt how to construct different types of visual aids which included tables, biographs, pictographs, line graphs, pie charts, photographs, drawings, diagrams, schedule charts and budget statements. I hope you can easily integrate these into your business communication. If you have any problems, please feel free to email us the addresses englishatvu.edu.pk Until next time, Allah Hafiz.


In this video, we provide the solution to question number 15 for practice exam 1 for math 1050. We're given the function f of x equals the square root of 2x plus 1 all over x squared minus 1. We're asked to compute the domain of this function. We need to report it in interval notation. So what are things that can go wrong? Because by the domain convention, we accept the domain to be as large as possible. We accept all real numbers for x, except for those that turn the expression into some non-real number. So we've been trained to watch out for a couple things. So first of all, square roots are problematic. If we take the square root of a negative number, that's not a real number. It's what we call an imaginary number. And so we have to make sure that the radicand, that is the expression inside of the square root, is not negative. So we have to consider the inequality 2x plus 1 is greater than or equal to 0. Equal to 0 is okay. If you take the square root of 0, that's just 0. Nothing non-real about that. So we need to solve this inequality. We can minus 1 from both sides. We get 2x is greater than or equal to negative 1. Divide both sides by 2. We get x is greater than or equal to negative 1 half. So as long as x is greater than negative 1 half, then the numerator, I should say the radicand will be non-negative, I should say. And then the square root of that would then be a real number. So as long as the input is greater than or equal to negative 1, no problems in the numerator. Well, what about the denominator, the x squared right here? One of the problems with denominators is if you divide by 0, right, that gives you a non-real number. So we have to make sure that x squared, excuse me, there, x squared minus 1 is not equal to 0. Well, the easiest way to do that is to solve the equation for 0 and go from there, for which x squared is equal to 1. You can take the square root of both sides. You have to remember when you take the square root, there's actually two square roots of positive and negative 1. And so you end up with plus or minus 1, or specifically 1 and negative 1. And these are the values that x should not be. x cannot be 1 or negative 1. So we have to put these together. Now, notice that negative 1 is less than negative 1 half. So this one already ruled that as right as a possibility. x has to be greater than or equal to negative 1 half, excuse me. So negative 1 is not even in the run at all. But 1 is greater than negative 1. So considering those together, we take the intersection here. We need that x is greater than and or equal to negative 1 half. So we would write something like this bracket, negative 1 half, because negative 1 half is acceptable. Go up to 1. Then we're going to put a parenthesis there, because 1 is not included. If you plug in x equals 1, that makes the denominator go to zero. That's not a real number. We take the union, we jump over to 1 again. So we jumped 1. The parenthesis means that 1 is not part of the domain here. But we include everything up to 1 from the left-hand side and from the right-hand side. And now once you get past 1, there's no more restriction on the domain. So we're going to put infinity right here. We always put a parenthesis on infinity, because infinity is not a real number, hence it's not included. So notice here that you need a parenthesis on infinity. You need a parenthesis on the 1, because it's not included, but you do need a bracket on the negative 1 half, because that is part of the domain. And therefore, the correct answer would be negative 1 half to 1, union 1 to infinity, where 1 is omitted and negative 1 half is included.


Let's take a look at this next one. Again, we're dealing with two triangles that we want to prove congruent. So let's name them So we can call them triangle TON and triangle FUN as in Proof is a ton of fun. All right, so we want to first prove our TF and OU bisect each other. What does that mean? Well, this word bisect Bisect means cut in half. So TF and OU cut each other in half In other words, TF refers to that long segment. TF is cut into two equal parts Similarly, OU is cut into two equal parts So now we have two pairs of sides. We've got the reds and the blues that's side and side and Right now we're missing a piece, right? We could either use side side side angle side And so we're missing a piece, but the way that the shape is drawn. We've got these vertical angles and Vertical angles are always congruent And so therefore we're going to prove that these two triangles are congruent using the side angle side theorem So let's get started So we know what it is we want to prove we want to prove that triangles TON and FUN are congruent and We agreed we're going to use the side angle side congruence theorem. So we can talk about a pair of sides that are congruent a pair of angles that are congruent and Another pair of sides that are congruent in each of those boxes. So let's start with the reds. I Know that UN and ON are congruent Next the other pair of sides the blues those blue sides are congruent and finally the angles the angles up the middle Angle TNO and FNU Okay, so why are all of these segments congruent the red segments UN and ON? Why are they congruent? Well, they're congruent because TF and OU bisect each other and What that means is I know that TF bisects OU Also OU bisects TF So for the red segments to be congruent which segment is doing the bisecting Which is being bisected. That's an important question In this case TF is doing the bisecting and OU is being bisected The opposite is true for the blue segments TN and FN in that case OU is doing the bisecting and TF is being bisected Now the connector reason you have to ask yourself. What does bisect mean? Well bisect means cuts in half And so for each of these reasons we're going to use the segment bisector definition And lastly we need to talk about where those green angles came from the green angles came from the vertical angles And always vertical angles are congruent So we need a box that says hey, I have vertical angles and then the box below it can say that they're congruent So there's the statement TNO FNU are vertical angles and as a result Vertical angles are congruent So the statement is if we have vertical angles Then those angles those vertical angles are congruent Super We're done


Chumps, the Yuletide version, and here it is. I have my Christmas pyjamas on. Isn't that nice? This is the first time anybody showed up on Channel 17 with the pyjamas on. My name is Bill Keele. I'm the host of this program. And my co-host is Joey Donovan. Joey? I just would like to say I hope this is the last time anybody shows up at Channel 17 in their pajamas. But after I saw Bill's, I took these off the Channel 17 Christmas tree so that I could match your festive mood. All right, we're even then. We're even. OK. Anyway, this is going to be a Yuletide version. Here it is, the 21st of December, the shortest day of the year, the first day of winter, welcome winter, not. And so anyway, so we're just going to talk about Christmas stories of family or incidents or whatever. So if you want to share a good Christmas story, call in 8-6-2-3-9-6-6. 