uh good morning maybe we shouldn't do this one after all it's not yeah probably shouldn't do this one let's um let's just go home yeah thanks for stopping by first human reacts to the idiots i am corbin this is rick we're just kidding we're here to do it uh please follow us on instagram juicy goblins it's so juicy good morning uh so today uh we'll be reacting to the official teaser for uh calonk yep calonk calonk calonk sounds good to me uh dropped while we were sleeping yes dropped at about one a.m. our time so we were fast asleep yeah but you guys obviously let us know that uh yes we woke up to a lot of messages yes we were asleep uh so we get into it now so we were dreaming of our stupid babies yeah uh starring alia bat and a bunch of other people bunch of other people uh i think it's the actor from sanju's in it as well i think there's a whole bunch of peeps i think it's a big name movie with lots of names so let's just dive in this is all about man come on we've been waiting for this the official trailer is apparently gonna drop on like the 21st as well which we prefer trailers to teasers but we'll take the teaser what a shock it's beautiful visuals in that are insane hey disney that's what a latin should have looked like yeah the gritty no grittyness but still incredibly beautiful not even close man american cinema needs to just take it up to that level because that at every freaking frame yeah that was gorgeous uh so i obviously it's a teaser so they're just teasing you which it worked yeah it did i'm i'm quite interested and it did exactly what a teaser should do because i left thinking what's this about yeah and now i'm but you i want to want to see it because of the visuals absolutely obviously the star power from it the star power but the it is visually it utterly gorgeous it was up there with um the padma vat yes and but but strongly musta not me you know what it reminded me of and these that always well it reminded me of the very first trailers we saw when we were first introduced to cinema in india gully boy that no gully boy gully boy was obviously the first one but the first visually stunning ones that we saw that was padma vat it was padma vat yeah where we just sat back and went what the freaking heck yeah so yeah and that's that if a laden had looked like that mm-hmm i would have been gobsmacked yeah this is gorgeous yeah this is what a great trailer so much i'm proud of i'm quite excited for the trailer now i think they said it comes out all around the 21st uh and then we get to sell this bad boy when it comes out in april april i i i do love how soon y'all's trailers are to your films yeah i do too in america it's there's there's a whole system it's about six months out if it's like a big film yeah even a year yeah depending on the the size of the film you could have a trailer six months out yep and then they have a second trailer and then a couple tv spots right they really do and i much prefer that wasn't the way it used to be yeah it used to be like this it used to be you'd get a trailer because they wanted it to stay in your mind within a few weeks yeah and i much prefer this because there's all that buzz of this movie's coming when we're gonna get the teaser when we're gonna get a trailer when we're gonna and then you get to carry this momentum now over the next several weeks into it's it's great yeah that's fantastic this fantastic this lived up to my expectations yeah they were high and we love alia bot we do we love you alia so what did you think of it and um if you don't agree with us you're an idiot you're stupid


Let us start our today's lecture for NPTEL video course on Geotechnical Earthquake Engineering. So, for this video course currently we are discussing with module number 6 which is Dynamic Soil Properties. So, a quick recap what we have learnt in our previous lecture as I have already mentioned the basic reference for this course. I will refer to my another video course of NPTEL that is on Soil Dynamics, module 4 of that course where I have discussed in detail about this Dynamic Soil Properties. Now, in the previous lecture we have discussed about what are the most important Dynamic Soil Properties. In that we need to know first about the density or unit weight of the soil then shear modulus which is a important Dynamic Soil Property and damping characteristics another important Dynamic Soil Property. We have seen how shear modulus is defined it is nothing but the behavior of any material for our case it is the soil when it is subjected to cyclic shear stress. So, this is the shear stress versus shear strain plot the slope of that curve will give us the shear modulus. So, it is defined as the ratio of shear stress to shear strain as we all know for a linear material that is a material which is behaving linearly we will find out that this tau versus gamma or shear stress versus shear strain relationship is something linear like this. Hence, we will get a single value of this shear modulus from this curve whereas for actual material or most of the material like for soil we will get it will behave like a non-linear material where the shear stress versus shear strain of the material will behave non-linearly like this and if we want to find out the slope of that curve that will keep on changing at different point the initial tangent of that will be considered as maximum shear modulus or G max. If we want to find out shear modulus at any higher shear strain value then from that point if we join to the initial point then the slope of that line will give us the second shear modulus and if we draw a tangent at that curvilinear portion of that material behavior the slope of that line will give us the tangent shear modulus. Now, where these three shear modulus are used for equivalent linear analysis we use G second modulus for linear analysis of course we use G max and for non-linear analysis we use the G tan then we have seen how we carry out the equivalent linear approach we generally define the material property in terms of G by G max non-dimensional parameter in the y axis. So, maximum value of this G by G max ratio will be obviously 1 and as the shear strain increases which is plotted in the x axis in the log scale the material degradation will show like this for a soil and we will say that it is almost close to 1 G by G max value for very low shear strain that is that is called the linear range where G equals to G max for low strain for that the low strain range should be within 3 into 10 to the power minus 4 percent. Now, how to find out this G max value or maximum value of shear modulus the measurement involves basically the three major methods one is direct field measurement by using this techniques or indirect field measurement by using this techniques or laboratory measurement using these techniques also from the shear wave velocity value v s value we can calculate the value of G max using this relationship using this density of the material or density of the soil as rho. Now, coming to various relationship of this G value with respect to SPT n value that is standard penetration test n value several researchers has obtained correlation between the corrected SPT n value which is n 160 with respect to G max also in the laboratory one can find out using resonant column test like this that what is the value of G max at the resonant frequency from this plot of the distortion at different frequency on the resonant column the torque applied on it also from cyclic triaxial test one can find out the G max value another way to obtain the G max value is from in situ test parameters using various empirical relationships which are available worldwide like SPT n value CPT on penetration value DMT dilatometer test value PMT pressure meter test value. So, from each of these test we can get these empirical relationships for different types of soil as proposed by various researchers worldwide, but we need to remember these relationships are developed for limited number of data set point and also for a typical soil of that region. So, acceptability of these empirical relationships is always needs to be checked when somebody is trying to do a rigorous analysis for obtaining the G max value and further use of that G max for dynamic analysis. The variation of this G by G max with respect to cyclic shear strain for different values of plasticity index was reported by Wussetik and Dobry in 1991 which shows like this type of variation and also with respect to the number of cycles that is if number of cycle increases then there will be a change in the G max value obviously it is going to decrease as can be seen from here. The another important parameter damping behavior which we have discussed in our previous lecture like at low strain we will expect low damping ratio whereas at high strain we will get high damping ratio value and the variation of damping ratio with respect to cyclic shear strain for a typical soil material will be something like this. This result is also given by Wussetik and Dobry in 1991 for different values of plasticity index for fine grain soils. Now, worldwide there are developments on the SPT N value versus the shear wave velocity relationship because many a times people cannot obtain the dynamic shear wave velocity at the soil site or field. So, instead of that people can reasonably use from the static test which is nothing but static penetration test SPT N value the value of V s which further can be used to compute the G max value and for further dynamic design. So, how it is done? The example we had discussed in our previous lecture for soil of Bangalore region in India Sitaram et al. in 2006 proposed this empirical relationship V s in terms of the corrected SPT N 160 value also corrected with respect to clean sand. So, N 160 C s in terms of that how one can obtain the V s value in absence of the actual data obtained through field test for the V s value also for other types of residual soil and silt sand and sand silt the upper bound and lower bound equations were proposed by this researchers. Another group of researchers they had proposed for soil of Chennai region in India Bhuminathan et al. in 2006 for clay soil and for sandy soil the relationship between V s and corrected SPT N value. Then the application of research for dynamic soil characterization of Mumbai city we had started in our previous lecture and the reference for that I have mentioned that Sumedh why maske's PhD thesis Dr. Sumedh maske who completed his PhD in 2011 at IIT Bombay under my supervision. So, first we had discussed what are the various hazards for Mumbai city that is shows the need of doing the study of geotechnical earthquake engineering for Mumbai city. This is the seismic zonation map of India as per IS 1893 part 1 of 2002 version which places Mumbai city in zone 3 and these are the seven original islands of Mumbai city which has been combined together by land filling et cetera over the period of time. And these are various earthquake history in and around Mumbai city which has been reported in this journal paper by maske and Chaudhary 2010 journal of applied geophysics Elsevier. This GIS based map shows the original seven island of Mumbai and the surrounding marshy land that is reclaimed land around Mumbai city which actually formed today's Mumbai city. Some other necessary information to study the Mumbai city area for this seismic hazard analysis first of all location of the Mumbai city is in seismic zone 3 as per IS 1893 2002 version part 1. So, earthquake of 6 to 6.5 intensity or magnitude is possible to occur. Past disasters earthquake occurred in the peninsular India not exactly in Mumbai, but in peninsular India and Mumbai is also a part of that. The Mumbai city population is more than 15 million as per the census of 2011. Active fault zone are present close to Mumbai like Panvel flexure, Thanek Creek and Dharmatar Creek as mentioned by these researchers. Also there are various 23 small active faults in and around Mumbai city as mentioned by Raghukanta Nayanagar in 2006. So, with these details we had completed our previous lectures. So, let us see how to take it forward today in our present lecture. Now, when we want to do any dynamic soil properties study on that for the Mumbai city first we should know the typical soil property of Mumbai. So, to find out the typical soil property you can see in this slide that for the entire Mumbai region various numbers of borehole data has been collected from reliable government and private agencies. So, that we get the soil profile of entire Mumbai city in this fashion and this GIS based map shows the locations of boreholes which are collected all around Mumbai city which is available in the paper of Musca and Choudhury 2011 geotechnical special publication of ASC. This is a typical soil profile for the Mumbai city at various locations like Girgao, Wadhala, Andheri etcetera. And this table shows the various stations borehole stations various soil types their depth SPTN value corrected originally recorded corrected groundwater depth and various other soil parameters like amount of gravel sand sealed clay liquid limit plastic limit specific gravity. So, all these informations were collected from reliable and authentic borehole data from various locations like Tilaknagar, Chambur, Mulund, Wadhala etcetera. This chart shows the worldwide used correlations available for SPTN value versus the shear wave velocity VS value like for all types of soil, Oba and Toru Munini in 1970 proposed this equation of VS versus N also several other researchers as shown in this table have mentioned different equations of VS versus M as reported over here which are used worldwide extensively and the details are available in this paper by Musca and Choudhury 2011 in the journal Natural Hazards published by Springer. So, in this you we should again remember that these empirical relationships were developed based on the soils collected from a particular region and also it is based on a certain number of data set points. So, application of these equations for any particular region one has to be very careful whether that type of soil exist at the same area or not or whether the soil behavior at the location is similar or not. If not then obviously this relationship should not be used and a new relationship is required to be developed for that particular area. So, now let us look at here that when we want to develop some particular relationship of that SPT N value versus the shear wave velocity VS for Mumbai region from the collected borehole data you can see over here. These are the typical average shear wave velocity values all around Mumbai city at various borehole locations which are digitized in the GIS map these details are available in this paper. Also these stations and soil type depth SPT N value are shown over here and this is the equation which has been developed for Mumbai city by Musca for his PHD work under my supervision and these are the ranges of values of SPT N value and corresponding shear wave velocity value in and around Mumbai at different locations. So, one can easily see it is not in the similar range in all the places it depends on location to location it varies from location to location. So, one needs to be very careful when somebody is planning to do any earthquake engineering study or analysis or design at different locations of Mumbai incorporating this dynamic soil properties. This result shows the correlation between shear wave velocity in the unit meter per second versus SPT N value and this is uncorrected SPT N value it may be noted. So, from the present study this is the observed and proposed equation that is V s equals to 72 n to the power 0.4 for entire Mumbai city soil and field observed or field measured actual shear wave velocity at three different locations are obtained at different values of SPT N value and corresponding V s values are plotted which are matching very well with the proposed entire region Mumbai soil. This result shows the correlations between shear wave velocity V s in meter per second unit versus the cleans and corrected SPT N 160 value. So, this is cleans and corrected SPT N value you can see the equation proposed here is V s equals to 40 n 160 C s to the power 0.47. So, there is a minor change from the uncorrected SPT N value to the corrected SPT N value. This is observed from the Dr. Mascis PhD thesis as reported over here and also it was concluded from the study that there is hardly any large variation or significant variation between the use of original raw SPT N value or corrected cleans and SPT N value also by previous other researchers why because there are very several uncertainties involved in the SPT N value corrections also. So, instead of adding up the uncertainties it is also suggested that for basic study or first step of study or design it is better to use the basic equation of V s equals to 72 n to the power 0.4 for the Mumbai city soil from the uncorrected SPT N value where it will give reasonably correct result for the shear wave velocity as it is also verified and authenticated from the field measurements as shown in this slide. Now, coming to this comparison of various Indian soils what are the different correlations available between shear wave velocity V s and uncorrected SPT N value as reported by four researchers group for four different city as on today it is available like this that is shear wave velocity V s in meter per second unit in the y axis and in x axis it is SPT N value uncorrected this line the top one shows the results for Delhi city which is given by Hanumanth Rao and Ramana 2008 that is the researchers from IIT Delhi they have developed this equation V s equals to 82.6 n to the power 0.43 for Delhi city this is the equation proposed to be used whereas Anbazagan and Sitaram in 2010 this lower most line with the circles symbol they proposed for the Bangalore city V s equals to 80 n to the power 0.33 as the proposed equation for the soil of Bangalore city. Maheshari et al in 2010 as shown by this dark triangles which is for Chennai city V s equals to 95.64 n to the power 0.301 is the proposed equation for soils of Chennai city and for Mumbai city the present study shows the results which is star marked over here this one V s equals to 72 n to the power 0.4 is the proposed correlation for Mumbai city soil that is from SPT N value how to calculate the shear wave velocity. So, it can be seen that for this four major cities in India like Delhi city Bangalore city Chennai city and Mumbai city these are the corresponding proposed correlations between SPT N value and the shear wave velocity V s value which will help finally to compute the maximum shear modulus G max value from this V s value for that particular region of soil and that will be finally used for the seismic design of any structures in that locality. In the similar fashion it is today's necessity that for most of the seismically active region or important locations where major construction is required or proposed there to carry out the seismic design or earthquake resistant design it is necessary to find out this kind of relationship of V s versus SPT N because at many places we will not be able to carry out the shear wave velocity test at field due to several reasons one of that as I have mentioned in previous lecture because of presence of several obstruction. If obstruction is present many a times s s w m a s w results will give us the wrong result because it is not giving the result of that V s value of that particular soil but it is giving a result of shear wave velocity passing through composite material that is whatever structure or hidden objects are present in the soil that material including the soil that does not capture the exact value of the stiffness or shear wave velocity etcetera of that particular soil. So, that is the reason why it is also proposed worldwide and also in India it is today's need to find out this kind of V s versus N relationship for most of the important cities and locations and the seismically active regions. This GIS based map shows the thematic map of average soil shear wave velocity more than 100 meter per second for the Mumbai city. You can see over here all these red color patches are nothing but those region where shear wave velocity average shear wave velocity to a particular depth is greater than 100 meter per second. So, one can say these are relatively stiffer soil compared to other regions. Musk and Choudhury 2011 this natural hazards journal paper in Springer is available with all these details. This GIS map shows the geospatial contour map of average soil shear wave velocity V s with interval of 50 meter per second for typical soil of Mumbai city. So, with 50 meter interval 50 meter per second interval the shear wave velocity values are plotted and shown over here like 140, 190, 290 etcetera 240 are shown over here which is useful why because if somebody is planning to do any construction at any region say at Malad region then they know at this location typical remember this is the typical representation. It may vary within that location also but typically the range of shear wave velocity will be within this value whatever is mapped over here. So, these details are very much useful for practicing engineers and design engineers to further carry out the design earthquake resistant design or earthquake related design at that site. This GIS based map shows the value of maximum shear modulus G max value where it is more than 20 MPa for Mumbai city. How it is obtained from the values of V s it can be easily computed the G max value knowing the density of the soil. So, through that the G max of more than 20 MPa those locations are marked over here that means these are relatively stiffer soil zone. Now, this table shows the classification of the soil site into 5 different categories like soil class A, B, C, D, E with their description as hard rock, rock, very dense soil and soft rock, stiff soil and soft soil based on the dynamic property of the soil which is expressed in terms of average shear wave velocity V s 30. What is 30? 30 indicates the average shear wave velocity up to a 30 meter depth from the ground surface. So, that is why this number 30 came here. So, V s 30 of the soil in the unit meter per second that value is used to classify the soil into different classes. This soil site classification is based on as per Nihar standard of 2000 which is nothing but codal guideline or provision as mentioned in USA and practiced worldwide. So, for soft soil V s 30 value will be less than equals to 180 meter per second. For stiff soil it is between 180 to 360 meter per second, for very dense soil it is between 360 to 760 meter per second and so on. So, Mumbai soil what study has been carried out Dr. K. for his PhD thesis at IIT Bombay. It has been observed that most of the soil of Mumbai region comes under soil class D or E that means stiff soil and soft soil because their V s 30 value found out to be within this range of between 100 to 360 meter per second as mentioned over here. Now, coming to the another important topic, subtopic that is soil liquefaction. First I will mention that for the detail basic understanding of soil liquefaction one should refer and listen to my another video course that is on soil dynamics which is developed for this NPTEL course once again and for that soil dynamics video course module number 4 also discusses about this soil liquefaction. What is soil liquefaction? As mentioned in this book by Kramer in 1996 it is nothing but the transformation from a solid state to a liquefied state as a consequence of increased pore pressure and reduced effective stress for soil. And as mentioned by Osinov in 2003 if the shear resistance of the soil becomes less than the static driving shear stress the soil can undergo large deformations and is say to liquefy. So in the state of soil liquefaction soil liquefaction can occur due to several dynamic loads but earthquake is of course one of the reason. So due to earthquake also soil liquefaction can occur. Now soil liquefaction due to earthquake modern liquefaction engineering developed after Nigata 1964 earthquake and great Alaskan earthquake of 1964 after that the research etcetera extensively started in this area of soil liquefaction. So key elements of soil liquefaction as proposed by C Detol in 2003 are mentioned over here and we should always remember that every new earthquake whenever another earthquake comes it creates a new research area because we get lots of more data lots of more understanding about how the soil behave during and after the earthquake. So all those information whether it got