8-6-2-3-9-6-6. And Joey, where do we go from here? Well, I think we have to, first of all, imagine that tomorrow the days will start to get longer. So that's the good news. Oh, more daylight, OK? More daylight. So but we really talked about Christmas memories, and you and I were filling each other with all sorts of stories about when we were kids. Can we tell them? I think for most part we can. OK, all right. I remember in our family on Thanksgiving night, an aunt and uncle of ours, who would be guests for Thanksgiving dinner, would pack up my brothers and sisters and me and take us downtown to City Hall Park, where the crash would have just been opened. And of course, I think you were probably maybe on the city council when we, because of respect for other cultures, decided that it wasn't really probably the best principle policy just to have a nativity scene on city property. So we no longer have that. But back when I was growing up, there was never any thought to those sensitivities that I think it's good that we respect. But I always remember that as a big kickoff to the Christmas season. And OK, so all you guys went down there and the mayor was there and? I don't think it was the mayor. It was just our family and other people that the lights had gone on in this beautiful life-sized manger scene that would be on display in City Hall Park. OK, now when you were kids, how about your disclosure of Santa Claus as a myth? Did you have any rules in your? Oh, I never heard of that. You never heard of Santa Claus? Well, what was Santa Claus in your family? Santa Claus was Santa Claus. Oh, but did he come down there chimney and all that kind of good stuff? I never knew, because we always felt that he had to use a door in our house. Oh, OK. But whatever, I always was very glad that whatever his method of entry, although there were sometimes some threats that perhaps he wasn't coming that year, he always did come. And it was always so very exciting. Well, why was it exciting? What was the excitement about it? I just think, don't you remember just building up all that time? Yeah, well, we had Santa Claus coming. This is the whole thing, and we put out cookies. Oh, yes. We put out cookies right near the fireplace in the South End. And we have cookies here. And I don't know if we put them. It wasn't a beer, but I had the beer. Santa Claus had the soda. We had milk. OK, it was milk. Yeah, that was good. And then in the morning, we ate the cookies or leave some crumbs around. And then we always left a note. Oh, yeah, OK. And the first thing in the morning, when I was younger, was to find, did he eat all the cookies? And did he leave any message? Did he write anything back? OK, all right. And oftentimes, he did. But I remember so many. I remember one Christmas, we always had the big tree decorating. And I'm not sure. Do you remember a thing that was called angel hair? Yeah. That you could put on the tree? Yeah. What it actually was, was fiberglass. And everyone who handled it broke out with this horrendous rash and itch and little cuts. I do remember that. And I think we only had angel hair that one Christmas. Oh, well, wasn't it flammable? It was probably flammable. It was probably poisonous. A little bad thing. And it certainly was irritating. But it was marketed as the next best thing for your Christmas tree. So we had to have it. Well, remember, we'd always just talk about the night before and assembling gifts. But the night before, we'd always have a, after the kids went to bed, we sort of chatted around and then probably stayed up too late. So the problem was keeping the kids in bed. Oh, early in the morning. Early. They didn't want to get up early.  Anyway, that worked out OK. And I would take pictures. And remember the big glaring lights in the old days of, whoa. Anyway, so but how about the night before, of assembling gifts? Tell me about your experience. I'm going back again when I was a very young child. In our family, I had two brothers, one of whom was an altar boy and the other a choir boy. So they always were committed to midnight mass, where my sisters and I were delegated to be made at home and go to bed early and be good children. So it was always sort of a problem, because I could hear my brothers going out to mass and coming back. And there'd be all sorts of happenings downstairs. But I was never privy to get up to see any of them. So why not?  Because they had to, because my mother and father, I never, I think they were too busy. They never went to midnight mass. But the two brothers were involved as altar servers and choir boy. No, in those days, you could have to not eat after midnight. If you're going to go to communion the next day, you had to fast after midnight. What's the story behind that? Well, that was such a problem, because when we would go to our stockings on Christmas morning, oftentimes Santa Claus would leave some really terrific treats that you did not see the likes of, except if you'd had a very good Halloween load. And you'd be looking at them and touching them, but you couldn't eat them, because you couldn't break your fast. You had to certainly wanted to go to communion on Christmas morning. And to do so, you had to fast. So that's another reason that I think I'll probably have a high place in heaven. All right. Now, how about getting a Christmas treat? Was that a big event, or was that just something that your family got a treat and came in and decorated it? I think my mother ordered it up. I don't remember. My husband and I, we would have those adventures with our children of getting a Christmas treat and choosing a treat. I remember one year, poor Tom got the treat on his own, brought it home, and there was a whole terrible mental breakdown. It was he probably hadn't brought his glasses to the tree store, but it was a very less than a Charlie Brown Christmas treat. And so we ultimately had to pass that on to another family and go out and buy the second one, because it was breaking hearts all over the house. Oh, wow. But I don't remember us. I just remember the Christmas treat coming, and whether my mom had just ordered it or my father did. Well, we went out and bought it for a couple of years. And then one year, much to the chagrin of my family, I got a silver tree. It was all a silver tree. It had a silver post that these things just stuck in the side. Man, that did not go over well. But we survived on it. If you have a story about your Christmas, give us a call, 8-6-2-3-9-6-6. We're just rambling here. And if you've got a story you want to share, let us hear it. Let us share it with others. 8-6-2-3-9-6-6.  Well, I figured, well, you could use it for many years.   I don't know. Maybe this was cheaper by $1.5. I like your honesty. It was cheaper by $1.5. It didn't last very long, but I still remember it as being in some of my family photographs. Another thing that we talked about, traditions of past years were Christmas cards. Oh, God. And it was, you know, you'd send many out, but it was such a pleasure to have the mailman come. And there would be piles of cards. And I remember the Christmas cards at my childhood home would be in a big basket. But when my father would come home from work, that would be the first thing he did. The cards that arrived that day would be held out of the basket for him to look at and read. And they meant so much to my parents. And, you know, many of them were from friends and neighbors. Oh, wow. But we kept on, you know, everybody sent Christmas cards. Well, they did. And a couple of nights before the mailing time, he's had to make sure they arrived the Christmas there before arriving after that is not so good. They would have a caller. Oh, great. Caller, you're on the air. What's your Yuletide story? My Yuletide remembrance is going to my parents' house with a bunch of different people, families, in-laws, and passing out caroling books. And we all sang, and it was great, except the two in-law children linked each other singing and said, do you realize we two, not blood relatives, are the only two singing on key? Are they still in the family? The whole rest of the family doesn't know where key is, but we all had just as much fun singing. Oh, that's wonderful. Are those in-laws still in the family? Are they still in the family? After criticizing your talent. My husband is still in the family. The wife of my brother is not. I got a question for you. I'm going to see if I can stump you. You had those traditional Christmas carol books. What was the name of the insurance company that sponsored those? Oh, God. Oh, Lord, you know, I don't think ours were free. I think they were some my mother bought. Oh, OK. Well, the answer is John Hancock. You can tell me the answer. I could have just gasped Hickok and Boardman, but we weren't from the Burlington area, so it wouldn't be Hickok and Boardman. Who wants to be? It was John Hancock. Were they the little blue ones? Yeah, little blue ones. Little blue ones, yeah. No, these were big, red, shiny, covered song books. Now, has your voice improved since then? Your singing voice improved since then? When we sing, I'm a dragon boat paddler in Burlington. I paddle with the Dragon Heart team. And when we sing a song, they give me a quarter to just mouth the words. OK, thank you for doing that, Dragon Heart. That's a great group. It's a wonderful group. Yeah, nice going. Thanks for calling in. We have another caller. You're ever so welcome. Bye-bye now. Merry Christmas. Thank you. You too. Caller, you're on the air. What's your story? Hi, my name is Joe Malia. I'm a cousin. Joe, can you sing? Not very well. But I have sung with Joey. OK, what's your story, Joe? Well, actually, your mother on Christmas Eve would always deliver a case of Coke to 21 Led Road, who's in one of those old cardboard boxes with six and a half ounce Coke bottles, which was a treasure to the Mealy family. And then we would spend the entire next day, Christmas, at the farmer's house. And that would include all the Mahoney's, double Mahoney's, Lettie's, Mealy's, lots and lots of cousins. And those are the memories that we all cherish a lot. And they were very, very beautiful, beautiful Christmas nights, weren't they, Joe? Do you hear Joey? Yes, they were. And all the girls were dressed in dresses made by Margaret Farmer. Right. That's Dr. Farmer's house, right? That's correct. Yeah, OK, good. Joe, anything funny happened to you?  Any farcical things go on in those days? At the Mealy house? At the Mealy house. Joe? Yeah, well, we were at Doc Farmer's house. And of course, we had eight Mahoney's and seven Mahoney's and nine Mahoney's, seven Mahoney's, eight Mealy's, six Lettie's. So you speak of farce. Yes, all singing and dancing to Mick McGilligan's ball was quite an adventure. Oh, good. It was a wonderful time. Well, thanks, Joe, for sharing that experience. Appreciate you calling in. Thank you. Have a good holiday. Thank you. How about that? So we're talking about Christmas trees when we're talking about how about shopping downtown. How did you tell us more about shopping downtown, Joey? I'm not a shopper, but. Well, back in the day, I think people did. Church Street was the main shopping place in the whole area. And there was no University Mall. There was no Williston. No Gaines, no. Oh, there was Gaines. There was Gaines, OK. Sid and Lou and the brothers, too, often had some good items. Yeah, Sid and Lou Mazel, yeah. But I think Christmas at Church Street, especially as I got older, and you might be in college or something, you'd go and you'd meet friends that you hadn't seen for a long time. You might get detoured into maybe a little pub for a drink or two to catch up with old times. But you couldn't really go a block or two without just having a good visit with somebody. And everybody was usually in good spirits. And just it felt like a real community. And I was down on Church Street last night with some of my grandchildren. And it was so beautiful. And yet, walking four blocks up to the big Christmas tree with the children, I don't think I said hello to anyone because I didn't know anybody. Of course, we were all bundled up because it was pretty cold. But it is, Burlington, we've really grown as a city and a town, which is good. But it's not quite the way it used to be, sort of a smaller community where you knew everyone. Before the show, we were talking about assembly. We were talking about assembling toys on Christmas Eve. To me, that was the nightmare. If you have a story of a nightmare of your house on Christmas Eve, call us 8-6-2-3-9-6-6. But the night before, you have to assemble some of those toys. And being impatient, like I was, I said, I'm not going to read those instructions. That's baloney. I'm not going to do this. And it was very frustrating. Did that same thing happen in your house? I think it was a problem. I was married to a man who always felt that he could probably erect the Eiffel Tower with his own knowledge and would often start without the three packages with the 48 nuts and bolts and all the other equipment. So we did have some times that tested our Christmas spirit. But overall, it turned out well. And sometimes by morning, whether it was a gift of Christmas itself, things were put together. We have a color. Color you're on the air with your Yuletide story. What is it? Yes, I have a story that I'd like to share. It's one of my childhood stories. I remember one time when I was young, maybe eight or nine, my dad put me in the car on Christmas Eve and said, come on, we're going to go get a gift. And I had no idea where we were going or what we were going to forget. And we ended up going to a store. It may have been a Gaines or something like that. But it was our first color television. Oh, wow. I remember I was so surprised to get our first color television and bring that back and set it up for Christmas Day. But it wasn't so much the gift that I was impressed with is the fact that my dad picked me to go out of five kids to go with him to get that television. That's a memory I'm always going to have till the day I die. And I just wanted to share that with a group. Tell Joey that I love her Christmas decorations. They look great. Thank you. And Merry Christmas to all. And Bill, I think you know who this is. Yes, I do. I'll see you on Christmas. Merry Christmas. Thanks for calling in. Yep. All right, bye-bye. Bye-bye. I was another thing as we were talking about Christmas Eve. I know in many French-Canadian families, they had the meat pies. Tortier. Tortier. That was so important to their culture. In our house, my father could never have Christmas Eve without oyster stew. And one night, when I was asking about it, they would have oysters when he was growing up in Underhill, Vermont, on Christmas Eve. And I always thought, how did they ever get oysters in Underhill on Christmas Eve? And then later, going to a Christmas holiday party at a big Irish farm family in Milton. And there, what did they serve? Oyster stew. So it may have been an Irish tradition, because I remember having this long conversation with these Irish folks in Milton, and they talk of they always had oysters on Christmas Eve. OK. The Kios did not? No, he did not. I have a story to share with it. Somehow, the family went to Morrisville or stole it to a maple syrup farm. That was the thing we'd go to just around Christmas time. Is that your kid? My kids, my kids. And somebody, one of my five kids, poured hot syrup in his sister's boot. Now, first of all, it's hot. OK, now we're out in Stowe or Morrisville, and it's cold. So when that sticky syrup gets in the boot. Was the sister's foot in the boot at the time? The boot at the time, along with the maple syrup. OK, so it's hot to begin with. And secondly, of course, she's crying and trying to come. What's wrong?  It's cold. And of course, by the time we get home, the maple syrup has cooled, and it's sticky.  