liquefied if it has been liquefied what type of soil was there if it has not liquefied during a magnitude of earthquake why it has not liquefied what are the characteristics of the soil and then try to correlate between various physical and engineering soil parameters with respect to the liquefaction estimation and so on. So this assessment of likelihood of triggering or initiation of soil liquefaction needs to be carried out then assessment of post liquefaction strength and overall post liquefaction needs to be studied because after the liquefaction how much strength of the soil is remaining whether it can be further used for construction of structure or not. These are important to know assessment of expected liquefaction induced deformations and displacements assessment of consequences of these deformations and displacement then finally the implementation and evaluation of engineered mitigation if necessary at that site. Now susceptibility of the soil to earthquake induced liquefaction these are the parameters which influences like of course earthquake intensity and duration what type of soil is present there then soil relative density particle size distribution of the soil presence or absence of the plastic fines ground water table location that is amount of degrees of saturation hydraulic conductivity or permeability of the soil placement conditions or depositional environment of the soil aging and cementation of the soil structure overburden pressure and finally the historical liquefaction. So all these factors which influences the earthquake induced soil liquefaction now liquefaction susceptibility criteria there are several methods as can be seen over here for fine grained soil there are various methods to obtain the liquefaction susceptibility criteria both for fine grained as well as coarse grained soil Chinese criteria and modified Chinese criteria Andrews and Martin criteria of 2000 Ude et al 2001 criteria Cde et al 2003 criteria Brae et al 2004 criteria Brae and Sancio 2006 criteria Bolange and Idris 2006 criteria and various other studies. So among these the most important or maximum widely used one is this one this is by Ude et al 2001 this is maximum used worldwide for the liquefaction susceptibility estimation. Coming to Chinese criteria and modified Chinese criteria Chinese criteria was developed by Wang in 1979 based on the study on Haitian and Sanxian in 1975 and 1976 the criteria says that the percent finer than 0.005 mm should be less than 15 percent liquid limit should be less than equals to 35 percent and water content should be greater than 0.9 times of liquid limit for that fine grained soil this criteria is basically for fine grained soil. So, this is the typical plasticity chart as we know liquid limit versus water content this is the a line equation. So, modified Chinese criteria says that the fine soils that the plot above this one the a line shows in the figure are considered to be susceptible to liquefaction if these three conditions that is percent of clay present in the soil is less than 15 percent liquid limit is less than equals to 35 percent and in situ water content is greater than or equals to 0.9 times liquid limit. Then the Ude et al 2001 criteria is a combination of several other researchers findings and this is a kind of report or methodology proposed which is accepted worldwide so far. So, seed et al in 1985 developed the ratio of the CRR versus CSR curve for granular soil. CRR is cyclic resistance ratio and CSR is cyclic stress ratio the details about these things I have already discussed in my another video course for NPTEL that is on soil dynamics. So, I request all the viewers of this course also to go through my another video course on soil dynamics like module number four of that course. So, we can get all these basic details I am not covering it here. So, it has been proposed how to estimate this cyclic resistance ratio or CRR from corrected blow count SPT N 160. So, these are all the collected historical data points of earthquake from actual field test results that where the soil got liquefied and where it did not liquefy that shows the three curves like percent fines less than equals to 5 percent another is between 5 to 15 percent this range between 15 to 35 percent and more than 35 percent. So, like that this percent fine if it is less than equals to 5 percent then it is called SPT clean sand based curve and clean sand correction also required to be computed to estimate the CRR value from this curve and seed et al in 2003 they proposed these are the recommendations for fine grain soil that is plasticity index versus water content or liquid limit water content to liquid limit ratio of in the x axis they have divided into different three zones that is susceptible to liquefaction moderately susceptible and non susceptible. So, accordingly if somebody wants to put their soil in this region and try to find out whether it is coming in susceptible range or non susceptible range this recommendation may be used further bre et al in 2004 they proposed another methodology using the similar concept of plasticity index versus water content to liquid limit ratio like this and mentioned that liquid limit is not considered as the authors observed that a number of specimens with liquid limit greater than 35 percent were found to be moderately susceptible to liquefaction another set of researchers like Bray and Sancio in 2006 they proposed a criteria based on 10 numbers of cyclic simple shear test performed for the same soil specimen in addition to the test carried out as we have explained in the previous slide just now some observations of Chichi earthquake of 1999 as reported by Chu et al in 2004 are also incorporated in that and in the same pattern of plasticity index versus the water content to liquid limit ratio have been zonified into three zones one is susceptible to liquefaction another further testing is needed and another is not susceptible to liquefaction zone. Another criteria is mentioned by Bolange and Idris in 2006 where CRR of clay like material and CRR of sand like material has been considered as two boundaries with respect to the x axis plot of plasticity index p i. So, for different values of plasticity index the typical range from transition of sand like to clay like soil behavior is mentioned by these researchers which is showing the exhibit of cyclic liquefaction. So, fine grain soils having plasticity index less than three are named as sand like and they can exhibit the cyclic liquefaction type response whereas for fine grain soils with plasticity index greater than 7 are named as clay like material where they are expected to exhibit cyclic mobility type response. So, in one case it is cyclic liquefaction and another case it is cyclic mobility depending on whether it is sand like behavior or clay like behavior and in between range that is when plasticity index is between 3 to 7 a transition between this sand like to clay like behavior is proposed to occur and this figure provides a schematic illustration how this transition from sand like to clay like behavior is occurring. So, this criteria in the CRR versus plasticity index domain without a scale and distinction of sand like and clay like fine grain soil is based on solely on the plasticity index on the specimen and r u value that is pore pressure ratio as it is mentioned over here excess pore pressure ratio for fine for sand like soils initial liquefaction is achieved when excess pore pressure ratio r u becomes equals to 1. So, if it is less than 1 then it is not liquefying whereas for clay like soil it go undergoes cyclic mobility when this r u value exceeds 0.8. So, for sand like material it has to be equals to 1. So, that one can say liquefaction is going to occur for clay like material it should be more than equals to 0.8 then one can say it is going through the cyclic mobility. So, this r u based liquefaction susceptibility definition it requires the determination of CSR levels and the duration of the excitation. So, these things are the topic of research even on today in this year 2013 still further researches are carrying out their various research from the collected field test data of liquefied zone, non liquefied zone, transition zone etcetera and also doing the laboratory test and combining this data set of results as many data set are possible. So, that this interpretation of results and then further proposing some new criteria will be valid and can be used by various researchers. Now, coming to the seismic liquefaction hazard map of Mumbai city using this GIS and GPS the details of this work can be obtained in this journal paper by Maske and Choudhury 2010 in journal of applied geophysics Elsevier publication volume 70 number 3 page number 2162225. The evaluation of soil liquefaction like how it was carried out from the entire set of borehole data collected for the entire Mumbai region as the dynamic soil properties the V s value has been estimated and reported just few slides back as we have discussed here. Then for each borehole locations the soil liquefaction susceptibility is estimated using the simplified procedure for evaluation of liquefaction potential. The simplified procedure was basically proposed by seed and idriss in 1971. Further it was modified by Ud and Idriss in 1997 and the final one is Ud et al. 2001 which is widely used worldwide as I have mentioned. So, discussion about this simplified procedure to evaluate the liquefaction potential is available in my another video course of NPTEL which is on soil dynamics module 4 of that discuss about the simplified procedure for liquefaction estimation. So, these are the stepwise procedure in step 1 the surface data used to access liquefaction should include location of ground water table SPTN value share wave velocity value unit weight of soil fines content of the soil moisture content in the step 2 evaluate the total vertical stress and effective vertical stress that is sigma v and sigma v dash for all potentially liquefiable layer within the deposit and then one need to calculate this cyclic stress ratio or CSR as induced by the design earthquake. So, for a particular region we all know what is design basis earthquake as per the codal provisions or the zonation map or the seismic micro zonation point of view one can find out what is the value of this m x for a region that m x value can be used this g is acceleration due to gravity. So, this is just a number r d is nothing but stress reduction factor due to the flexibility of the soil and the sigma v is total vertical stress and sigma v dash is effective vertical stress. So, with that a non-dimensional parameter CSR cyclic stress ratio can be obtained which is proposed by C Dan Idris in 1971. Now, how to select this stress reduction coefficient r d there are various researchers who had proposed different ranges or values or equations for r d we can see over here as we go deeper and deeper inside the ground from the ground surface. So, from this ground surface where the depth is 0 if we go deeper below the ground as the depth increases in meter unit it is shown over here the r d value reduces also from 1 to this 1. So, 1 means there is no correction or stress reduction coefficient in this equation is not required 1 means that is at ground surface, but as we go deeper because of flexibility of the soil this stress reduction coefficient needs to be incorporated. One can see Idris in 1999 proposed this line to calculate the value of r d for a certain value of m s which is certain value of v s shear wave velocity 120 meter per second magnitude of moment magnitude of earthquake is about 6.5 p g a value of 0.2 g for that this is the line whereas, shittija etol 2009, kishida etol 2009 mentioned this line to be used whereas, set in etol 2004 mentioned this value to be used whereas, for another range of same soil that is v s value 120 meter per second, but under higher magnitude of earthquake when higher magnitude of earthquake m w of 7.5 is coming at that location these are the values of r d as proposed by different researchers. It is adopted from the research paper of Idris and Bolanje of 2010 one can easily see that there is a wide variation in the value of this r d which can influence this calculated value of this C s r. So, it is a question which is the correct value to calculate this r d. So, in absence of the correct value one can easily use the method proposed by you etol 2001 to calculate the reasonable range or value of r d. In cyclic resistance ratio another parameter C r r at the reference magnitude of 7.5 can be calculated using SPT data of n 160 using this expression which is known as black's equation as proposed by black which is available in the paper of you and Idris 1997 and also you did all 2001. These are the x is in this equation x is nothing but n 160 corrected SPT a n value and various coefficients a b c d e f g h all are mentioned over here. And factor of safety against liquefaction is computed using this expression factor of safety against liquefaction is nothing but C r r 7.5 7.5 is nothing but at the moment magnitude of 7.5. If the earthquake of that zone for which the design is considered is different than 7.5 then corrections due to the magnitude correction needs to be carried out. So, C r r by C s r will give the factor of safety with respect to liquefaction. Using this concept this paper of musk and choudhury in journal of applied geophysics Elsevier classified the three ranges of factor of safety with respect to liquefaction their value to identify or remark their soil as critically liquefiable, moderately liquefiable and non-liquefiable soil. What is critically liquefiable when factor of safety less than 1? When factor of safety is between 1 to 1.3 it is mentioned as moderately liquefiable soil and when factor of safety is greater than 1.3 it is considered as non-liquefiable soil. So, using this ranges of factor of safety for entire Mumbai city in this paper the calculations for factor of safety against liquefaction with respect to depth and with respect to all boreholes were carried out. Finally, this is the GIS based map for entire Mumbai which shows the critically liquefiable area. Critically liquefiable means in these locations in these patches as shown by this color that shows a critically liquefiable area in Mumbai at a moment magnitude of 6. So that means if a moment magnitude of 6 earthquake comes in Mumbai these are the region where soil is going to liquefy as the present scenario is concerned. Critically liquefiable means factor of safety with respect to liquefaction will be less than 1. This is another liquefaction hazard map for Mumbai city which shows for moment magnitude of 7.5 if it comes in Mumbai these are the region where it is going to critically liquefy and one can easily see these are nothing but the areas where it is the reclaimed land. So that is why it is another kind of validation that as we know the reclaimed land or filled up land are more prone or susceptible for liquefaction during an earthquake which is also got validated from the property and from this liquefaction hazard map. So, how people can use this hazard map for further design? So, if somebody is planning to construct any big high rise building in these locations which is quite possible in Mumbai city extra design care and design measure needs to be taken for the foundation design and other design like if somebody is constructing a pile foundation in these locations then pile foundations need to be designed with respect to the liquefiable zone which I am going to discuss in subsequent lecture modules in this course. This table shows the various values of factor of safety against liquefaction for entire Mumbai city as obtained in this journal paper by Mascan Choudhury 2010 in journal of Applied Geophysics in Elsevier publication. One can see these are the different site address like Andheri, Bhandu, Baurivali, Bandra, Mallard, Dahisar etcetera. So, factor of safety against liquefaction for different moment magnitude are mentioned over here like moment magnitude of 5, 5.5, 6, 6.5, 7, 7.5 all values are given over here. One can see the non-bold values are perfectly fine. That means if in Mumbai magnitude of 5, 5.5 earthquake comes there is absolutely no problem in terms of liquefaction is concerned in these mentioned 10 locations in these 10 sites. But if magnitude of 6, 6.5 or more than that comes the soil tends to start liquefying at certain locations like one can see at Bhandu West if a magnitude of earthquake magnitude 7.5 comes the soil is going to critically liquefy that is it is going to fully liquefy that means the factor of safety less than one even at magnitude 7 also. Similarly, for other locations also the values are given over here which is very much useful for any designers to utilize this concept to further take protection and necessary design steps and methodology and construction steps for earthquake resistant design in and around Mumbai city using this data. So in concluding remarks we can say that typical shear wave velocity what we have obtained for the soil in Mumbai region between 3 meter to 10 meter range varies typically between 140 to 350 meter per second. And typical areas like Kandivali, Borivali, Goregaon, Malad of Mumbai city can be prone to critically liquefiable condition when an earthquake magnitude of 7.5 hits in and around Mumbai. The soil amplification factor for Mumbai which has been obtained earlier it can range between 2.5 to 3.5 if a similar type of Bhuj earthquake motion of 2001 hits Mumbai city. And from this known knowledge of geotechnical earthquake engineering one can further take precautionary measure to find out what are the significant effect of depth of this liquefying layer which needs to be considered for design of pile foundation and any other type of foundation even for shallow foundation also which are necessary to incorporate in the design. So with this we have come to the end of present module, module number 6. We will continue further in our next lecture.


Hey, thank you. Thank you everyone for coming to the last talk of this lecture series Was it there a bit of echo? Yeah, I'm trying to take care of it. Okay And then I'm trying to go guy I Still see some people getting into the room. So maybe I'll wait in another In a minute, okay, so Last time we did we just we discussed the cologne the definition of the Coulomb branch due to a problem in Finkelberg Nakajima So let me just recall the notations. We had G a reductive group and in a representation of G And then we produced from this first a space called RGN and Then we studied the homology of the space which was an algebra a GN so the Geovo like a variant homology of the space RGN We also had a quantized version Where we Which was defined by adding C star covariance to the homology So that this H bar quantization parameter is the C star equivalent parameter And then we defined the space the Coulomb branch MC GN to be spec Of this algebra We also considered a few variants of it. Um, maybe I'm not Need them today, but we had some variants of it where we were able to make for example a Candidate for a resolution of the space using project of a graded ring Okay, so today We're going to be interested in in quiver what's called a quiver gauge theories so what that means Is we fix a gamma directed graph cover And at the moment I make no assumptions on gamma can have multiple edges can have loops so We just find And then we have two dimension vectors v and w so they've been Natural numbers the i i is the vertices of the graph And the reason why we have two is because we have this frames picture in mind So we have a circle for every vertex of the graph here. We have some edge of the graph And then for each we have a frame Here and I put wi in the frame and from this data of the graph the dimension vectors We construct a group g which is simply the product of the gls of the vertices and a representation and which is simply the Direct sum of first hams inside the graph And then hams to the framing I don't know why there's so much drugs in this Can't at least get rid of one of them. So I remind you Let me discuss this before but there's two spaces. We're going to consider one is the Higgs branch Um, I'm sorry, maybe one piece of the notation before I get there associated to gamma. We have G gamma that cat's moody the algebra symmetric cat's moody the algebra So it it doesn't care about the orientation of the edges in the graph Um, mostly I'll be concerned with the case when there's no loops when there are loops Maybe this exactly what the algebra we associate. Maybe it comes a little bit more complicated. I'm not Too much expert. So mostly we can say the case without loops Um associated and then um associated this v and w we build a dominant weight lambda And mu just a weight And they are related to v and w by these familiar equations lambda Is the sum of fundamental weights with coefficients given by the w i's And lambda minus mu is the sum of simple roots with coefficients given by the odds So Yeah, so as I said, we have two spaces now associated to this data. We have the Higgs branch All right, m h lambda mu and this is just a Nakajima core for variety by definition I take the cotangent bundle of n and I take the Hamiltonian reduction by At the moment map level zero and at the g it parameter chi and I use my By by the group g And then I have the Higgs Sorry, then I have a Coulomb branch just the Coulomb branch for this gauge theory to find above I just write lambda mu for the instead of g and n Okay, so the first question of course is like what can you say what is this space? What is this? um Coulomb branch associated to the quiver gauge So this is some well That's that's going to be based on the topic of today's lecture. So, um Let's start with the case when g gamma is a finite type so finite type means it's a The dink and it's the dink and diagram of a ad e the algebra in this case, um, I'm going to write g gamma for the corresponding group Or maybe slightly more precisely the langland dual group. But again, this langland duality is going to be irrelevant because we're in this simply this type So g gamma is going to be the corresponding group And then we're going to study the affine-gross mining of g of gamma Okay, so let let me just be clear that there is a Um two affine-gross mining's in the story one affine-gross mining which was used to define the Coulomb branch And that's the affine-gross mining of g the gauge group and now i'm talking about the affine-gross mining of the group whose Dink and diagram is given by gamma When I first heard about the bfm construction it was at a conference in paris. It was in January of 2015 and and sasha was giving a talk It was organized by vastero, I think And and at some point in the talk I interrupted sasha and I said you mean to tell me there's two affine-gross mining in this story And they're not related as sasha said yes okay, so this affine-gross mining is affine-gross mining of this ade group g gamma And I write uh g gamma so it's g gamma k my g gamma o And I just recall notation. We have gr lambda. So that's the g gamma of o orbit through the point t lambda So it's going to be turned to be the same lambda that i'm using above so this lambda Was obtained using just this dimension vector and it encoded a dominant weight for for g gamma and And then I also will have This slice w mu I mentioned him before but let me mention him again. So how's he defined I take? G gamma of t inverse and then I take this this first congruent subgroup condition And take its orbit through t mu. So it's the kernel of the evaluation. So give me these transverse orbits. They go around this And then I define this affine-gross mining slice w lambda mu to be girl lambda bar intersect w mu So it's a it's a subs It's a finite dimensional sub scheme of of w lambda. So it's an affine scheme So that's the definition Of the below. So here's a the first the first theorem about this these coulomb branches Is that they are these affine-gross mining slice? So Well, sorry in this circumstance. So again, we have the assumption that g gamma is a finite type And also the assumption is that mu is dominant Okay, so recall the construction of this um Of this thing didn't require mu to be dominant. We just took mu and used it to I mean mu is obtained using relate to this v's but nothing to do with Being dominant or otherwise. So here only if it's dominant, we have this now. So if mu is dominant then The coulomb branch is isomorphic to that affine-gross mining slice Okay, and and If you've been following my talks then you should find this theorem not