Now, those were. I do remember one green Christmas growing up, and I thought the world had come to an end. Why? We had no snow on this Christmas, and it seemed so strange. And I wish I could remember, and I don't know if any of our viewers could remember what year that was. I think it may have been in the late 50s or early 60s. OK. And now, we've had several Christmases, I think, that have been green. Thank God we're not going to have one this year. I know. But do you don't remember that Christmas? No, I don't. I don't think so, no. So on Christmas Day, what did you do for activities after you opened the, one other thing about gifts? There's one thing I did. You give all these gifts to these kids, and they're in the data with the stuff. I used to hide them. I put them away for, they were playing with three or four different gifts, and I put away five or six and give them to them later on. Oh, that's all something new. They forget about it. So we sort of give them a long life. Yes, or if you really thought you bought the perfect thing. And when I was trying to think like my pretty pony and the cabbage patch dolls, I used to have a connection of an old friend in New York who would call me and say, I'm on to a pretty pony. I think I can buy three next Thursday. And it would be, because you couldn't get a cab. There'd be lines to get a cabbage patch doll. Oh, sure.  And finally, I would score one or do something almost illegal to get one. And then the joy of the cabbage patch doll would last 15 minutes or so on Christmas morning. And you'd just go, oh, I can't believe what I did to get that thing. Not appreciated. Well, I think it was, but not quite as much. I'm not sure what would have happened if the cabbage patch doll had not been under the tree. Oh, OK. If you have a Christmas story or a Christmas related story, give us a call, 8-6-2-3-9-6-6. Now, what'd you do on Christmas day? Go sliding, go skating, or what'd you do on Christmas day when you get kicked out of the house after a while? No, as my cousin Joe said, we would do, we'd go to my father's aunt's house for Christmas dinner. Then we'd go to my father's sister's house. And then we would end up at my grandfather's home, which he lived with Dr. and Mrs. Farmer on North Avenue. And that's where we'd meet all our Mahoney cousins. And we would dance and sing and be downstairs playing ping-pong. And just would have a fabulous, fabulous Christmas night with all our relatives. Oh, wow. OK. And we had uncles who loved to sing and loved to harmonize. We always thought they were going to go on the Ed Sullivan show, but one uncle always had to have his head in so he could hear the harmony. We knew Ed Sullivan wouldn't like that. So he kept the group from going nationwide. But it was just a warm time, you know? And I think one thing we should talk about, though, is Bill tonight, is I think you and I and so many others have such wonderful memories of this time of year. But I think too many others don't. You know, I remember when my children were small, we adopted a single gentleman for Christmas and we stopped at his home, which was very humble, on Christmas Eve. And he was so happy that we had came and we had bought some little gifts and a fruit basket for him and everything. And he was warm and wonderful and told my children some stories. And we left that man that night. And I realized how little we remember some of those people who don't have families and who are struggling financially often times. And some of our older people who have health issues. And so I hope that we all will sort of remember those people, remember the food shelf, remember, I think they're still taking turkeys. I know that they like cheques and cash. And they can really spin that into more food than just if you bring cans, but they welcome anything. But I think it's a special time for us to remember those who are not as lucky as we are. That's a good point, Joy, thinking of bringing it up because nothing beats a personal visit. And on the other side of that coin, to be alone, a Christmas Eve or the Christmas day, just a five or 10 minute visit to somebody who has maybe no one or some related, that's the perfect gift for them. They may not remember it 20 minutes afterwards, but at least you were there and at least you felt good, you did something and they appreciated your visit. So that's so often that we do have to forget that. And it is a time where I feel as though we have the automatic wallet going 24 hours a day and thinking, oh my gosh, I have two more trips to the grocery store. You know, one of the things we were talking about, Christmas cards, I think it was high school or it was going to St. Michael or whatever, but we got these jobs at Christmas time, a little patronage jobs at the post office. Oh wow. Christmas cards were a big thing, so you had to put on additional mail and carriers, and that paid well. It did paid very well. And you had to have some political connection in those days while maybe even today. But I remember we used to deliver mail in the Burlington area and you couldn't come back before a certain time. And when you're talking about we, you're really only talking about your gender. That's true, that is exactly true. Thank you, I thought, I knew that it was. That's a good point, that's a good point. I knew that it was. But we boys, we delivered the mail and we would get back early, so we can't go to the post office and check out. So we'd go to Memorial Auditorium and watch the basketball team practice.  But there was also some jobs when I lived in Barrie for a while. We'd open doors for the customers in the hardware store. You ever hear those stories? No, I mean, but I don't know if those jobs don't exist today, but that's one way we try to earn Christmas money to do that kind of thing. We were talking earlier about all the different men's stores on Church Street. It was Hayes and Carney. And Shepherd and Amell, anything, John Riley, Miles and Riley, and others. Humphreys, a favorite store, a smart man. I remember Hayes and Carney's in particular because my father, you'd go in there and Peter Carney, God love him, would know exactly my father's size, shirt size, hat size, everything. And also I might pick up a tie and he'd say, I don't think he'd like that. That was helpful. It was extremely helpful. I defy that anybody could get that service today. Amazon doesn't do that. Amazon does not give that personal care. Perhaps Michael Kio Shop might do that. Michael still does a good job there at the corner of colleges on Church Street. Yes. So what else can we remember? Oh, how about sliding downhill? How about sliding areas when we had snow and winter? Like what were some of the places we used to slide? I went to Mount St. Mary's for elementary school and right outside the convent there was a wonderful sliding hill. And we'd go to the convent after and just get cardboard boxes and just would slide down that hill and just think it was just wonderful. I drive by that now if I'm ever up at Mount St. Mary's and I go, there's not much of a hill there at all. I lived across the street from there. And so we'd go there. But also I used to slide down Fletcher-Allen Hill. Yes. And the challenge there for me was slide down Fletcher-Allen Hill, go across Colchester Road and getting back of Ira Allen School Land. That was a whoa kind of a thing. While dodging cars going across Colchester Avenue. I told that to my mother once and she bawled me out. I did it again but I didn't tell her again. But you remember the controversy about when the hospital was building that parking lot? Oh yeah. Because they took away the sliding hill. And there were so many Burlington people that really didn't want that to happen. And I guess in the theme of progress you can't compare a parking garage to a hospital sliding hill. No, you can't. Joey, this has been a great 30 minutes. It has been fun. And I think we've got some callers who added. Some callers. Thank you for calling in and sharing your experiences with us and listening to us. We know you're out there. And Joey, have a good holiday. And I'd like your, I'd like your attire. I gotta say that. Thank you very much. Thank you. I love baubles. Thanks for tuning in to Channel 17 and Stump the Chumps. We'll see you on January 9th, our next session. Have a good holiday. Merry Christmas. Merry Christmas.