surprising at all because I already said many times That the coulomb branch is supposed to be some psychic dual to the Higgs branch And I've also said a few times that the affine-gross mining slice is some psychic dual to the necrogen macro variety so This theorem just a confirmation of those two facts So what happens when mu is not dominant but still in the finite type situation Then we can define something called a generalized affine-gross mining slice The definition looks a little bizarre First of all rather than working in the affine-gross mining actually worked inside the group g of k and I formed the following thing So I considered all um major all elements of g of k which emitted Gauss decomposition like this So inside of any so maybe I should fix some notation before I write this So here, um u and u minus in g gamma to note opposite unipotent subgroups And so inside any Inside the group of matrices or any reductive group you can consider those elements which emit Gauss decomposition like upper triangular diagonal lower triangular decomposition So upper triangular diagonal lower triangular So I'm looking at those matrices which admit or those elements of g of k which admit such a decomposition such that um, the lower triangle and upper triangular piece are one my t inverse and the diagonal piece is like t the mu times this diagonal thing which is one much So at first time seems like a slightly strange thing to do Here's some reasons why um, it's a good idea. Oh the meaning of the subscript So if g is any group and then I have g if t inverse and it emits an Morphism to g which takes t inverse to one and the kernel of this map is g1 t inverse Sometimes called the first congruent subgroup So you can the matrices so for example this u1 t inverse would look something like this one one one down the diagonal here And here up up in the upper part would be like t inverse plus t of minus two So So Where was it? Yes, so this is the definition of This w mu and it looks a little bizarre But here's a few reasons why it's not so strange. So so first of all if mu is dominant It's not very obvious But it's not hard to see that that in map Well, of course this w maps to the afringer's mine in just because g of k maps the afringer's mine in So you map if you if mu is dominant in your map w mu into the afringer's mine in Then um, I realize my g sub gamma looks a little like my g r Hopefully no one gets confused besides me. Okay. If mu is dominant then w mu maps into that Maps isomorphically on to mu w the thing I previously called w mu in the afringer's mine in So there's not two different things called w mu. So that's good to Um, this space w mu has a modular interpretation. I won't recall it um It's not really Um, it's in terms of well, let me vaguely recall it. It's in terms of principal g bundles on p1 With a reduction to a burrell into an opposite burrell Um, I don't find it any more enlightening than this matrix definition In fact, I find the matrix definition more enlightening, but anyway, this is how it's originally defined by by one chemical burden epigema And three which is closely related to and sort of also closely related to this matrix definition Um, and this I think the first incarnation of this w mu. It's some kind of modulate space of singular monopoles and and I say this in This is appearing in the physics literature. So I don't not sure if there's Yeah, a precise mathematical statement here, but this is in the physics literature a little more Demofted Okay, okay, so that's some reasons why why this w is not as like random as it appears Anyway, once you have this w mu then we define the generalized affine-garse-mining slice w and the mu to simply be this W mu well, normally we would take w intersect or lambda bar But now we're not in affine-garse-mining. We're inside this g of k So we take w mu intersect the double coset space and put the closure If I don't if I write w lambda mu without a bar on top, it just means the same thing without the closure So this thing's called So one, um One funny feature about this space is it's often not It's I started the talk that my series of talks by talking about conical symplemic singularities Well, I haven't even talked about or conical some type of resolution I haven't even talked about a resolution space, but actually this space is not actually Usually conical. So it's not conical Or at least doesn't actually admit an obvious cease to action making it conical I don't know exactly if it's possible to prove that it's literally not conical But at least doesn't admit an obvious cease to action making it conical not conical Unless you might think it's unless mu is dominant, but it's actually unless mu is almost dominant Unless mu plus row usual row is dominant Um, I mentioned last time the physicists and and Nakajima sort of introduced this notation in the mathematical literature have a division of these theories Um, it's a good bad and ugly. So good in this case good means mu is dominant Bad is mu plus row is not dominant and ugly is the intermediate case when mu plus row is dominant from mu is not dominant Okay, let's see some examples Oh, I mean, maybe I should say to But okay this theorem With this definition of w lambda mu I won't rather writing it again With this definition of w lambda mu this generalized slice I mean, obviously I wouldn't be telling you this but this if it wasn't true this theorem extends So now we can erase No, I'm very sorry, but we I miss uh question in the chat That was a question of six minutes ago But I already answered it I really okay I saw it and I I answered it Okay, it's fine. There's a for me No problem um, so this theorem Of bfn extends to the case of mu is not dominant when we use this definition of the generalized that fungus minus Again in the finite case If it's not finite type, it's not totally clear. I guess what the meaning of anything I wrote here is Okay, um So let's see some examples So the first example Was was the with this g gamma being sl2 And so my quiver just says one vertex and I'm going to take Let's consider the general case. So I have Framing in in the m here. I discussed this last time a few times when I was talking about Example when I was talking about the bfn construction So in this case lambda is n times the first fundamental weight Mu is n times the first I mean the only fundamental way minus m times the only Simple alpha like if you like you can just think of this as n and this is n minus 2 n is perfectly good way to give it um The corresponding Nakajima quick variety we discussed before is just the cotangent bundle of the grismanian so the mh And then Okay, and what's our coolant branch or our generalized ethnograph money and slice Well, it's actually not very hard to read off like this thing looks like a big mess Guesty composition business, but it's actually not very hard to to analyze it Especially in this as all two case and when you do it if after maybe possibly taking transpose in your matrix or something You reach the following nice description. So it's the set of all matrices a b c d And these are all matrices over polynomial ring and t so Got rid of all powers of t and rest just by scaling through And a is monic Of degree m exactly degree m b and c are of degree Less than m d. I impose no condition at all and ad well except for that ad minus bc must be equal to t to then So it's a it's a nice space You can have fun playing around with what it looks like for different values of m and n for example, you might try to figure out what How the degree of d is sort of affected by the degrees there is in this space At least when When this In the dominant case in the m minus 2m. It's greater than zero Then this space is isomorphic to a m n minus m slow to assist Okay, so that's one nice one nice example and this Generalizes in a few ways. Well one way we could say that this generalizes is that um There's a similar kind of matrix description of it for any glm but also For any glm or any finite or any even affine type a so g gamma's Or affine They pay this this space or or precisely I mean in the app in the affine in the affine type. I don't usually write Wm I just write mc lambda mu This space is a bow variety Joe, I think we have a Might have a question in the q and a please great, um Oh, okay, it's nice of you. I always David. Okay. Let me I'll come to that in one second So let me just finish this right if g gamma's is finite or affine type a then this is a bow variety This is proven by Nakajima and takayama and we'll hear more About bow varieties in richard romani's talks talk next week So it's some explicit description of it, which is um Sort of different than this matrix description Okay, there was a question about does symphlectic duality statement still hold if it's not conical um Well, it depends exactly which part of the symphlectic duality statement um But but the short answer is yes like the um If we focus on like the third part the part about the category Oh Or even the second part about the part about the Homology Those parts still those parts do hold as I'll say in a few minutes Um, even even when mu is not dominant Okay, um So that's I guess it's a sort of second example or a meta example a third example um is um Interestingly enough how to get the the gress mining a finite potential bundle of gress mining as a as a Coulomb branch And and it's like this if you take lambda to be If you take g gamma to be sln and you take lambda to be omega m plus omega n minus m Then you take mu to be zero if you convert that to quiver data what it means is you have a one here two framings and an increasing Sequence like so And then a constant sequence like so so that's the quiver data coming from this lambda nu And this affine gress mining slice in fact this case it's not even a generalized affine gress mining slice, but mu is zero It's dominant this affine gress mining slice is just a cotangent one And you can see this just from the geometry of affine gress mining um, oh and another example I was inspired to add this example because of Eugene's talks, uh, he's like Said I've been neglecting the Hilbert scheme indeed. I have been neglecting the Hilbert scheme So here he is in this for this quiver data so or for this, uh generalized or generalized affine gress mining slice if you like of type a one one or Yeah, um Anyway, for this for this quiver data the the corresponding Coulomb branch is the Um, is the Hilbert scheme as is the I mean, technically the Coulomb branch is singular. So it's really not really the Hilbert scheme. It's really it's c2 I mean see Just symmetric and Just sim and Is another question. No, no, it's the same question okay, so Yeah, that's that's it for this examples. Oh, maybe one more one more example another another classic example We take this quiver data, which is the same as produces the cotangent bundle of the fly variety as the As the as the Higgs branch It also produces the cotangent bundle of the fly variety as the Coulomb branch This is Same as W and oh my god N minus one zero Usually Okay, great. So What are some properties? What are some results we know about these? These generalized affine gress mining slices and more generally these these Coulomb branches associated to cover data All right, sorry Joe, is it true that cotangent bundle to fly varieties are self dual? for any group And yeah, simply quality. Yes. Yeah but only only in type air they realized this as Higgs and Coulomb branches of of gauge theories Yeah Oh, sorry outside in the general type. Maybe it's better to say that the cotangent bundle of gmod b is the Is the langans dual group? Okay, so here's some general properties. So So for the first one, we go back to the case of the finite type And I mentioned of course that there's this more there's this map from W mu to the affine gress mining and if we restricted the to W lambda mu will land inside girl lambda bar Well, this map is of course far from injected Maybe maybe I should have mentioned this from the very beginning But one fact is that the dimension of this W lambda mu is twice rho lambda minus mu Whereas the dimension of girl lambda bar is just twice row of lambda So if mu is not dominant like this negative mu will actually be contributing to the dimension So the dimension will become bigger than girl lambda bar, right? Like if you imagine the