third lecture around the inverse Galois problem. Thank you. Can you hear me in the back? OK, thank you. So let me recall where we are. So in the first lecture, we talked about the quotient an mod g. And I've explained that when this is rational, you get a positive solution to the inverse Galois problem. But I've also explained that this is not always rational. So then in the second lecture, we discussed the rigidity method, which is another way to apply Hilbert's irreducibility theorem to construct extensions with given Galois group. So this applies in many cases. But it's still a limited scope of application. So for example, it doesn't apply to p-groups. I mean, it applies to many simple groups. But such things as p-groups are generally outside of the scope of the rigidity method. So OK, and what was it about? We were looking for families over p1. So because of the versatility property I mentioned, looking for families over p1 is the same as looking for curves p1 inside of this space. So if this is not rational, maybe we can find a rational curve in it. And if we cannot find a rational curve in it, then maybe we can find rational points directly. So that will be the theme of today's lecture. So find interesting rational points in this question. OK, so actually, we're not going to work with this variety. It's slightly more convenient to work with another one that I'm going to introduce now. But it will make no difference. It's just more convenient. So let's fix a finite group G. So recall here, I chose an embedding of G into a symmetric group. And I let it act on the A and by permuting the coordinates. Now let me do something slightly different. I embed G into SNN of k. So I work over some field k, so number field. So you embed G into SNN of k in some way. And I mean, you can always do this. First, embed G into SN, for example. For some n, embed SN into some SNN of k. And then let's consider G as a subgroup of SNN. So let's consider SNN as an algebraic group of k. So that's an algebraic variety with a group law. And I view G as a subgroup. And I consider the quotient of the group by the subgroup. OK, so if you like, it's the quotient of SNN by the right action of G by multiplication. And why is it better than this one? Well, because the action is free, so this is smooth. Here you have singularities. So it's more convenient to work with this. So here we have SNN goes to SNN mod G. So this is smooth. So this is a g-torsor. And it turns out to be a versatile g-torsor just in the same way that we got a versatile g-torsor with affine space. I emphasize these are varieties of a k. So we're going to work with this. And constructing rational points here is really the same as constructing rational points there. I mean, in technical terms, these varieties are stably barational equivalent. This is not so important. And it's explained in the exercises. So let me recall the relationship between the rationality of this quotient and Grunwald's problem. So recall. So sorry, I will go here. So if SNN mod G is rational, then it satisfies weak approximation. So I recall weak approximation means that if you are given a finite set of places and local points at these places, you can find a rational point, which is as close as you want to these local points at these places. OK, so because it's true for a1, it's easy to deduce it's true for rational varieties. And this, remember, implied the positive solution to Grunwald's problem for g over k for all sets of places S. And just to refresh your memory, so this is asking for the existence of extensions of k with group g, sorry, Galois extensions of k with group g, whose completions at the places of S are prescribed. And just to recall how this works, the local prescribed extensions give you kv points on this quotient by versatility. Then you approximate them. And in this way, you get a rational point, which by IKDAL, you can assume to have a connected fiber. So this gives you a Galois extension of k with the prescribed completions. OK, so it's actually not difficult to see that this implication is in equivalence. OK, so that's the situation. And I've also explained that this fails, for example, for z mod 8z over q. So now, after Bianca's lecture, the natural question is, what is this failure of weak approximation on this variety? Is it a Brahmanian obstruction? Is this a Brahmanian obstruction to weak approximation? And, well, the answer is yes. So to state this, let me introduce some notation. So unfortunately, I cannot use the same notation as Bianca did because her varieties were proper. But these are not proper varieties. And I don't want to introduce smooth compactifications. So let me introduce slightly different notation, but it's the same thing. So x of variety, I will denote by x of k omega, the product of the local points, product of x of kv. So this is exactly what Bianca considered, except she wrote x of ak, the adelic points. It's the same when x is proper. OK, and so then she considered the Brouwer group. And here, the same, it's not the Brouwer group of x, which is going to be interesting, but the Brouwer group of a smooth, projective variety by Russian equivalent to x. So this is the so-called Unramified Brouwer group, denoted Brouwer nr. So this is going to be the Brouwer group of any smooth, proper variety containing x. Any smooth, proper variety containing x as an open subset. So it's an untrouble fact, but it's true that it does not depend on the variety you choose. It really only depends on x. OK, and so then Bianca explained that you have the Bramann in set. So yes? No, no, no. Yes, you can. But I mean, we're in character 6-0 anyway. If you're in characteristic p, you can do other things, but you have to be careful with the p-primary torsion subgroup, where p is the characteristic. So basically, if you look only at what's prime to p, you can do, as we say. OK, so recall. We have these inclusions, the rational points, contained in the Bramann in set, x of k omega, exponents un-ramified, contained in x of k omega. OK, and so recall, this is closed. So the closure of the rational points in the set of collections of local points has to be contained in there. And so if this inclusion on the right is strict, then weak approximation cannot hold. This is a Bramann in obstruction to weak approximation. And so the question here is, take x equal SNN mod g, is this inclusion strict or is it not? So the answer is yes. And more generally, here's the theorem. And please don't hesitate to ask any question you might have. And let me know, because I can't see you. So here's theorem 0. So this is due, I think, to Voskresensky and Sonsuq. So you take x equal SNN mod g. So we are working over a field, a number field, over k, a number field. And g is a finite, a billion group. OK, then the rational points are dense in the Bramann in set. So this is a situation where the Bramann in obstruction controls everything. You can do weak approximation of a collection of local points as soon as they satisfy the Bramann in conditions. So