extreme case and mu is anti-dominant then this is going to be making a positive number Okay, so So this map is not injective, but if we restrict to the attracting locus And I should say this attracting locus only exists If mu is actually a this this w other mu exists even if mu is not a weight of The lambda representation But the tracking locus only exists if mu is a way of lambda presentation You can prove in this finite type case pretty easily that That this torus has a fixed point if and only if mu is a weight So if we was the next and when when it has a fixed point and then we take this attracting locus Then the map is injective this Gives a nice morphism between the attracting locus and And I say s mu which I never defined before but this s mu Probably I mean that may be s mu minus to be technically correct. I think People have maybe seen it before it's the u minus k orbit And this this this so this result the fact that this thing gives this restriction gives a nice morphism as a theorem of pre-life And as I mentioned before this is true for even if mu is not a weight both sides will be empty if mu is not a weight of the lambda So in particular if we take the top homology of this Attracting locus, I mentioned like a long time ago that I really like these top homology of attracting locus because that's what enters this decomposition of the homology If we take this top homology of attracting locus, it will be the same As a top homology here Which will be isomorphic to The view of lambda mu and this is by Mirko Richelon So one way to think about this generalized afringer's mining slice Is it something which contains this Mirko Richelon and locus as the attracting locus It's just something cooked up to be of twice the dimension of this This girl in the bar intersection Um, and this leads to the following ejecture Which I think is very important still open trajectory of well formulated by the fn They call it the geometric's attack a conductor. So it's the statement that for any Gamma not necessarily a finite type So in finite type case, it's true There is some kind of vector space isomorphism between the homology of the attracting set in this Coulomb branch associated to lambda mu and the V100 And then I should of course point out that this we already know Um by Nakajima's work is the homology of this central fiber F-land and the central fiber in the nakajima variety So that we get this isomorphism from left to right is what's Predicted by some flexibility. This is like my second Homological some Homological some flexibility prediction But this is not known The only case as far as I know outside of finite type is where it is known in affine typing And to show you that it's how unknown it is. Um, we don't even we would expect that this torus um, so There's always a torus c star to the i which I was calling t I hope it wasn't too confusing because it's the torus of this g gamma not the torus of the gauge group g so this torus So one like part of this conjecture would be that this torus and has a fixed point in this Coulomb branch if and only if The mu is actually a weight Um, and that's not even that even that's not known And I should say that this this conjecture I would say is just a conjecture about an isomorphism vector spaces um We wouldn't expect sort of a geometrically defined action of our Lie algebra on the direct sum of these homologies Because it's sort of not what we see in geometric sateke but There is a sort of partial I don't know result or understanding of what should be going on. Um, so In the usual geometric sateke you can say something about restricting to a levy subalgebra and even in in this story You can also say something about what you expect about restricting to a levy So you would expect this isomorphism to be compatible with levy restriction. And that's the statement you would like So sir, uh, this is defined in terms of slices in some, uh, affine-grasmaya or some cat smoothie group No, no, this this is defined in as Coulomb as a Coulomb branch So not not as a Coulomb So, so yeah, maybe I was a little okay. Let me just back up to the beginning then so I said two things at the beginning one one right here So we define this um mc lambda mu as a Coulomb branch so SPAC of this, uh Homology like this official Coulomb branch definition over here So SPAC of the homology of this Vfn space That's the definition of mc lambda mu When gamma is a finite type When gamma is a finite type then And mu is not necessarily dominant then this Coulomb branch is a generalized affine-grasmaya and slice um, I don't think anyone knows Or has tried to work it out whether this theorem really holds Outside the finite type Whether whether we can consider these general study these generalize affine-grasmaya and slices for cat smoothie groups um At least I don't think there's any published results in that direction So for now this this thingy is just defined as a Coulomb branch. Okay, any other question? Okay, so this is all I don't know property one So property one was this in the finite type of business and it led to this connection. Okay property two um So property two, um Well, it's just a special case of this but it's sort of very interesting special case So I'm going to highlight it. So what happens when we take lambda to be zero Let's let's go back to the finite type situation So whenever another way of knowing that I'm working in finite type is usually I'm the finite type I'll write w lambda view whereas in the general type I'll write mc lambda Okay, so in the finite type situation in the finite type with lambda zero I mean, there's no representation to think about there's no gray lambda bar really to think about just a point But this w zero new is actually still very interesting. It's actually nothing but the space of base maps Uh from p1 so I mentioned here because this is uh Uh To to g my to g to g my b g gamma So I mentioned here since we're in uh It's supposed to be about numeric geometry and this is obviously an objective study in numeric geometry So base maps of degree of a minus mu from p1 to g gamma mod b um And this space has no no attracting set. I mean no fixed point. No, there's no no attractions But one feature it does have Is it has well as every coolant branch has which I didn't emphasize too too much. It has this map this integral system map to T mod w so careful this t may be better to write just like let's see uh Okay, so this is the this every coolant branch has a map to the spec of the Um Geo covranc homology of a point And and which I identify in this case with this c to the mu c to the minus Um, so every coolant branch has that map. It's always an integrable system And in this case this map is well studied for these Oh sometimes these spaces are called open zestatum spaces based on base maps and one one interesting feature Is that if we take we're maybe not interested in the We can't look at the homology of the attracting set because there is no attracting set But we can look at the homology of this fiber of this integrable system map And this is actually something you can do with any coolant branch But w any w or any coolant branch study this Zero locus of the integrable system, which will contain the attracting set always when it exists So this is something of general interest But in particular this case is quite interesting and it turns out this is actually esomorphic to the weight space of a verma module So here this m is a verma module And m mu denotes the weight space of the verma module not not the Not the not the mu a mere module. Hi, so I knew it's the notation looks like a little bit So m is a verma module of highest weight zero So interesting enough. We got these weight space of verma modules coming up this way and actually there's a generalization of this Um That the homology of the zero locus in w lambda mu will be a weight space of a verma tensor of u so another Property or sorry generalization of this story Is of course we started the story by with we might be interested in g gamma not of simply this type Maybe finite type and not simply laced Um Sorry type b bc and so on And um for that there's a modification Of this There's a modification of the Coulomb branch story um, so that's due to naka jima in weeks This is called Coulomb branches with symmetrizes And with this modified version of Coulomb branches you can recover Uh these w w lambda mu say that these generalize aphangous mining slices for any finite type G gamma Um, and then another another result I'd like to mention Is okay, so bad. So still of this The g gamma of finite type simply laced So ad situation um But now but new not necessarily dominant then the symplectic leaves So in general it's a very interesting question to study symplectic leaves of Coulomb branches for any g and n and um, there's not any general results in that direction But in this case, um, there is a result. So the symplectic leaves of this w Lambda mu bar r What you might expect there w new mu for New between lambda and mu in new dominant Okay, and that's a theorem of with dia in weeks okay any questions about uh these results and these uh Coulomb branches and these generalized slices So what is the Higgs branch that is taken in the case of non-sympleased? Oh, that's a very good question. I mean there's no, um There's no exact definition of a Nakajima quiver right in the nonsense of this case I mean, it's probably no there was a recent work of geist Leclerc and shore on this topic and There's a relationship between the definition they use for the Coulomb branch and this geist Leclerc and shore but There's not exactly a Higgs branch. I think and there's not exactly a statement of symplectic duality So when you say that the homology of this middle fiber is a vermin module weight space Yeah How much of the do you see like the action on the vermin module or the just the dimensions or what is how much of the vermin module do you see? um just just the dimensions um although from the Quantization which I'll talk about soon um, so so this um Maybe there's a good time to segue into the next topic. So let and slightly answer your questions. So let me let me get back to it um, so suppose I have Um Sympathetic resolution although for my story now the the resolution parts can be sort of irrelevant So let's just stick with just the x the the conical symplectic singularity I forget about the resolution for a second And um, as I mentioned before we have this maybe attracting a look to say it but say it has a with a torus action And actually I suppose I'm gonna have to erase the word conical That example certainly wasn't conical. Okay, so I'm collecting singularity with a torus action and I have this attracting locus x plus And then maybe I also have um Uh Some integrable system map So I don't know if this is exactly a general feature of some glyphic singularities, but these Um cool um branches come with come with this integral system as I mentioned. So we had this integral system map Brandy cool um branch to expect Jacob and cool Back to another question, uh The relation between sympathy duality and t duality and Are some familiar examples of integral systems the list like kitchen system, etc. Um So first of all the Um I don't think there's a relationship between the t duality for these integral systems In simplect duality um Probably I mean there is some things you can say about like some kind of t duality for these integral systems But let me not try to say anything because I'll probably say something that's not true. Um, are there familiar examples of integral systems? I think so. Um um Hitching system, I don't Know of any exact way of the hitching system appearing this way um But for for example One example of this picture, maybe it's a good example. So put it right here um, is if we take the nilpot and cone of sln and It has an integral system called some I guess usually called the gelfand setland integral system and it goes to Uh c to the N choose two And how is it defined you take your matrix a to the characteristic polynomial Of the upper i by i minor of a A ai So for all i i equals one Up to n minus one ai is the upper nine by i minor And notice if you take the zero level of this Um It will contain the upper triangular matrices, which is the attracting set for the Hamiltonian torus action Um, okay. I'm just going back to my general