this doesn't directly tell you that you have a Bramann in obstruction, but because you know that weak approximation fails and because you have this density, the only possibility is that there is a Bramann in obstruction. OK, so more generally, we hope that such a density statement is true in much greater generality. So Bianca alluded to the case of geometrically rational surfaces. So let me state the general conjecture, conjecture 1. So due to Colliot-Delen, so this should work for so-called rationally connected varieties. So k is a number field for all the lecture. I will not repeat it. So x is smooth rationally connected, which I will abbreviate by RC, variety of k. Then the same density statement should hold. It's dense. So in other words, the Bramann in obstruction tells you about the existence and weak approximation of rational points. So OK, what is rationally connected? So it's a purely geometric notion. So you can extend the scalars to the complex numbers if you like. So x rationally connected. So let's say you embed k into c. So this means that if you take two general points, you can connect them by a rational curve over c. Two general c points can be connected by a rational curve over c, a rational curve, the image of a p1, some open in p1. So for example, if x is geometrically rational, that's true because if it's geometrically rational, an open will be an open in p1. And in p1, you can use lines to connect two points. So that's very easy. But this notion is much better behaved than rationality. And so for example, also our SLN mod g is rationally connected because it's unirrational. So unirrational means it's dominated by a rational variety. And SLN itself is rational. And again, if you have two complex points, you can leave them. And because it's rational, use lines in the projective space to connect them. OK, so this conjecture, as Bianca mentioned, is very much open even for surfaces. But still, there are some significant positive results. So just to give an example, let me say that this is known to be true. So that's a theorem of Borovoy. True if x is a homogeneous space of a linear group, linear algebraic group with connected stabilizers. So what does this mean? So you have a linear algebraic group which is acting on your variety. And homogeneous space means that the action is transitive on the k bar points. And OK, but you require that the stabilizers be connected. So for example, SLN mod g is a homogeneous space of SLN. SLN acts on the left. But the stabilizers, they are isomorphic to g, which is a finite group. It's not connected at all. So it doesn't apply to this. OK, but this is still quite a large class of varieties. So let's come back to our motivation. So this conjecture is directly related to Grunwald's problem because of the same situation as in the first lecture. So namely, there's this theorem of Ikadal, which is telling us that the Hilbert's irreducibility theorem is true not only over P1 or rational varieties, but also over rationally connected varieties that satisfy the conjecture. So let me state it like this. Take a connected g torso. Then if you look, sorry, an x is a smooth rationally connected variety satisfying conjecture one. Satisfying conjecture one. Then not only can you find rational points approximating any points in the Brahmanian set, but you can even find one such that the fiber of pi is connected. So if you take the set of rational points such that the fiber of pi is connected, then even this is still dense in the Brahmanian set. Sorry, k omega brah nrm phi x. OK, so what does this tell us? Well, it tells us that if you want to solve Grunwald's problem, what you really have to do is to prove this conjecture for SNN mod g and then understand the Brahmanian abstraction. So consequence, if you have conjecture one for SNN mod g and you can compute the Brahmanian set, then you get a complete answer to Grunwald's problem for g over k. You get a complete answer to Grunwald's problem. Again, because your local prescribed extension give you local points here. And then the only question you have is can you complete this collection of local points at the other places in such a way that the resulting family is in the Brahmanian set? If you can do it, then you have a positive answer. And otherwise, you don't. So in a sense, theorem 0 in this way is a reformulation of the Grunwald Wang theorem that tells the complete answer to Grunwald's problem for a billion groups. It's a much cleaner statement. But on the other hand, you still have to compute the Brahmanian set. And I mean, that's where the part which is not so clean goes in the computation of this set in this case. But it can be done in this way. You can recover the statement of the Grunwald Wang theorem. So in practice, if you have another group and you know the conjecture, well, you still have to compute this Brahmanian set. And as Bianca explained, this can be hard. So well, the first thing you would do is to compute the Brouwer group, the Unramified Brouwer Group here. And well, in general, this is hard. For this variety, SNN mod G, at least we have formula that we can try to apply. So there's the part of the Brouwer group that survives over K bar, for which there's a formula due to Bogomolov. And there's the algebraic part of the Brouwer group, the part which dies over K bar, for which there's another formula due to Arari. So in practice, usually you can use them to compute the Unramified Brouwer group in this case. Still, this doesn't tell you what the Brahmanian set is. So this formula only gives you the structure of the group to compute the Brahmanian sets. You need representatives, and that's not so easy. But still, in some cases, it gives you the answer. For example, it can be that you find that the Brouwer group here is just reduced to the constant classes. So then there is no Brahmanian abstraction at all. And then you have a complete positive answer to Grunwald's problem without excluding any place. So this happens. So there's an example like this in the exercises, where you apply the two formula and you find this conclusion. So in the lecture, the only thing I will say is there's one general fact that gives you some information about Grunwald's problem starting from this. It's a theorem due to Luchini-Arteche theorem, which says if you have conjecture 1 for SNN mod G, then you have a positive solution to Grunwald's problem for G as soon as you avoid the places that divide the order of G. Positive solution to Grunwald's problem for G, for any S that does not contain places dividing the order of G. So this is the hope that I formulated at the beginning of the first lecture. But we hope that this is always true. So this is the explanation why. So the proof of this theorem is really a study of the evaluation of the classes in the Brouwer group of such a space at the local points. And well, he proves that nothing happens at the places that don't divide the order of G. OK. So the theorem that I wanted to state in this lecture is a positive result for conjecture 1. Sorry, so this was theorem 3. This was theorem 2. And this is theorem 4, which I proved with Jonathan Harpass, which is that conjecture 1 holds for SNN mod G. If G is nilpotent, or even supersolvable, if G is, say, supersolvable. So let me remind you what this is. So supersolvable is in between nilpotent and solvable. It means there is a filtration, G equals G1, G2, and so on, Gn, such that each GI is normal in G, and the quotients are cyclic, normal in G, and the GI mod GI plus 1 are cyclic. So this includes nilpotent groups, so all P groups, for example. And so as I've explained, this gives a positive answer to Grunwald's problem, at least if you exclude these places, the places that divide the order of G. And in theory, if you are able to compute the Brahman in abstraction, you can get more precise answers. So in a sense, this is the best approximation you can get to the Hilbert-Knotup method. I mean, they wanted to construct rational points on the An mod G or SNN mod G by requiring that this be rational. OK, it's not rational, but you can still construct enough rational points to make the method work, in a sense. So OK, is there any question about this? OK, if there's none, so now what I will try to do is to explain how you would go about proving such things, what type of tools you can use. And that's what I will do in the remainder of this talk. So OK, so the main tool, one of the main tools is so-called Descent theory, so I will try to explain this. So for this, I need to introduce the notion of torsos in a slightly more general situation. So let X be a smooth variety, and G an algebraic group. I'm not assuming that it's a finite group anymore, an algebraic group of a K. So it could be a finite group, but it need not be. OK, and K is a field of characteristic 0. So a G torsor over X, the definition is essentially the same as I gave for finite groups. So it is a smooth, surjective map pi from Y to X. Y is some variety of a G. Smooth is telling you that the fibers are smooth. And an action of G on Y, such that this is a covariant, so such that this is a covariant, and such that G acts simply transitively on the fibers at the level of K bar points. Pi is G equivalent. So again, here the action of G on X is trivial. I really mean that it's invariant if you're pre-composed by the action of G. And G of K bar acts simply transitively on the fibers of Y of K bar goes to X of K bar. OK, so when you have this, in particular, X is the question of Y by G. So the notion of torsor like this for algebraic groups is very convenient, for example, for stating the Hilbert 90 theorem. So example, so Hilbert 90. So one reformulation of the Hilbert 90 theorem is telling you that GLN torsors over a field over K are isomorphic to GLN, are trivial. GLN torsors over the points, over K. And are isomorphic to GLN. And SLN torsors over K, that's a consequence. Are isomorphic to SLN. I mean, you can just take it for granted, it's reformulation. So next thing I need to talk about descent is the notion of twist. So definition, so if you take a G torsor over K like this, and sorry, a G torsor over X and a G torsor over K, then it makes sense to twist pi by P. And this means the following, so the twist, which we denote by Y sub P, but sub on the left. Well, that's just you take the product of Y by P, and you mod out by the diagonal action of G. And this still maps to X. I mean, you have the first projection here. And because X is Y mod G, you still get the map to X. Right, so now I have everything I need to state the conjecture of descent in this context. So it doesn't look like it's very much, but it's really the same as descent in the context of elliptic course. It's a generalization. So conjecture 5, descent. So suppose you have a smooth, rationally connected variety over a number field, X, over a number field K. And suppose you have a G, some linear group, some over K, linear algebraic group over K. And suppose you have a G torsor over X, a G torsor. So the conjecture, well, I should maybe call it a hope. If conjecture 1, the rational points are dense in the Brahman insets, holds for Y, and for all of its twists, then it holds for X, for all the twists of Y. So it means for all G torsors over K, P like this. No, so this is a slightly subtle point that I was hoping to avoid. If G is a billion, then yes, they are still G torsors. Otherwise, they are torsors under an inner form of G. But here I'm not using this fact anyway. I'm just looking at them as varieties. OK, so if conjecture 1 holds for all of the twists of Y, then it holds for X. So this is a way to descend conjecture 1. I mean, it can happen, and that's what happens in many situations, that you have your over ITX, and you are able to produce some G torsor over X. Whose geometry is simpler, and whose arithmetic is simpler, and maybe you can actually solve conjecture 1 for Y and for all of its twists, and then you're done. So it's not clear at all a priori why such a thing should be useful. But in practice, it turns out to be. OK, so here's a theorem about descent, which is due to Cooley-Othelan, Sonsuc, Arpas, and myself. Is that this descent procedure, the conjecture 5, holds is true if G is a torus. It's true if G is a torus. So let me recall what the torus is. So it's a commutative algebraic group. So torus means it's a commutative algebraic group, which geometrically over k bar is isomorphic to a product of GM, GM cross GM, cross GM, cross GM. Or if you like, it's the group of homomorphisms from a lattice L to GM. OK, L is Z to the power n, but maybe there's some Galois action on it. OK, so with this, let me show you how using this, you can prove theorem 0. So sketch of proof of theorem 0. So recall theorem 0 says that conjecture 1, density of rational points in the Brahman inset, is true for SLN mod G when G is finite a billion. OK, so this is just a sketch. So first, there's a fact, which is not obvious, but it's not difficult. And it's an exercise in the list of exercises. So let's just take this for granted. Fact that you can find an exact sequence of commutative groups of a k. There exists an exact sequence. 