picture then um, I had would have This example is a very good example to keep in mind. So I have x and it has this integral system Well, if it's a cruel average it would go to this this spec of the this guy So in general it goes to I don't know c to some m and um then This attracting set will be contained inside of the zero level of the integral system And if we now quantize x So here a That's we can use this a a here is it going to be a quantization of x and I mentioned the last Beginning basically that we consider a category o for a And that would be thought of as a categorification of the Topomology of this attracting set well to be a bit more precise I said before that I should consider this category as a categorification of the topomology in the tracking set in the resolution so it's really sort of the um Top quotient of this category which catifies this but roughly speaking categorizes There is a map from the growth in the group to there and on the other hand we can consider um Because of this integrable system We get it quantizes to a gulf and sentlin subalgebra to a to a to a maximal commutative subalgebra a um Which maybe I don't find you to choose some special notation for it. Maybe I'll just write as a polynomial So this is a maximum commutative subalgebra and then we can consider the um Recall this the gulf and settlin subalgebra and then we study gulf and settlin modules for a So modules for a which are uh locally finite for the action of this maximal commutative subalgebra And we think of this as categorifies This zero level So now um, so for for example in this case of nilpot and cone um, this was just Then this would just be the gulf and settlin modules or uh, just universal developing ultravesolin one of those central character um So to answer michael's question now finally, uh, michael asked category of finds michael asked In this uh situation I wrote above with the base maps um Can we say something about the the zero level? It could be something about the action on that when I think of it as a vermin module And the answer is well, not not directly on the homology, but once I categorify it. Yes so When I categorify it and think of these modules for this gulf and settlin modules for the quantization then Then it says yes Okay, so this takes me into the last topic which I have a few minutes to explain 10 minutes so, um So the last topic is about the quantizations Of these uh coolum branches for quiver gauge series these Which is generalized african-gryspanian slices. Um, so Um Well, I'm going to explain basically two results in five ten minutes. So I guess five minutes each so the first result was that um And now we're going to go for the finite type case for the first result So then there's this algebra called uh, why new which is a called a shifted union So it's an algebra with this explicit generators and relations That's a so related to the union. I mean when mu is dominant, it's a subalgebra the union and when mu is not dominant it's not a subalgebra, but it's closely related to the union and um It it acts um by difference operators on a big polynomial ring Okay, so this this actually this thing only depended on on Mu, but this polynomial ring depends on lambda. So the R ranges here From one up to Vi or vi's are calculated using lambda in the usual way or lambda in new usual So it's a finitely many variables whereas this is why mu has like infinitely many generators And acts by difference operators in this polynomial ring and finally many variables In the image Of of of it inside this ring of difference operators And this called by lambda and we call it truncated shifted union So here's a theorem which was proved in an appendix and by the following Authors I'll just put their initials because there's too many of them But I guess they've all appeared already in this lectures except for one of the k's not me. It's codera Like one of the k's is me, but the other one is codera. So I write isn't it? so otherwise, you can guess who everyone is and The the theorem is that this quantized Bfn algebra Associated to the same where gnn are associated to the same quiver data from lambda mu is isomorphic to lambda And I just want to very briefly explain the idea of the proof so that we can embed Using localization We can embed in localization in homology echo variant homology We can embed this Coulomb branch algebra inside The Coulomb branch algebra for the torus and of the same time We can also sort of forget about the representation just sort of convenient to do so but it's not that important and But the cost of inverting some stuff and and I guess I should mention that it doesn't really act on this polynomial ring, but really on this polynomial ring More precisely on the ring of rational functions So anyway, you have to invert some stuff. I won't bother writing. We have to invert and this guy so this is just the Coulomb branch algebra, of course, planning through a torus and the Coulomb branch of a torus What with trivial matters they call it no representation is just the cotangent bundle of of its Lie algebra so And so this can can be quantized to a ring of difference operators isn't that so um Yeah, we can go into a lot of detail here. There's like a lot of crazy formulas you can write down, but let me not do that now um, but that I just want to explain that the basic link between this shifted union and Quantized Coulomb branch is that both of them act on a ring of difference operators. This is a very important idea Leads to this isomorphism And maybe just as a quick example of this if we take this quiver associated to the cotangent bundle of the five variety Remember both as Coulomb branch and sx branch then The corresponding truncated shifted union is pretty easy to see. It's a you know some developing other sln module of the central character There's also a whole story with some parameters and so on and so on So you can get any central character this way very good parameters So that's what you get as the quantized Coulomb branch algebra for this theory And then the last results That I want to mention So Um, so this is my work And again a one new initial. This is peter tingley This is a story. Um, so we we studied these We studied these these Truncated shifted unions are more generally these quantized Coulomb branch algebras for any for any gigana So we proved the fun first thing that there is a categorical problem gigana action on The direct sound of these category os and this categorifies Well, I categorize or The representation it categorifies depends on these parameters that I suppressed but it usually categorifies the tensor product representation Thank you for the generic parameters of course tensor product. You can also categorify other representations um, and so this works for any g gamma and the second result we proved is that this category o Is uh, it seems like a dual To the category o for the quantized Coulomb right and the while the key to proving both results um We We related this category o to modules For uh Kovanov lauda ruke webster algebra Okay, so to some category of modules over this diagram some diagrammatic algebra And this diagrammatic algebra was already linked to categorification and linked to these quantized covariates and um Okay, I guess I stopped there. Sorry for rushing through the last results Any question or comment I had a question about this shifted the young and I saw you say that the g gamma was a finite type So is there makes sense to ask similar question when it's a fine type? Like a fine type a something like this um, yes um There there's um, yes, um I'm trying to remember what I know or don't know I think that I mentioned that the way that this isomorphism works is that both algebras act both this um Coulomb branch algebra and and the shifted hanging algebra they act on by different operators here And they have the same And that's how we identify them And I think if I remember correctly that in general it's this truncated this shifted yang will have a sort of smaller image than this bfn algebra at least um, this is some Thing we discussed a little bit with alivia At some point um that maybe one of these bigger we should use some bigger yang And and alex weeks thought about this a bit, but I think there was no Like definite conclusion here So I think there may be some generalization this is all but I haven't thought about it recently and I don't know if anyone has so um, I think it's not completely clear which Kind of yang and you should use to generalize this result outside of finance it Okay, thanks. Any other question? Yeah, I'm in the q&a you have a You have two questions in the q&a Okay Are these shifted the anions related to the usual union some cases like those can be defined by rgt relations. I mean Yes, I mean in this finite type case um, maybe I should have said this before that y0 is just the usual union so, um It's in the finite type case. We just have a the usual union and if mu is dominant Then why mu is a subalgebra of y0 Of what which is just the usual union, but if mu is non-dominant then it's Something it's not a subalgebra um So continuing to sorry I said another and and eugenius is a related to molecule kukov yang yin I mean in the finite type case again, this is the only one I think there's only one And so it's the the we can use what I mean these y0 is just the usual union to find however you like um If if mu is not dominant, I mean in the finite type case When mu is not dominant these shifted the anions are not the usual union They're not some other buddy's union, but they're nothing that complicated. They have a pretty explicit description by generous relations is generalizing the Drinfeld new presentation of the union In outside of finite type again, I don't know You know, uh, alex thought about it when we talked about it with maybe other people Um, I don't know how to generalize this whether these molecules could be useful in general outside of finite type There is another question If you change the cool What is the quantization? Uh, I certainly never thought about that. I can answer the question if you change the cosmology theory to k theory So if you've changed it, um everything here, there's a there's a almost everything goes through if you've changed things to k theory There's a k theory ready cool own branches Which end up being um well So so this this version of what happens when you go to k theory was studied a lot by um finkelberg and finkelberg uh zimbalook In fact, I remember the last time I was at ihs I was in the hallway just outside the room where you guys are and talking to sasha about this um So sasha zimbalook that is so if you work, um, if you do everything in k theory We'll just back up to this step Let's go to this step. So this part Can be done. Um, you can change things to k theory and uh The coolant branch makes sense coolant branch algebra makes sense And the corresponding coolant branch can be identified in the in the finite type k's can be identified In finite type and dominant case can be identified with a slice in the affine flag fountain And the corresponding coolant branch algebra can be identified with instead of a truncated shifted yangon with a truncated shifted um Quantum affine algebra, so instead of truncated shifted yangons here So this step here, so this, everything here goes through in the K-theoretic setting where the shifted Yangons are placed by quantum affine algebras. And again, this is like some long papers of Finkelberger and Symboluk. What happens in elliptic case? I have no idea. So in the K-theoretic case, is it non weather if there is a microcalor structure on the places? I think in some special cases, yes, but in general, I don't think no. I mean, even in the non-K-theoretic cases, also, I mean, in general, it's conjecture that these Coulomb branches would have, in general, would have a microcalor structure. In the conological case, there are also moduli of manifolds, so maybe... Oh, okay, sorry, sorry. You're right. Yeah, okay, sorry. So, yeah, okay, so if mu is dominant, then I think it's known that they have a microcalor. And I think also in the K-theoretic case, I think that there is a, okay, I'm not completely sure. I think it's known that, yeah. If mu is dominant, then it's fine, it's fine, it's fine. Any further question? Let's see. No? Okay, let's thank Joy again.