1 g t q 1, where t is a torus over k, and q is also a torus of a special kind. So where q t are a tori over k. And sorry, I have to go there. And q is a so-called quasi-trivial torus. Sorry, and q is quasi-trivial. So what does that mean? Well, it means that if you write q geometrically as home of a lattice to gm, there is a basis of the lattice which is stable under the action of Galois. So that means really q over k bar is home of some permutation module to gm. So for example, gm is quasi-trivial. The product of gm is quasi-trivial. The restrictions are quasi-trivial. OK, let me not insist on this. I'll just tell you why it's useful just for two things, quasi-triviality. Well, first, it implies that q is rational. It's just like gm is rational. And also, q torsors over k over a field are isomorphic to q over a field are isomorphic to q. And this is again Hilbert's theorem 90 in yet another guise. So all of this, if you like, is an exercise. If we take it for granted, it's rather formal. And let me explain how using this, you can prove these statements for SNN mod g. I'm sorry? Yeah, it's the two of them. I mean, you need Shapiro to get to gm, and then you apply Hilbert 90 for gm. Yeah, sorry. Thank you. So OK, so here's what you do. You have SNN mod g, and you consider SNN. So we've embedded our billion group into this torus t. So that takes a product with t. And let me take the quotient by g, by the diagonal action. OK, so I claim, well, the first projection induces a map here. And now, if you think about it, of course, t acts on this by multiplication on this factor on the left, right? You mod it on the right. But there's still an action on the left. And it's obvious that it's going to act simply transitively on the fibers. So this is a torsor. This is a t-torso. OK, so here's our t-torso. And we're going to apply descent to it. So theorem 6 tells us we can do it if you want to prove conjecture 1. For this variety, it's enough to prove it for this one and all of its twists. So apply descent to this. OK, but then why should it be easier for the top space, the top variety? OK, so to understand this better, let's look at the second projection. SLN cross t goes to t. So I mod out by g. This goes to t mod g, which is q. But now, for the very same reason, this is an SLN torsor. You still have an action of SLN on this, on the left. And it's clear that SLN is going to act simply transitively on the fibers here. SLN torsor. OK, but now here's the geometry. It's very simple. We have this base, which is q. And q is a quasi-trivial torus. In particular, it's rational. And here, we have this SLN torsor. And let's look at the generic fiber. It's an SLN torsor over the function field of q. And by Hilbert 90, this is SLN itself. So there are rational points in it, which tells us there are rational sections here. Rational sections. So all in all, this top space is by rational equivalent to q times SLN. So it's rational. SLN cross t mod g is rational. And so it satisfies weak approximation. And conjecture one, for obvious reasons, satisfies conjecture one. OK, so you still need to look at the twist, not just this. But if you look at the twist, take a torsor, say p over spec k, a torsor. If you twist this by p, you will get the same with p instead of t here. So maybe I just replace it. And then everything else is the same.  So p mod g is not q, a theory. It's just a q torsor. But q torsors are q. So in fact, you get q. And then exactly the same argument applies. So that's how you do it. So you see, it's a typical example where we don't have like this a natural torsor to apply descent. But it turns out that you can construct one in some way, then apply descent to reduce yourself to another variety whose geometry is simpler. So OK, so here's the actual theorem that is behind the theorem I stated on super solvable groups. So theorem seven, which we proved with unit on our pass. So in fact, it's that the descent method is true, not just when g is a torus, but also when it's a finite super solvable group. Conjecture five is true if g is a finite super solvable group. So this is what we call a super solvable descent. So I'm sorry, I just erased the statement of conjecture five. So let me just repeat it. You start with a torsor over a rationally connected variety, a g torsor where g is a finite super solvable group. If rational points are dense in the Brahman insets for the top space and all of its twists, then the same holds for x. And it's very easy to see that theorem seven implies the conjecture for SLN mod g. So this implies this was theorem four, I think. You just apply it to this torsor, SNN goes to SNN mod g. So this is a g torsor, but the top space is rational. So obviously it satisfies the conjecture. And the twists, well, they are exactly as here. If you twist, you still have the action of SNN on the other side. And the twists are SNN torsors, so they are SNN. So the twists are isomorphic to SNN. So they are rational. So I will not say much about the proof of this. I just want to say that the basic idea starts exactly like this. So in fact, now g is a super solvable group, so we are not going to embed it into a torus. But OK, we have this filtration with cyclic quotients. You take the first quotient, and you embed it into, and you apply this fact to it. So you embed it into a torus with quasi-trivial quotients. And then you do this construction, exactly this, except that, so is there a color? No? So OK, so I'm just going to rewrite here, except that here, of course, because you're looking at the quotient, so now this is going to be, sorry. So we had this g equals g, sorry, g is 0. I forget, g1. I forget what I wrote, g2, and so on. So we embed g mod g2, which is cyclic into a torus. And so here, we look at the silent mod g2. So, OK, cross t. And we mod out by the action of g mod g1, or g, if you like. So, and then you have the same diagram, and you apply descent for torsos and dottora on the left. The only thing is, then, the analysis of the arithmetic of the top space is more complicated. It's not just rational. But you still get an action of SLN on the left, but it's not anymore a torsor under SLN. It becomes a homogeneous space, and the stabilizers are g2. So at least you've gained something. The stabilizers are smaller than what you had at the beginning. It's not g. Now it's g2. But that's how the proof starts. Then you have to study the arithmetic of this vibration and work inductively. So I don't want to talk more about this. Instead, I just want to point out that super survival descent is actually quite more general than just this fact. So you can apply it to other variants of the inverse problem. Let me just give one example. So theorem 7 has other applications. So for example, it's somewhat of an anecdotal statement, but it's a challenge to prove it. So here's the theorem. So g, a finite super survival group, survival group. And you fix k and number field and some elements of k, alpha 1, alpha n in k, k star. Then there exists a Galois field extension capital K of k, Galois with group g, such that the alpha i's are norms from k. So this is something that had been proved first for a billion groups by Fry, Laugren, and Rachel Newton. They proved it by actually doing, by counting the number of a billion extensions that do this. And they found a positive number. But to do it for a super survival group, well, I mean, first you have to find a way to construct Galois extensions with given super survival Galois group. So the only known ways are Shafferovich's proof. And this one, as far as I know, only this one can give you such a statement. So how you deduce this from super survival lessons, well, that's one of the exercises. So I encourage you to go to the exercise sessions. OK, so we have a bit of time for questions if you like. And I'll stop here. Thank you for your attention.


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