Hey there, Chad Bond, you're here for High University Libraries. Just the other day I got a question from a student who was looking to do research on interest in the National Football League in a particular area, in this case, Los Angeles, California. The best database that we have to actually address that kind of question is SimplyMap. And this video shows you how to look for people who are interested or percentage of people who are interested in National Football League or watching football in the Los Angeles area or any area that you choose. So here's how you do it. So in order to get this kind of data, we're looking for, once again, information about people who like football in the Los Angeles, California area. So what we're going to do is, once we log in to SimplyMap, we're going to click on the plus tab there, and we're just going to create a new tab. And for this, we're actually going to do what they call a standard report. And I'm just going to launch the standard report wizard here. That's usually one of the best ways to do that, just to get started that way. So I'm going to ask this first for location. Let's do locations. Let's do what I want to do by city. What I'm going to do by state, we'll do California. And then once it loads there, you can actually start typing in Los Angeles and then use this location. Next, we're going to click on variables. And when it loads here, it actually gives you a way to browse. I'm actually going to close out of this because I want to search for NFL. So you can actually go down and click on search here. And you can just search for NFL. I've already done this before, so I brought 280 results for NFL.  So if you want to use that, we can actually click use this variable. And we'll just go down and click use this variable quite a bit. You'll have all kinds of different information down here that you can look at post season watch occasionally. And you can see as we add these, they're adding over here on the left hand side of our page there. So here we have things like watch frequently, if we go down some more. Interest level, someone interested, use this variable. A little bit interested, use this variable. You can kind of see, very interested, use this variable. And so that gives us a nice list there. And if we close out of this now, that will give us our list of variables for Los Angeles, California compared to USA. Okay, so what if we want to compare Los Angeles to another city? We can actually go and click on locations, find city by whatever you want to. We'll just do city there again, and let's just do Alabama. And I'm just going to choose Birmingham. Click use this location and then close that. And then once it refreshes, you can actually have a nice report there. You can download the data if you want to, lots of different things you can do with it there. So if you want to add more variables, once again, go back to variables. And let's say you just want to add census data and you want some information about people in the household and the average age. For example, let's just do use all variables here. For example, close this and then it's going to add all those variables that we selected there for Birmingham and Los Angeles once it crunches the data there for you. And if we scroll down below there, we see where our variables are added. So we've got all kinds of good data there that we can manipulate, download, etc. If you'd like to download the data, just go over to actions, download table, as an Excel file or CSV. You can also email it to yourself as an Excel file, etc. So also if you're working with a group, you can click on share tab. And that will give you a URL that you can actually share to your friends. And this will only work on campus at Ohio University, but would be a good way to share that with your colleagues if you're working on a group project. Hopefully this helps you understand how to better use SimplyMap to try to find variables for sports interests in a particular area. If you need more help, look for the contact link on the business blog. I'll be glad to help you any way you can. Take care.


You got to do the opposite of what everybody else is doing. So when I was a kid I would go on vacation. I think this vacation is awesome and I'd be going at the same time as everybody else and I thought why don't I just live on vacation all the time. So when I graduated high school I moved out to snowboard full-time because it's my favorite thing in the world. When I was an adult I always try to be going the opposite way of rush hour traffic. I try to be going on vacation and everybody else is back at work. You can see them relaxing here in Cabo. I try to do the things that other people don't or aren't willing to do so that I can live life that other people aren't willing to have because if you do the things that other people do you're going to have the exact same thing to have. And if you look at the average person it's not something most of us aspire to. So you have to learn to relax and to be uncomfortable. To do things that don't fit in with other people and most people can't bear to not fit in and that's why they're never going to get extraordinary results. So learn to do the unconventional and the unconventional will find you.


to the session. I'll help you with the following question. The question says, the cost of 2KG of apples and 1KG of grapes on a day was found to be Rs.160. After a month, the cost of 4KG of apples and 2KG of grapes is Rs.300. Represent the situation algebraically and geometrically. If we know that a pair of linear equations, variables x and y, will be represented algebraically as b1y plus c1 equal to 0 and a2x plus b2y plus c2 is equal to 0. And geometrically, we present straight lines. Key idea to be used for this question. Let's move on to the solution. Let the cost of 1KG of apples be equal to Rs.x and also let the cost of 1KG of grapes be equal to Rs.y. According to the question, we have that 2KG of apples and 1KG of grapes on a day was found to be Rs.160. So, we have that is the cost of 2KG of apples plus the cost of 1KG of grapes that is y is equal to Rs.160 or we can say 2x plus y is equal to 160. Let this be equation 1. Again in the question we have that after a month, the cost of 4KG of apples and 2KG of grapes is Rs.300. So, we get according to the question we have 4x plus 2y is equal to Rs.300 or we can say 4x plus 2y is equal to 300. Let this be equation number 2. Thus, the situation is represented algebraically as 2x plus y is equal to 160 and 4x plus 2y is equal to 300. Consider the equation 2x plus y is equal to 160. Now, we shall find at least two solutions for this equation. If we put x as 80 in this equation, we get y as 0 and when x is taken as 30 then we get y as 100. So, the two points that we get for this equation is point A with coordinates 80 comma 0, point B with coordinates 30 comma 100. Now, let's consider the equation 4x plus 2y equal to 300. If we put x as 20 in this equation, we get y as 110 and when x is taken as 40 and the value of y is 70. So, the two points that we get here are point C with coordinates 20 comma 110 and point D with coordinates 40 comma 70. Now, we shall plot these four points on the graph. On this graph, we have taken scale on x axis as 1 centimeter and the scale on y axis is taken as 1 centimeter equal to 20 units. First, let's plot the point A with coordinates 80 comma 0. This point represents the point A with coordinates 80 comma 0. Next, we have the point B with coordinates 30 comma 100. This point corresponds to the point B with coordinates 30 comma 100. Now, we shall join the points A and the point B. As you can see on joining the point A and point B, we get a straight line. This straight line represents the equation 2x plus y equal to 160. Now, we plot the point C with coordinates 20 comma 110. This point corresponds to the point C with coordinates 20 comma 110. Then, we have the point D with coordinates 40 comma 70. This point corresponds to the point D with coordinates 40 comma 70. Now, we shall join the point C and D. As you can see on joining the point C and D, we get a straight line. This straight line represents the equation 4x plus 2y equal to 300. So, we get two parallel lines. Thus, we say the lines and CD are parallel lines. So, this is the geometrical representation of the situation. Hence, the final answer is algebraically the two situations can be represented as follows. 2x plus y equal to 160, 4x plus 2y equal to 300, where x and y are respectively the prizes in rupees per kg of apples and grapes. So, hope you enjoyed the session. Have a good day.